Quantum Integrable Model of an Arrangement of Hyperplanes

The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumpt...

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Дата:	2011
Автор:	Varchenko, A.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2011
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Quantum Integrable Model of an Arrangement of Hyperplanes / A. Varchenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 29 назв. — англ.

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spelling	irk-123456789-1468072019-02-12T01:24:27Z Quantum Integrable Model of an Arrangement of Hyperplanes Varchenko, A. The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe algebra of the corresponding Gaudin model. It is expected that this correspondence holds in general (without the assumptions). As a byproduct of constructions we show that in a Gaudin model (associated to an arbitrary simple Lie algebra), the Bethe vector, corresponding to an isolated critical point of the master function, is nonzero. 2011 Article Quantum Integrable Model of an Arrangement of Hyperplanes / A. Varchenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 82B23; 32S22; 17B81; 81R12 DOI:10.3842/SIGMA.2011.032 http://dspace.nbuv.gov.ua/handle/123456789/146807 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe algebra of the corresponding Gaudin model. It is expected that this correspondence holds in general (without the assumptions). As a byproduct of constructions we show that in a Gaudin model (associated to an arbitrary simple Lie algebra), the Bethe vector, corresponding to an isolated critical point of the master function, is nonzero.
format	Article
author	Varchenko, A.
spellingShingle	Varchenko, A. Quantum Integrable Model of an Arrangement of Hyperplanes Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Varchenko, A.
author_sort	Varchenko, A.
title	Quantum Integrable Model of an Arrangement of Hyperplanes
title_short	Quantum Integrable Model of an Arrangement of Hyperplanes
title_full	Quantum Integrable Model of an Arrangement of Hyperplanes
title_fullStr	Quantum Integrable Model of an Arrangement of Hyperplanes
title_full_unstemmed	Quantum Integrable Model of an Arrangement of Hyperplanes
title_sort	quantum integrable model of an arrangement of hyperplanes
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/146807
citation_txt	Quantum Integrable Model of an Arrangement of Hyperplanes / A. Varchenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 29 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT varchenkoa quantumintegrablemodelofanarrangementofhyperplanes
first_indexed	2025-07-11T00:38:43Z
last_indexed	2025-07-11T00:38:43Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 032, 55 pages Quantum Integrable Model of an Arrangement of Hyperplanes? Alexander VARCHENKO Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA E-mail: anv@email.unc.edu Received July 19, 2010, in final form March 19, 2011; Published online March 28, 2011 doi:10.3842/SIGMA.2011.032 Abstract. The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe algebra of the corresponding Gaudin model. It is expected that this correspondence holds in general (without the assumptions). As a byproduct of constructions we show that in a Gaudin model (associated to an arbitrary simple Lie algebra), the Bethe vector, corresponding to an isolated critical point of the master function, is nonzero. Key words: Gaudin model; arrangement of hyperplanes 2010 Mathematics Subject Classification: 82B23; 32S22; 17B81; 81R12 To Sabir Gusein-Zade on the occasion of his 60-th birthday Contents 1 Introduction 3 1.1 Quantum integrable models and Bethe ansatz . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Gaudin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Gaudin model as a semiclassical limit of KZ equations . . . . . . . . . . . . . . . . . . . . 4 1.4 Bethe ansatz in the Gaudin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Gaudin model and arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Bethe algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7 Algebra of geometric Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.8 Quantum integrable model of a weighted arrangement (or dynamical theory of arrangements) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.9 Bethe ansatz for the quantum integrable model of an arrangement . . . . . . . . . . . . . 7 1.10 Geometric interpretation of the algebra of Hamiltonians . . . . . . . . . . . . . . . . . . . 8 1.11 Byproducts of constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.12 Exposition of the material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Arrangements 9 2.1 An affine arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Orlik–Solomon algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 ?This paper is a contribution to the Special Issue “Relationship of Orthogonal Polynomials and Spe- cial Functions with Quantum Groups and Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/OPSF.html mailto:anv@email.unc.edu http://dx.doi.org/10.3842/SIGMA.2011.032 http://www.emis.de/journals/SIGMA/OPSF.html 2 A. Varchenko 2.4 Space of flags, see [23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Duality, see [23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6 Contravariant map and form, see [23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.7 Remarks on generic weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.8 Orlik–Solomon algebra as an algebra of differential forms . . . . . . . . . . . . . . . . . . 11 2.9 Critical points of the master function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.10 Special vectors in Fk(A) and canonical element . . . . . . . . . . . . . . . . . . . . . . . . 13 2.11 Arrangements with normal crossings only . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.12 Real structure on Ap(A) and Fp(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.13 A real arrangement with positive weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.14 Resolution of a hyperplane-like divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 A family of parallelly translated hyperplanes 15 3.1 An arrangement in Cn × Ck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Good fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Bad fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Conservation of the number of critical points 16 5 Hamiltonians of good fibers 17 5.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.2 Key identity (5.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.3 An application of the key identity (5.2) – proof of Theorem 5.1 . . . . . . . . . . . . . . . 19 5.4 Another application of the key identity (5.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.5 Hamiltonians, critical points and the canonical element . . . . . . . . . . . . . . . . . . . . 21 6 Asymptotic solutions and eigenvectors of Hamiltonians 21 6.1 Asymptotic solutions, one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.2 Critical points of the master function and asymptotic solutions . . . . . . . . . . . . . . . 22 7 Hamiltonians of bad fibers 23 7.1 Naive geometric Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 7.2 Space Fk(A(z0)) and operators LC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.3 Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.4 Positive (aj)j∈J , real (gj)j∈J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.5 Proof of Theorem 7.5 for z0 ∈ ∆ ∩ Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7.6 Proof of Theorem 7.5 for any z0 ∈ ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7.7 Critical points and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7.8 Hamiltonians, critical points and the canonical element . . . . . . . . . . . . . . . . . . . . 27 8 Geometric interpretation of the algebra of Hamiltonians 28 8.1 An abstract setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 8.2 Proof of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.3 Remark on maximal commutative subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 30 8.4 Interpretation of the algebra of Hamiltonians of good fibers . . . . . . . . . . . . . . . . . 30 8.5 Interpretation of the algebra of Hamiltonians of bad fibers if Assumption 7.4 is satisfied . 32 9 More on Hamiltonians of bad fibers 33 9.1 An abstract setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 9.2 Hamiltonians of bad fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 9.3 Remark on critical points of real arrangements . . . . . . . . . . . . . . . . . . . . . . . . 39 Quantum Integrable Model of an Arrangement of Hyperplanes 3 10 Arrangements with symmetries 39 10.1 A family of prediscriminantal arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . 39 10.2 Discriminantal arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 10.3 Symmetries of the family of prediscriminantal arrangements . . . . . . . . . . . . . . . . . 41 10.4 The Sk-action on geometric Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 10.5 Functions K∂xb (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 10.6 Naive geometric Hamiltonians on SingW−(z0) . . . . . . . . . . . . . . . . . . . . . . . . 43 10.7 Sk-symmetries of the canonical element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 11 Applications to the Bethe ansatz of the Gaudin model 46 11.1 Gaudin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 11.2 Master function and weight function, [23] . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 11.3 Bethe vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 11.4 Identification of Gaudin and naive geometric Hamiltonians . . . . . . . . . . . . . . . . . 49 11.5 Bethe algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 11.6 glr+1 Bethe algebra and critical points of the master function . . . . . . . . . . . . . . . . 51 11.7 Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 11.7.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 11.7.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 11.7.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 References 54 1 Introduction 1.1 Quantum integrable models and Bethe ansatz A quantum integrable model is a vector space V and an “interesting” collection of commuting linear operatorsK1,K2, . . . : V → V . The operators are called Hamiltonians or transfer matrices or conservation laws. The problem is to find common eigenvectors and eigenvalues. The Bethe ansatz is a method to diagonalize commuting linear operators. One invents a vector-valued function v(t) of some new parameters t = (t1, . . . , tk) and fixes the parame- ters so that v(t) becomes a common eigenvector of the Hamiltonians. One shows that v(t) ∈ V is an eigenvector if t satisfies some system of equations, Ψj(t) = 0, j = 1, . . . , k. (1.1) The equations are called the Bethe ansatz equations. The vector v(t) corresponding to a solution of the equations is called a Bethe vector. The method is called the Bethe ansatz method. 1.2 Gaudin model One of the simplest and interesting models is the quantum Gaudin model introduced in [7] and [8]. Choose a simple Lie algebra g, an orthonormal basis {Ja} of g with respect to a nonde- generate g-invariant bilinear form, and a collection of distinct complex numbers x = (x1, . . . , xN ). Then one defines certain elements of the N -th tensor power of the universal enveloping algebra of g, denoted by K1(x), . . . ,KN (x) ∈ (Ug)⊗N and called the Gaudin Hamiltonians, Kb(x) = ∑ b6=c ∑ a J (b) a J (c) a xb − xc , b = 1, . . . , N. Here we use the standard notation: if J ∈ g, then J (i) = 1⊗(i−1) ⊗ J ⊗ 1⊗(N−i). 4 A. Varchenko The Gaudin Hamiltonians commute with each other and commute with the diagonal subal- gebra (Ug)diag ⊂ (Ug)⊗N , [Kb(x),Kc(x)] = 0, [Kb(x), (Ug)diag] = 0. Let VΛ denote the finite-dimensional irreducible g-module with highest weight Λ. Decompose a tensor product VΛ = ⊗Nb=1VΛb into irreducible g-modules, VΛ = ⊕Λ∞VΛ∞ ⊗WΛ,Λ∞ , (1.2) where WΛ,Λ∞ is the multiplicity space of a representation VΛ∞ . The Gaudin Hamiltonians act on VΛ, preserve decomposition (1.2), and by Schur’s lemma induce commuting linear operators on each multiplicity space W = WΛ,Λ∞ , K1(x), . . . ,KN (x) : W →W. These commuting linear operators on a multiplicity space constitute the quantum Gaudin model. Thus, the Gaudin model depends on g, ( , ), highest weights Λ1, . . . ,ΛN ,Λ∞ and complex numbers x1, . . . , xN . 1.3 Gaudin model as a semiclassical limit of KZ equations On every multiplicity space W = WΛ,Λ∞ of the tensor product VΛ = ⊕Λ∞VΛ∞ ⊗WΛ,Λ∞ one has a system of Knizhnik–Zamolodchikov (KZ) differential equations, κ ∂I ∂xb (z) = Kb(x)I(x), b = 1, . . . , N, where x = (x1, . . . , xN ), I(x) ∈ W is the unknown function, Kb(x) are the Gaudin Hamilto- nians and κ ∈ C× is a parameter of the differential equations. KZ equations are equations for conformal blocks in the Wess–Zumino–Novikov–Witten conformal field theory. For any value of κ, KZ equations define a flat connection on the trivial bundle CN×W → CN with singularities over the diagonals. The flatness conditions,[ κ ∂ ∂xb −Kb(x), κ ∂ ∂xc −Kc(x) ] = 0, in particular, imply the conditions [Kb(x),Kc(x)] = 0, which are the commutativity conditions for the Gaudin Hamiltonians. Thus, we observe two related problems: (1) Given a nonzero number κ, find solutions of the KZ equations. (2) Given x, find eigenvectors of the Gaudin Hamiltonians. Problem (2) is a semiclassical limit of problem (1) as κ tends to zero. Namely, assume that the KZ equations have an asymptotic solution of the form I(x) = eP (x)/κ ( w0(x) + κw1(x) + κ2w2(x) + · · · ) as κ → 0. Here P (x) is a scalar function and w0(x), w1(x), w2(x), . . . are some W -valued functions of x. Substituting this expression to the KZ equations and equating the leading terms we get ∂P ∂xb (x)w0(x) = Kb(x)w0(x). Hence, for any x and b, the vector w0(x) is an eigenvector of the Gaudin Hamiltonian Kb(x) with eigenvalue ∂P ∂xb (x). Thus, in order to diagonalize the Gaudin Hamiltonians it is enough to construct asymptotic solutions to the KZ equations. Quantum Integrable Model of an Arrangement of Hyperplanes 5 1.4 Bethe ansatz in the Gaudin model The Gaudin model has a Bethe ansatz [2, 3, 19]. The Bethe ansatz has special features. Namely, (i) In the Gaudin model, there exists a function Φ(t1, . . . , tk) (the master function) such that the Bethe ansatz equations (1.1) are the critical point equations for the master function, ∂Φ ∂tj (t1, . . . , tk) = 0, j = 1, . . . , k. (ii) In the Gaudin model, the vector space W has a symmetric bilinear form S (the tensor Shapovalov form) and Gaudin Hamiltonians are symmetric operators. (iii) In the Gaudin model, the Bethe vectors assigned to (properly understood) distinct critical points are orthogonal and the square of the norm of a Bethe vector equals the Hessian of the master function at the corresponding critical point, S(v(t), v(t)) = det ( ∂2Φ ∂ti∂tj ) (t). In particular, the Bethe vector corresponding to a nondegenerate critical point is nonzero. These statements indicate a connection between the Gaudin Hamiltonians and a mysterious master function (which is not present in the definition of the Gaudin model). One of the goals of this paper is to uncover this mystery and show that the Bethe ansatz can be interpreted as an elementary construction in the theory of arrangements, where the master function is a basic object. 1.5 Gaudin model and arrangements For a weighted arrangement of affine hyperplanes, we will construct (under certain assumptions) a quantum integrable model, that is, a vector space W with a symmetric bilinear form S (called the contravariant form), a collection of commuting symmetric linear operators on W (called the naive geometric Hamiltonians), a master function Φ(t) and vectors v(t) (called the special vectors or called the values of the canonical element) which are eigenvectors of the linear operators if t is a critical point of the master function. Then for a given Gaudin model (W,S,K1(x),K2(x), . . . : W → W ), one can find a suitable (discriminantal) arrangement and identify the objects of the Gaudin model with the correspon- ding objects of the quantum integrable model of the arrangement. After this identification, the master function Φ(t) and the special vectors v(t) of the arrangement provide the Gaudin model with a Bethe ansatz, that is, with a method to diagonalize the Gaudin Hamiltonians. 1.6 Bethe algebra Let WΛ,Λ∞ be the vector space of a Gaudin model. It turns out that the subalgebra of End(WΛ,Λ∞) generated by the Gaudin Hamiltonians can be extended to a larger commutative subalgebra called the Bethe algebra. A general construction of the Bethe algebra for a simple Lie algebra g is given in [4]. That construction is formulated in terms of the center of the universal enveloping algebra of the corresponding affine Lie algebra ĝ at the critical level. As a result of that construction, for any x one obtains a commutative subalgebra B(x) ⊂ (Ug)⊗N which commutes with the diagonal subalgebra Ug ⊂ (Ug)⊗N . To define the Bethe algebra of VΛ or of WΛ,Λ∞ one considers the image of B(x0) in End(VΛ) or in End(WΛ,Λ∞). The Gaudin Hamiltonians Kb(x) are elements of the Bethe algebra of VΛ or of WΛ,Λ∞ . The construction of the Bethe algebra in [4] is not explicit and it is not easy to study the Bethe algebra of WΛ,Λ∞ for a particular Gaudin model. For example, a standard difficult question is if the Bethe algebra of WΛ,Λ∞ is a maximal commutative subalgebra of End(WΛ,Λ∞), cf. [12, 5]. 6 A. Varchenko 1.7 Algebra of geometric Hamiltonians In this paper we address the following problem. Is there a geometric construction of the Bethe algebra? For the quantum integrable model (W,S,K1,K2, . . . : W → W ) of a given weighted arrangement, can one define a geometric “Bethe algebra” A, which is a maximal commutative subalgebra of End(W ), which contains the naive geometric Hamiltonians K1,K2, . . . , and which can be identified with the Bethe algebra of [4] for discriminantal arrangements? In this paper (under certain assumptions) we construct an algebra A called the algebra of geometric Hamiltonians and identify it (in certain cases) with the Bethe algebra of the Gaudin model. 1.8 Quantum integrable model of a weighted arrangement (or dynamical theory of arrangements) To define the quantum integrable model of an arrangement we consider in an affine space Ck (with coordinates t = (t1, . . . , tk)) an arrangement of n hyperplanes, k < n. Each hyperplane is allowed to move parallelly to itself. The parallel shift of the i-th hyperplane is measured by a number zi and for every z = (z1, . . . , zn) ∈ Cn we get in Ck an affine arrangement of hyperplanes A(z) = (Hj(z)), Hj(z) = { t ∈ Ck \| gj(t) + zj = 0 } , where gj(t) are given linear functions on Ck. We assign nonzero numbers a = (aj) to the hyperplanes of A(z) (the numbers do not depend on z) and obtain a family of parallelly translated weighted hyperplanes. For generic z ∈ Cn, the arrangement A(z) has normal crossings only. The discriminant ∆ ⊂ Cn is the subset of all points z such that A(z) is not with normal crossings only. The master function Φ on Cn × Ck is the function Φ(z, t) = ∑ j aj log(gj(t) + zj). Let A(A(z)) = ⊕kp=0Ap(A(z)) be the Orlik–Solomon algebra and F(A(z)) = ⊕kp=0Fp(A(z)) the dual vector space. We are interested in the top degree components Ak(A(z)) and Fk(A(z)). The weights a define on Fk(A(z)) a symmetric bilinear form S(a) (called the contravariant form) and a degree-one element ν(a) = ∑ j aj(Hj(z)) ∈ A1(A(z)). Denote SingFk(A(z)) ⊂ Fk(A(z)) the annihilator of the subspace ν(a) · Ak−1(A(z)) ⊂ Ak(A(z)). For z1, z2 ∈ Cn − ∆, all combinatorial objects of the arrangements A(z1) and A(z2) can be canonically identified. In particular, the spaces Fk(A(z1)), Fk(A(z2)) as well as the spaces SingFk(A(z1)), SingFk(A(z2)) can be canonically identified. For z ∈ Cn − ∆, we denote V = Fk(A(z)), Sing V = SingFk(A(z)). For any nonzero number κ, the hypergeometric integrals∫ γ(z) eΦ(z,t)/κω, ω ∈ Ak(A(z)), define a Gauss–Manin (flat) connection on the trivial bundle Cn×Sing V → Cn with singularities over the discriminant. The Gauss–Manin differential equations for horizontal sections of the connection on Cn × Sing V → Cn have the form κ ∂I ∂zj (z) = Kj(z)I(z), j ∈ J, (1.3) Quantum Integrable Model of an Arrangement of Hyperplanes 7 where I(z) ∈ Sing V is a horizontal section, Kj(z) : V → V , j ∈ J, are suitable linear operators preserving Sing V and independent of κ. For every j, the operator Kj(z) is a rational function of z regular on Cn−∆. Each operator is symmetric with respect to the contravariant form S(a). These differential equations are our source of quantum integrable models for weighted ar- rangements. The quantum integrable models are the semiclassical limit of these differential equations similarly to the transition from KZ equations to the Gaudin model in Section 1.6. The flatness of the connection for all κ implies the commutativity of the operators, Ki(z)\|Sing VKj(z)\|Sing V = Kj(z)\|Sing VKi(z)\|Sing V for all i, j and z ∈ Cn −∆. Let V ∗ be the space dual to V . If M : V → V is a linear operator, then M∗ : V ∗ → V ∗ denotes the dual operator. Let W ⊂ V ∗ be the image of V under the map V → V ∗ associated with the contravariant form and SingW ⊂W the image of Sing V ⊂ V . The contravariant form induces on W a nondegenerate symmetric bilinear form, also denoted by S(a). The operators Kj(z) ∗ preserve the subspaces SingW ⊂ W ⊂ V ∗. The operators Kj(z) ∗\|W : W → W are symmetric with respect to the contravariant form. The operators Kj(z) ∗\|SingW : SingW → SingW , j ∈ J , commute. For z ∈ Cn − ∆, we define the quantum integrable model assigned to (A(z), a) to be the collection( SingW ; S(a)\|SingW ; K1(z)∗\|SingW , . . . ,Kn(z)∗\|SingW : SingW → SingW ) . The unital subalgebra of End(SingW ) generated by operators K1(z)∗\|SingW , . . . , Kn(z)∗\|SingW will be called the algebra of geometric Hamiltonians of (A(z), a). It is clear that any weighted arrangement with normal crossings only can be realized as a fiber (A(z), a) of such a construction and, thus, a weighted arrangement with normal crossings only is provided with a quantum integrable model. If z0 is a point of the discriminant, the construction of the quantum integrable model assigned to the arrangement (A(z0), a) is more delicate. The operator valued functions Kj(z) may have first order poles at z0. We write Kj(z) = K0 j (z) + K1 j (z), where K0 j (z) is the polar part at z0 and K1 j (z) the regular part. By suitably regularizing the operators K1 j (z0), we make them (under certain assumptions) preserve a suitable subspace of Sing V , commute on that subspace and be symmetric with respect to the contravariant form. The algebra of regularized operators K1 j (z0) on that subspace produces the quantum integrable model assigned to the fiber (A(z0), a). It may happen that some linear combinations (with constant coefficients) Kξ(z) = ∑ j ξjKj(z) are regular at z0. In this case the operator Kξ(z 0) preserves that subspace and is an element of the algebra of Hamiltonians. In this case the operator Kξ(z 0) is called a naive geometric Hamiltonian. (It is naive in the sense that we don’t need to go through the regularization procedure to produce that element of the algebra of Hamiltonians. In that sense, for z ∈ Cn−∆ all elements of the algebra of geometric Hamiltonians of the arrangement (A(z), a) are naive.) For applications to the Gaudin model one needs a suitable equivariant version of the described construction, the corresponding family of parallelly translated hyperplanes has a symmetry group and the Gaudin model is identified with the skew-symmetric part of the corresponding quantum integrable model of the arrangement. 1.9 Bethe ansatz for the quantum integrable model of an arrangement The Hamiltonians of the model are (suitably regularized) right hand sides of the Gauss–Manin differential equations (1.3). The solutions to the equations are the integrals ∫ γ(z) e Φ(z,t)/κω. By taking the semiclassical limit of the integrals as κ tends to zero, we obtain eigenvectors of the Hamiltonians, cf. Section 1.3. The eigenvectors of the Hamiltonians are labeled by the critical points of the phase Φ of the integrals due to the steepest descent method. 8 A. Varchenko 1.10 Geometric interpretation of the algebra of Hamiltonians It turns out that the solutions ∫ γ(z) e Φ(z,t)/κω to equations (1.3) produce more than just eigenvec- tors of the geometric Hamiltonians. They also produce a geometric interpretation of the whole algebra of geometric Hamiltonians. It turns out that the algebra of geometric Hamiltonians is naturally isomorphic (under certain conditions) to the algebra of functions on the critical set of the master function Φ. The isomorphism is established through the semiclassical limit of the integrals. Moreover, this isomorphism identifies the residue bilinear form on the algebra of functions and the contravariant form of the arrangement. This geometric interpretation of the algebra of geometric Hamiltonians is motivated by the recent paper [15] where a connection between the algebra of functions on the critical set of the master function and the Bethe algebra of the glr+1 Gaudin model is established, cf. [14]. 1.11 Byproducts of constructions The general motive of this paper is the interplay between the combinatorially defined linear objects of a weighted arrangement and the critical set of the corresponding master function (which is a nonlinear characteristics of the arrangement). As byproducts of our considerations we get relations between linear and nonlinear characteristics of an arrangement. For example, we prove that the sum of Milnor numbers of the critical points of the master function is not greater than the rank of the contravariant form on Sing V . As another example of such an interaction we show that in any Gaudin model (associated with any simple Lie algebra) the Bethe vector corresponding to an isolated critical points of the master function is nonzero. That result is known for nondegenerate critical points, see [27], and for the Gaudin models associated with the Lie algebra glr+1, see [15]. 1.12 Exposition of the material In Section 2, basic facts of the theory of arrangements are collected. The main objects are the space of flags, contravariant form, master function, canonical element. In Section 3, a family of parallelly translated hyperplanes is introduced. In Section 4, remarks on the conservation of the number of critical points of the master function under deformations are presented. In Section 5, the Gauss–Manin differential equations are considered. The quantum inte- grable model of a weighted arrangement with normal crossings only is introduced. The “key identity” (5.2) is formulated. In Section 6, the asymptotic solutions to the Gauss–Manin differential equations are dis- cussed. In Section 7, the quantum integrable model of any fiber (A(z0), a), z0 ∈ ∆, is defined under assumptions of positivity of weights (aj) and reality of functions (gj(t)). A general con- jecture is formulated. In Section 8, it is shown that the algebra of geometric Hamiltonians is isomorphic to the algebra of functions on the critical set of the master function. That fact is proved for any (A(z), a), z ∈ Cn under Assumption 7.4 of positivity of (aj) and reality of (gj(t)). In Section 9, more results in this direction are obtained, see Theorems 9.16 and 9.17. In Section 10, an equivariant version of the algebra of geometric Hamiltonians is introduced and in Section 11 relations with the Gaudin model are described. Quantum Integrable Model of an Arrangement of Hyperplanes 9 2 Arrangements 2.1 An affine arrangement Let k and n be positive integers, k < n. Denote J = {1, . . . , n}. Let A = (Hj)j∈J , be an arrangement of n affine hyperplanes in Ck. Denote U = Ck − ∪j∈JHj , the complement. An edge Xα ⊂ Ck of the arrangement A is a nonempty intersection of some hyperplanes of A. Denote by Jα ⊂ J the subset of indices of all hyperplanes containing Xα. Denote lα = codimCkXα. We always assume that A is essential, that is, A has a vertex, an edge which is a point. An edge is called dense if the subarrangement of all hyperplanes containing it is irreducible: the hyperplanes cannot be partitioned into nonempty sets so that, after a change of coordinates, hyperplanes in different sets are in different coordinates. In particular, each hyperplane of A is a dense edge. 2.2 Orlik–Solomon algebra Define complex vector spaces Ap(A), p = 0, . . . , k. For p = 0 set Ap(A) = C. For p > 1, Ap(A) is generated by symbols (Hj1 , . . . ,Hjp) with ji ∈ J , such that (i) (Hj1 , . . . ,Hjp) = 0 if Hj1 ,. . . ,Hjp are not in general position, that is, if the intersection Hj1 ∩ · · · ∩Hjp is empty or has codimension less than p; (ii) (Hjσ(1) , . . . ,Hjσ(p)) = (−1)\|σ\|(Hj1 , . . . ,Hjp) for any element σ of the symmetric group Sp; (iii) ∑p+1 i=1 (−1)i(Hj1 , . . . , Ĥji , . . . ,Hjp+1) = 0 for any (p+1)-tuple Hj1 , . . . ,Hjp+1 of hyperplanes in A which are not in general position and such that Hj1 ∩ · · · ∩Hjp+1 6= ∅. The direct sum A(A) = ⊕Np=1Ap(A) is an algebra with respect to multiplication (Hj1 , . . . ,Hjp) · (Hjp+1 , . . . ,Hjp+q) = (Hj1 , . . . ,Hjp , Hjp+1 , . . . ,Hjp+q). The algebra is called the Orlik–Solomon algebra of A. 2.3 Weights An arrangement A is weighted if a map a : J → C, j 7→ aj , is given; aj is called the weight of Hj . For an edge Xα, define its weight as aα = ∑ j∈Jα aj . We always assume that aj 6= 0 for every j ∈ J . Define ν(a) = ∑ j∈J aj(Hj) ∈ A1(A). Multiplication by ν(a) defines a differential d(a) : Ap(A) → Ap+1(A), x 7→ ν(a) · x, on A(A). 10 A. Varchenko 2.4 Space of flags, see [23] For an edge Xα, lα = p, a flag starting at Xα is a sequence Xα0 ⊃ Xα1 ⊃ · · · ⊃ Xαp = Xα of edges such that lαj = j for j = 0, . . . , p. For an edge Xα, we define Fα as the complex vector space with basis vectors Fα0,...,αp=α labeled by the elements of the set of all flags starting at Xα. Define Fα as the quotient of Fα by the subspace generated by all the vectors of the form∑ Xβ , Xαj−1⊃Xβ⊃Xαj+1 Fα0,...,αj−1,β,αj+1,...,αp=α. Such a vector is determined by j ∈ {1, . . . , p− 1} and an incomplete flag Xα0 ⊃ · · · ⊃ Xαj−1 ⊃ Xαj+1 ⊃ · · · ⊃ Xαp = Xα with lαi = i. Denote by Fα0,...,αp the image in Fα of the basis vector Fα0,...,αp . For p = 0, . . . , k, set Fp(A) = ⊕Xα,lα=p Fα. 2.5 Duality, see [23] The vector spaces Ap(A) and Fp(A) are dual. The pairing Ap(A) ⊗ Fp(A) → C is defined as follows. For Hj1 , . . . ,Hjp in general position, set F (Hj1 , . . . ,Hjp) = Fα0,...,αp where Xα0 = Ck, Xα1 = Hj1 , . . . , Xαp = Hj1 ∩ · · · ∩Hjp . Then define 〈(Hj1 , . . . ,Hjp), Fα0,...,αp〉 = (−1)\|σ\|, if Fα0,...,αp = F (Hjσ(1) , . . . ,Hjσ(p)) for some σ ∈ Sp, and 〈(Hj1 , . . . ,Hjp), Fα0,...,αp〉 = 0 otherwise. Denote by δ(a) : Fp(A) → Fp−1(A) the map dual to d(a) : Ap−1(A) → Ap(A). An element v ∈ Fk(A) is called singular if δ(a)v = 0. Denote by SingFk(A) ⊂ Fk(A) the subspace of all singular vectors. 2.6 Contravariant map and form, see [23] Weights a determine a contravariant map S(a) : Fp(A)→ Ap(A), Fα0,...,αp 7→ ∑ aj1 · · · ajp (Hj1 , . . . ,Hjp), where the sum is taken over all p-tuples (Hj1 , . . . ,Hjp) such that Hj1 ⊃ Xα1 , . . . , Hjp ⊃ Xαp . Identifying Ap(A) with Fp(A)∗, we consider the map as a bilinear form, S(a) : Fp(A)⊗Fp(A)→ C. Quantum Integrable Model of an Arrangement of Hyperplanes 11 The bilinear form is called the contravariant form. The contravariant form is symmetric. For F1, F2 ∈ Fp(A), S(a)(F1, F2) = ∑ {j1,...,jp}⊂J aj1 · · · ajp〈(Hj1 , . . . ,Hjp), F1〉〈(Hj1 , . . . ,Hjp), F2〉, where the sum is over all unordered p-element subsets. The contravariant form was introduced in [23]. It is an analog of the Shapovalov form in representation theory. On relations between the contravariant and Shapovalov forms, see [23] and Section 11. 2.7 Remarks on generic weights Theorem 2.1 ([23]). If weights a are such that none of the dense edges has weight zero, then the contravariant form is nondegenerate. Theorem 2.2. If weights a are such that none of the dense edges has weight zero, then Hp(A∗(A), d(a)) = 0 for p < k and dimHk(A∗, d(a)) = \|χ(U)\|, where χ(U) is the Euler charac- teristics of U . In particular, these statements hold if all weights are positive. Proof. The theorem is proved in [28, 18]. It is also a straightforward corollary of some results in [23]. Namely, in [23] a flag complex d : Fp(A)→ Fp+1(A) was considered with the differential defined by formula (2.2.1) in [23]. By [23, Corollary 2.8] the cohomology spaces of that flag complex are trivial in all degrees less than the top degree. In [23], it was also proved that the contravariant map defines a homomorphism of the flag complex to the complex d(a) : Ap(A)→ Ap+1(A). Now Theorem 2.2 is a corollary of Theorem 2.1. � Corollary 2.3. If weights a are such that none of the dense edges has weight zero, then the dimension of SingFk(A) equals \|χ(U)\|. 2.8 Orlik–Solomon algebra as an algebra of differential forms For j ∈ J , fix a defining equation for the hyperplane Hj , fj = 0, where fj is a polynomial of degree one in variables t1, . . . , tk. Consider the logarithmic differential form ωj = dfj/fj on Ck. Let Ā(A) be the C-algebra of differential forms generated by 1 and ωj , j ∈ J . The map A(A)→ Ā(A), (Hj) 7→ ωj , is an isomorphism. We identify A(A) and Ā(A). 2.9 Critical points of the master function Given weights a : J → C, define the (multivalued) master function Φ : U → C, Φ = ΦA,a = ∑ j∈J aj log fj . (2.1) Usually the function eΦ = ∏ j f aj j is called the master function, see [25, 26, 27], but it is more convenient to work with definition (2.1). A point t ∈ U is a critical point of Φ if dΦ\|t = 0. We can rewrite this equation as ν(a)\|t = 0 since ν(a) = dΦ. (2.2) Theorem 2.4 ([26, 17, 21]). For generic exponents a all critical points of Φ are nondegenerate and the number of critical points equals \|χ(U)\|. 12 A. Varchenko Denote C(t)U the algebra of rational functions on Ck regular on U and IΦ = 〈∂Φ ∂ti \| i = 1, . . . , k 〉 ⊂ C(t)U the ideal generated by first derivatives of Φ. Let AΦ = C(t)U/IΦ be the algebra of functions on the critical set and [ ]: C(t)U → AΦ, f 7→ [f ], the canonical homomorphism. If all critical points are isolated, then the critical set is finite and the algebra AΦ is finite- dimensional. In that case, AΦ is the direct sum of local algebras corresponding to points p of the critical set, AΦ = ⊕pAp,Φ. The local algebra Ap,Φ can be defined as the quotient of the algebra of germs at p of holomorphic functions modulo the ideal Ip,Φ generated first derivatives of Φ. Denote by mp ⊂ Ap,Φ the maximal ideal generated by germs of functions equal to zero at p. Lemma 2.5. The elements [1/fj ], j ∈ J , generate AΦ. Proof. If Hj1 , . . . ,Hjk intersect transversally, then 1/fj1 , . . . , 1/fjk form a coordinate system on U . This remark proves the lemma. � Define a rational function Hess(a) : Ck → C, regular on U , by the formula Hess(a)(t) = det 16i,j6k ( ∂2Φ ∂ti∂tj ) (t). The function is called the Hessian of Φ. Let p be an isolated critical point of Φ. Denote by [Hess(a)]p the image of the Hessian in Ap,Φ. It is known that the image is nonzero and the one-dimensional subspace C[Hess(a)]p ⊂ Ap,Φ is the annihilating ideal of the maximal ideal mp ⊂ Ap,Φ. Let ρp : Ap,Φ → C, be the Grothendieck residue, f 7→ 1 (2πi)k Resp f k∏ i=1 ∂Φ ∂ti = 1 (2πi)k ∫ Γp f dt1 ∧ · · · ∧ dtk k∏ s=1 ∂Φ ∂ti , where Γp = {(t1, . . . , tk) \| \|∂Φ ∂ti \| = εi, i = 1, . . . , k} is the real k-cycle oriented by the condition d arg ∂Φ ∂t1 ∧ · · · ∧ d arg ∂Φ ∂tk > 0; here εs are sufficiently small positive real numbers, see [9]. It is known that ρp : [Hess(a)]p 7→ µp, where µp = dimCAp,Φ is the Milnor number of the critical point p. Let ( , )p be the residue bilinear form, (f, g)p = ρp(fg). (2.3) That form is nondegenerate. Quantum Integrable Model of an Arrangement of Hyperplanes 13 2.10 Special vectors in Fk(A) and canonical element A differential top degree form η ∈ Ak(A) can be written as η = fdt1 ∧ · · · ∧ dtk, where f is a rational function on Ck, regular on U . Define a rational map v : Ck → Fk(A) regular on U . For t ∈ U , set v(t) to be the element of Fk(A) such that 〈η, v(t)〉 = f(t) for any η ∈ Ak(A). The map v is called the specialization map and its value v(t) is called the special vector at t ∈ U , see [27]. Let (Fm)m∈M be a basis of Fk(A) and (Hm)m∈M ⊂ Ak(A) the dual basis. Consider the element ∑ mH m⊗Fm ∈ Ak(A)⊗Fk(A). We have Hm = fmdt1∧· · ·∧dtk for some fm ∈ C(t)U . The element E = ∑ m∈M fm ⊗ Fm ∈ C(t)U ⊗Fk(A) will be called the canonical element of the arrangement A. It does not depend on the choice of the basis (Fm)m∈M . For any t ∈ U , we have v(t) = ∑ m∈M fm(t)Fm. Denote by [E] the image of the canonical element in AΦ ⊗Fk(A). Lemma 2.6. We have [E] ∈ AΦ ⊗ SingFk(A). Proof. By formula (2.2), if e ∈ d(a) · A(A)k−1, then 〈1⊗ e, E〉 ∈ IΦ. Hence 〈1⊗ e, [E]〉 = 0 (2.4) in AΦ. Let e1, . . . , el be a basis of d(a) · Ak−1(A). Extend it to a basis e1, . . . , el, el+1, . . . , e\|M \| of Ak(A). Let e1, . . . , el, el+1, . . . , e\|M \| be the dual basis of Fk(A). Then el+1, . . . , e\|M \| is a basis of SingFk(A). Let [E] = \|M \|∑ i=1 [gi]⊗ ei for some [gi] ∈ AΦ. By (2.4) we have [gi] = 0 for all i 6 l. � Theorem 2.7 ([27]). A point t ∈ U is a critical point of Φ, if and only if the special vector v(t) is a singular vector. Proof. The theorem follows from Lemma 2.6. � Theorem 2.8 ([27]). (i) For any t ∈ U , S(a)(v(t), v(t)) = (−1)k Hess(a)(t). (ii) If t1, t2 ∈ U are different isolated critical points of Φ, then the special singular vec- tors v(t1), v(t2) are orthogonal, S(a) ( v ( t1 ) , v ( t2 )) = 0. 14 A. Varchenko 2.11 Arrangements with normal crossings only An essential arrangement A is with normal crossings only, if exactly k hyperplanes meet at every vertex of A. Assume that A is an essential arrangement with normal crossings only. A subset {j1, . . . , jp} ⊂ J will be called independent if the hyperplanes Hj1 , . . . ,Hjp intersect transversally. A basis of Ap(A) is formed by (Hj1 , . . . ,Hjp) where {j1 < · · · < jp} are independent or- dered p-element subsets of J . The dual basis of Fp(A) is formed by the corresponding vectors F (Hj1 , . . . ,Hjp). These bases of Ap(A) and Fp(A) will be called standard. In Fp(A) we have F (Hj1 , . . . ,Hjp) = (−1)\|σ\|F (Hjσ(1) , . . . ,Hjσ(p)) (2.5) for any permutation σ ∈ Sp. For an independent subset {j1, . . . , jp}, we have S(a)(F (Hj1 , . . . ,Hjp), F (Hj1 , . . . ,Hjp)) = aj1 · · · ajp and S(a)(F (Hj1 , . . . ,Hjp), F (Hi1 , . . . ,Hik)) = 0 for distinct elements of the standard basis. 2.12 Real structure on Ap(A) and Fp(A) We have definedAp(A) and Fp(A) as vector spaces over C. But one can define the corresponding spaces over the field R so that Ap(A) = Ap(A)R ⊗R C and Fp(A) = Fp(A)R ⊗R C. If all weights a are real, then the differential d(a) : Ap(A) → Ap+1(A) preserves the real subspaces and one can define the subspace of singular vectors SingFk(A)R ⊂ Fk(A)R so that SingFk(A) = SingFk(A)R ⊗R C. 2.13 A real arrangement with positive weights Let t1, . . . , tk be standard coordinates on Ck. Assume that every polynomial fj , j ∈ J , has real coefficients, fj = zj + b1j t1 + · · ·+ bkj tk, where zj , b i j are real numbers. Denote UR = U ∩ Rk. Let UR = ∪αDα be the decomposition into the union of connected components. Each connected component is a convex polytope. It is known that the number of bounded connected components equals \|χ(U)\|, see [29]. Theorem 2.9 ([26]). Assume that weights (aj)j∈J are positive. Then the union of all critical points of the master function ΦA,a is contained in the union of all bounded components of UR. Each bounded component contains exactly one critical point. All critical points are nondegene- rate. Corollary 2.10. Under assumptions of Theorem 2.9 let t1, . . . , td ∈ UR be a list of all distinct critical points of the master function ΦA,a. Then the corresponding special vectors v(t1), . . . , v(td) form a basis of SingFk(A)R. That basis is orthogonal with respect to the contravariant form S(a). Note that the contravariant form on SingFk(A)R is positive definite. Quantum Integrable Model of an Arrangement of Hyperplanes 15 2.14 Resolution of a hyperplane-like divisor Let Y be a smooth complex compact manifold of dimension k, D a divisor. The divisor D is hyperplane-like if Y can be covered by coordinate charts such that in each chart D is a union of hyperplanes. Such charts will be called linearizing. Let D be a hyperplane-like divisor, U be a linearizing chart. A local edge of D in U is any nonempty irreducible intersection in U of hyperplanes of D in U . An edge of D is the maximal analytic continuation in Y of a local edge. Any edge is an immersed submanifold in Y . An edge is called dense if it is locally dense. For 0 6 i 6 k − 2, let Li be the collection of all dense edges of D of dimension i. The following theorem is essentially contained in Section 10.8 of [25]. Theorem 2.11 ([22]). Let W0 = Y . Let π1 : W1 → W0 be the blow up along points in L0. In general, for 1 6 s 6 k − 1, let πs : Ws → Ws−1 be the blow up along the proper transforms of the (s − 1)-dimensional dense edges in Ls−1 under π1 · · ·πs−1. Let π = π1 · · ·πk−1. Then W = Wn−1 is nonsingular and π−1(D) has normal crossings. 3 A family of parallelly translated hyperplanes 3.1 An arrangement in Cn × Ck Recall that J = {1, . . . , n}. Consider Ck with coordinates t1, . . . , tk, Cn with coordinates z1, . . . , zn, the projection Cn × Ck → Cn. Fix n nonzero linear functions on Ck, gj = b1j t1 + · · ·+ bkj tk, j ∈ J, where bij ∈ C. Define n linear functions on Cn × Ck, fj = zj + gj = zj + b1j t1 + · · ·+ bkj tk, j ∈ J. In Cn × Ck define an arrangement Ã = {H̃j \| fj = 0, j ∈ J}. Denote Ũ = Cn × Ck − ∪j∈JH̃j . For every fixed z = (z1, . . . , zn) the arrangement Ã induces an arrangement A(z) in the fiber over z of the projection. We identify every fiber with Ck. Then A(z) consists of hyperpla- nes Hj(z), j ∈ J , defined in Ck by the same equations fj = 0. Denote U(A(z)) = Ck − ∪j∈JHj(z), the complement to the arrangement A(z). In the rest of the paper we assume that for any z the arrangement A(z) has a vertex. This means that the span of (gj)j∈J is k-dimensional. A point z ∈ Cn will be called good if A(z) has normal crossings only. Good points form the complement in Cn to the union of suitable hyperplanes called the discriminant. 3.2 Discriminant The collection (gj)j∈J induces a matroid structure on J . A subset C = {i1, . . . , ir} ⊂ J is a cir- cuit if (gi)i∈C are linearly dependent but any proper subset of C gives linearly independent gi’s. For a circuit C = {i1, . . . , ir}, let (λCi )i∈C be a nonzero collection of complex numbers such that ∑ i∈C λ C i gi = 0. Such a collection is unique up to multiplication by a nonzero number. 16 A. Varchenko For every circuit C fix such a collection and denote fC = ∑ i∈C λ C i zi. The equation fC = 0 defines a hyperplane HC in Cn. It is convenient to assume that λCi = 0 for i ∈ J −C and write fC = ∑ i∈J λ C i zi. For any z ∈ Cn, the hyperplanes (Hi(z))i∈C in Ck have nonempty intersection if and only if z ∈ HC . If z ∈ HC , then the intersection has codimension r − 1 in Ck. Denote by C the set of all circuits in J . Denote ∆ = ∪C∈CHC . Lemma 3.1. The arrangement A(z) in Ck has normal crossings only, if and only if z ∈ Cn−∆. Remark 3.2. If all linear functions gj , j ∈ J , are real, then for any circuit C ∈ C the numbers (λCi )i∈C can be chosen to be real. Therefore, in that case every hyperplane HC is real. 3.3 Good fibers For any z1, z2 ∈ Cn −∆, the spaces Fp(A(z1)), Fp(A(z2)) are canonically identified. Namely, a vector F (Hj1(z1), . . . ,Hjp(z 1)) of the first space is identified with the vector F (Hj1(z2), . . . , Hjp(z 2)) of the second. Assume that weights a = (aj)j∈J are given and all of them are nonzero. Then each arrange- ment A(z) is weighted. The identification of spaces Fp(A(z1)), Fp(A(z2)) for z1, z2 ∈ Cn −∆ identifies the corresponding subspaces SingFk(A(z1)), SingFk(A(z2)) and contravariant forms. For a point z ∈ Cn − ∆, denote V = Fk(A(z)), Sing V = SingFk(A(z)). The triple (V,Sing V, S(a)) does not depend on z ∈ Cn −∆ under the above identification. 3.4 Bad fibers Points of ∆ ⊂ Cn will be called bad. Let z0 ∈ ∆ and z ∈ Cn − ∆. By definition, for any p the space Ap(A(z0)) is obtained from Ap(A(z)) by adding new relations. Hence Ak(A(z0)) is canonically identified with a quo- tient space of V ∗ = Ak(A(z)) and Fp(A(z0)) is canonically identified with a subspace of V = Fp(A(z)). Let us consider Fk(z0) as a subspace of V . Let S(a)\|Fk(z0) be the restriction of the contravari- ant form on V to that subspace. Let S(a)(z0) be the contravariant form on Fk(A(z0)) of the arrangement A(z0). Lemma 3.3. Under the above identifications, S(a)\|Fk(z0) = S(a)(z0). 4 Conservation of the number of critical points Let A = (Hj)j∈J be an essential arrangement in Ck with weights a. Consider its compactification in the projective space Pk containing Ck. Assign the weight a∞ = − ∑ j∈J aj to the hyperplane H∞ = Pk − Ck and denote by A∨ the arrangement (Hj)j∈J∪∞ in Pk. The weighted arrangement (A, a) will be called unbalanced if the weight of any dense edge of A∨ is nonzero. For example, if all weights (aj)j∈J are positive, then the weighted arrangement (A, a) is unbalanced. Clearly, the unbalanced weights form a Zarisky open subset in the space of all weights of A. Lemma 4.1. If (A, a) is unbalanced, then all critical points of the master function of the weighted arrangement (A, a) are isolated. Quantum Integrable Model of an Arrangement of Hyperplanes 17 Proof. Let π : W → Pk be the resolution (described in Theorem 2.11) of singularities of the divisor D = ∪j∈J∪∞Hj . Let Φa be the master function of (A, a). Then locally on W the function π−1Φa has the form π−1Φa = m∑ i=1 αi log ui + log φ(u1, . . . , uk), φ(0, . . . , 0) 6= 0. Here u1, . . . , uk are local coordinates, 0 6 m 6 k, the function φ(u1, . . . , uk) is holomorphic at u = 0, the equation u1 · · ·um = 0 defines π−1(D) in this chart. If the image of a divisor ui = 0, 1 6 i 6 m, under the map π is an s-dimensional piece of an s-dimensional dense edge of A∨, then αi equals the weight of that edge. In particular, αi, i = 1, . . . ,m, are all nonzero. Let U(A) = Ck − ∪j∈JHj . The critical point equations of π−1Φa on π−1(U(A)) are αi ui + 1 φ(u) ∂φ ∂ui (u) = 0, i = 1, . . . ,m, (4.1) 1 φ(u) ∂φ ∂ui (u) = 0, i = m+ 1, . . . , k. If the critical set of π−1Φa on π−1(U(A)) is infinite, then it contains an algebraic curve. The closure of that curve must intersect π−1(D). But equations (4.1) show that this is impossible. � Denote by µ(A, a) the sum of Milnor numbers of all of the critical points of Φa on U(A). Lemma 4.2. If (A, a) is unbalanced, then µ(A, a) = \|χ(U)\|. Proof. Assume that a(s), s ∈ [0, 1], is a continuous family of unbalanced weights of A. Then µ(A, a(s)) does not depend on s. Indeed, using equations (4.1) one shows that the critical points cannot approach π−1(D) as s ∈ [0, 1] changes. For generic weights a we have µ(A, a(s)) = \|χ(U)\| by Theorem 2.4. Hence, µ(A, a) = \|χ(U)\| for any unbalanced weights a. � 5 Hamiltonians of good fibers 5.1 Construction Consider the master function Φ(z, t) = ∑ j∈J aj log fj(z, t) as a function on Ũ ⊂ Cn × Ck. Let κ be a nonzero complex number. The function eΦ(z,t)/κ defines a rank one local system Lκ on Ũ whose horizontal sections over open subsets of Ũ are univalued branches of eΦ(z,t)/κ mul- tiplied by complex numbers. For a fixed z, choose any γ ∈ Hk(U(A(z)),Lκ\|U(A(z))). The linear map {γ} : Ak(A(z))→ C, ω 7→ ∫ γ eΦ(z,t)/κω, is an element of SingFk(A(z)) by Stokes’ theorem. It is known that for generic κ any element of SingFk(A(z)) corresponds to a certain γ and in that case the integration identifies SingFk(A(z)) and Hk(U(A(z)),Lκ\|U(A(z))), see [23]. The vector bundle ∪z∈Cn−∆Hk(U(A(z)),Lκ\|U(A(z)))→ Cn −∆ 18 A. Varchenko has a canonical (flat) Gauss–Manin connection. The Gauss–Manin connection induces a flat connection on the trivial bundle Cn × Sing V → Cn with singularities over the discriminant ∆ ⊂ Cn. That connection will be called the Gauss–Manin connection as well. Theorem 5.1. The Gauss–Manin differential equations for horizontal sections of the connection on Cn × Sing V → Cn have the form κ ∂I ∂zj (z) = Kj(z)I(z), j ∈ J, where I(z) ∈ Sing V is a horizontal section, Kj(z): V → V , j ∈ J, are suitable linear operators preserving Sing V and independent on κ. For every j, the operator Kj(z) is a rational function of z regular on Cn−∆. Each operator is symmetric with respect to the contravariant form S(a). Theorem 5.1 is proved in Section 5.3. A formula for Kj(z) see in (5.3). The flatness of the connection for all κ implies the commutativity of the operators, Ki(z)\|Sing VKj(z)\|Sing V = Kj(z)\|Sing VKi(z)\|Sing V for all i, j and z ∈ Cn −∆. Let V ∗ be the space dual to V . If M : V → V is a linear operator, then M∗ : V ∗ → V ∗ denotes the dual operator. Let W ⊂ V ∗ be the image of V under the map V → V ∗ associated with the contravariant form and SingW ⊂ W the image of Sing V . The contravariant form induces on W a nondegenerate symmetric bilinear form, also denoted by S(a). Lemma 5.2. For z ∈ Cn −∆, the operators Kj(z) ∗ preserve the subspaces SingW ⊂W ⊂ V ∗. The operators Kj(z) ∗\|W : W → W are symmetric with respect to the contravariant form. The operators Kj(z) ∗\|SingW : SingW → SingW , j ∈ J , commute. For z ∈ Cn − ∆, we define the quantum integrable model assigned to (A(z), a) to be the collection( SingW ; S(a)\|SingW ; K1(z)∗\|SingW , . . . ,Kn(z)∗\|SingW : SingW → SingW ) . (5.1) The unital subalgebra of End(SingW ) generated by operators K1(z)∗\|SingW , . . . ,Kn(z)∗\|SingW will be called the algebra of geometric Hamiltonians of (A(z), a). If the contravariant form S(a) is nondegenerate on V , then this model is isomorphic to the collection( Sing V ; S(a)\|Sing V ; K1(z)\|Sing V , . . . ,Kn(z)\|Sing V : Sing V → Sing V ) . It is clear that any weighted essential arrangement with normal crossings only can be realized as a good fiber of such a construction. Thus, every weighted essential arrangement with normal crossings only is provided with a quantum integrable model. 5.2 Key identity (5.2) For any circuit C = {i1, . . . , ir} ⊂ J , let us define a linear operator LC : V → V in terms of the standard basis of V , see Section 2.11. For m = 1, . . . , r, define Cm = C − {im}. Let {j1 < · · · < jk} ⊂ J be an independent ordered subset and F (Hj1 , . . . ,Hjk) the corresponding element of the standard basis. Define LC : F (Hj1 , . . . ,Hjk) 7→ 0 if \|{j1, . . . , jk} ∩ C\| < r − 1. If {j1, . . . , jk} ∩ C = Cm for some 1 6 m 6 r, then using the skew-symmetry property (2.5) we can write F (Hj1 , . . . ,Hjk) = ±F (Hi1 , Hi2 , . . . , Ĥim , . . . ,Hir−1Hir , Hs1 , . . . ,Hsk−r+1 ) Quantum Integrable Model of an Arrangement of Hyperplanes 19 with {s1, . . . , sk−r+1} = {j1, . . . , jk} − Cm. Define LC : F (Hi1 , . . . , Ĥim , . . . ,Hir , Hs1 , . . . ,Hsk−r+1 ) 7→ (−1)m r∑ l=1 (−1)lailF (Hi1 , . . . , Ĥil , . . . ,Hir , Hs1 , . . . ,Hsk−r+1 ). Lemma 5.3. The map LC is symmetric with respect to the contravariant form. Proof. For l = 1, . . . , r, denote Fl = F (Hi1 , . . . , Ĥil , . . . ,Hir , Hs1 , . . . ,Hsk−r+1 ). It is clear that LC is symmetric with respect to the contravariant form if and only if S(a)(LCFl, Fm) = S(a)(Fl, LCFm) for all 1 6 l,m 6 r. But both sides of this expression are equal to (−1)l+mai1 · · · airas1 · · · ask−r+1 . � On Cn × Ck consider the logarithmic differential 1-forms ωj = dfj fj , j ∈ J, ωC = dfC fC , C ∈ C. For any circuit C = {i1, . . . , ir}, we have ωi1 ∧ · · · ∧ ωir = ωC ∧ r∑ l=1 (−1)l−1ωi1 ∧ · · · ∧ ω̂il ∧ · · · ∧ ωir . Lemma 5.4. We have∑ independent {j1<···<jk}⊂J (∑ j∈J ajωj ) ∧ ωj1 ∧ · · · ∧ ωjk ⊗ F (Hj1 , . . . ,Hjk) = ∑ independent {j1<···<jk}⊂J ∑ C∈C ωC ∧ ωj1 ∧ · · · ∧ ωjk ⊗ LCF (Hj1 , . . . ,Hjk). (5.2) Proof. The lemma is a direct corollary of the definition of maps LC . � Identity (5.2) is a key formula of this paper. Identity (5.2) is an analog of the key Theo- rem 7.2.5 in [23] and it is a generalization of the identity of Lemma 4.2 in [27]. 5.3 An application of the key identity (5.2) – proof of Theorem 5.1 Fix κ ∈ C×, z ∈ Cn − ∆, γ ∈ Hk(U(A(z)),Lκ\|U(A(z))). Let {γ} : Ak(A(z)) → C, ω 7→∫ γ e Φ(z,t)/κω, be the corresponding element of V . We have {γ} = ∑ independent {j1<···<jk}⊂J (∫ γ eΦ(z,t)/κωj1 ∧ · · · ∧ ωjk ) F (Hj1 , . . . ,Hjk). Let z 7→ γ(z) ∈ Hk(U(A(z)),Lκ\|U(A(z))) be a locally constant section of the Gauss–Manin connection. Then {γ(z)} = ∑ independent {j1<···<jk}⊂J (∫ γ(z) eΦ(z,t)/κωj1 ∧ · · · ∧ ωjk ) F (Hj1 , . . . ,Hjk). 20 A. Varchenko Lemma 5.5. The differential of the function {γ(z)} is given by the formula κd{γ(z)} = ∑ C∈C LC{γ(z)}ωC . Proof. The lemma follows from identity (5.2) and the formula of differentiation of an integ- ral. � Lemma 5.6. For every circuit C, the operator LC preserves the subspace Sing V . Proof. The values of the function {γ(z)} belong to Sing V . Hence, the values of its derivatives belong to Sing V . Now the lemma follows from Lemma 5.5. � Recall that ωC = dfC/fC and fC = ∑ j∈J λ C j zj . Denote Kj(z) = ∑ C∈C λCj fC(z) LC , j ∈ J. (5.3) Then ∑ C∈C ωC ⊗ LC = ∑ j∈J dzj ⊗Kj(z). (5.4) Lemma 5.7. Let z 7→ γ(z) ∈ Hk(U(A(z)),Lκ\|U(A(z))) be a locally constant section of the Gauss– Manin connection. Then κ ∂ ∂zj {γ(z)} = Kj(z){γ(z)}, j ∈ J. (5.5) Lemmas 5.3, 5.6, 5.7 prove Theorem 5.1. 5.4 Another application of the key identity (5.2) Recall that Ũ is the complement to the union of hyperplanes (H̃j)j∈J in Cn×Ck, see Section 3.1. Denote by C(z, t)Ũ the algebra of rational functions on Cn × Ck regular on Ũ . For any basis vector (Hj1 , . . . ,Hjk) of V ∗, let us write ωj1 ∧ · · · ∧ ωjk = fj1,...,jk(z, t)dt1 ∧ · · · ∧ dtk + z-part, where fj1,...,jk ∈ C(z, t)Ũ and the z-part is a differential form with zero restriction to any fiber of the projection Cn ×Ck → Cn (in coordinates t1, . . . , tk, z1, . . . , zn, that form has at least one of dz1, . . . , dzn as factors in each of its summands). Define the canonical element Ẽ ∈ C(z, t)Ũ ⊗V by the condition 〈Ẽ, 1⊗ (Hj1 , . . . ,Hjk)〉 = fj1,...,jk , for any independent {j1, . . . , jk} ⊂ J . Theorem 5.8. For any j ∈ J , there exist elements h1, . . . , hk ∈ C(z, t)Ũ ⊗ V such that (1⊗Kj(z)) Ẽ(z, t) = ( aj fj(z, t) ⊗ 1 ) Ẽ(z, t) + k∑ i=1 ( ∂Φ ∂ti (z, t)⊗ 1 ) hi(z, t). Quantum Integrable Model of an Arrangement of Hyperplanes 21 Proof. We have ν(a) = k∑ i=1 ∂Φ ∂ti (z, t) dti + ∑ j∈J aj fj(z, t) dzj (5.6) and ∑ independent {j1<···<jk}⊂J ωj1 ∧ · · · ∧ ωjk ⊗ F (Hj1 , . . . ,Hjk) = Ẽ(z, t) (dt1 ∧ · · · ∧ dtk ⊗ 1) + z-part, (5.7) where z-part is a V -valued differential k-form with zero restriction to each fiber of the projection Cn × Ck → Cn. Then identity (5.2) and formulas (5.6), (5.7), (5.4) imply the theorem. � 5.5 Hamiltonians, critical points and the canonical element Fix z ∈ Cn − ∆. Recall that in Section 5.1 we have defined the quantum integrable model assigned to (A(z), a) to be the collection( SingW ; S(a)\|SingW ; K1(z)∗\|SingW , . . . ,Kn(z)∗\|SingW : SingW → SingW ) . Let p ∈ U(A(z)) be an isolated critical point of the master function Φ(z, ·) : U(A(z)) → C. Let Ap,Φ be the local algebra of the critical point and [ ] : C(t)U(A(z)) → Ap,Φ the canonical projection. Denote by Hess(a) the Hessian of Φ(z, ·) with respect to variables t1, . . . , tk. Let E ∈ C(t)U(A(z)) ⊗ V be the canonical element associated with A(z), see Section 2.10. Denote by [E] its projection to Ap,Φ ⊗ V . By Lemma 2.6 we have [E] ∈ Ap,Φ ⊗ Sing V . Let V → W be the map associate with the contravariant form and [E] the image of [E] under the induced map Ap,Φ ⊗ Sing V → Ap,Φ ⊗ SingW , [E] ∈ Ap,Φ ⊗ SingW. Theorem 5.9. We have (i) S(a)([E], [E]) = (−1)k[Hess(a)], (ii) (1⊗Kj(z) ∗)[E] = ([aj/fj(z, ·)]⊗ 1)[E] for j ∈ J . Proof. Part (i) follows from Theorem 2.8. Part (ii) follows from Theorem 5.8. � Remark 5.10. The elements [aj/fj(z, ·)], j ∈ J , generate Ap,Φ due to Lemma 2.5 and the assumption (aj 6= 0 for all j). 6 Asymptotic solutions and eigenvectors of Hamiltonians 6.1 Asymptotic solutions, one variable Let u be a variable, W a vector space, M(u) ∈ End(W ) an endomorphism depending holomor- phically on u at u = 0. Consider a differential equation, κ dI du (u) = M(u)I(u) (6.1) depending on a complex parameter κ ∈ C∗. 22 A. Varchenko Let P (u) ∈ C, (wm(u) ∈ W )m∈Z>0 be functions holomorphic at u = 0 and w0(0) 6= 0. The series I(u, κ) = eP (u)/κ ∞∑ m=0 wm(u)κm (6.2) will be called an asymptotic solution to (6.1) if it satisfies (6.1) to all orders in κ. In particular, the leading order equation is dP du (u)w0(u) = M(u)w0(u). (6.3) Assume now that M(u) = M−1 u +M0 +M1u+ · · · , Mj ∈ End(W ), has a first order pole at u = 0 and I(u, κ) is a series like in (6.2). The series I(u, κ) will be called an asymptotic solution to equation (6.1) with such M(u) if it satisfies (6.1) to all orders in κ. In particular, the leading order equation is again equation (6.3). Equation (6.3) implies w0(0) ∈ kerM−1, dP du (0)w0(0) = M0w0(0) +M−1 dw0 du (0). (6.4) 6.2 Critical points of the master function and asymptotic solutions Let us return to the situation of Section 3. Let t(z) be a nondegenerate critical point of Φ(z, · ) : U(A(z)) → C. Assume that t(z) depends on z holomorphically in a neighborhood of a point z0 ∈ Cn. Fix a univalued branch of Φ in a neighborhood of (z0, t(z0)) (by choosing arguments of all of the logarithms). Denote Ψ(z) = Φ(z, t(z)). Let B ⊂ Ck be a small ball with center at t(z0). Denote B− = { t ∈ B \| Re Φ ( z0, t ( z0 )) > Re Φ ( z0, t )} . It is well known that Hk(B,B −;Z) = Z, see for example [1]. There exist local coordinates u1, . . . , uk on Ck centered at t(z0) such that Φ(z0, u) = −u2 1 − · · · − u2 k + const. Denote δ = { (u1, . . . , uk) ∈ Rk \| u2 1 + · · ·+ u2 k 6 ε } , where ε is a small positive number. That δ, considered as a k-chain, generates Hk(B,B −;Z). Define an element {δ}(z, κ) ∈ V by the formula {δ}(z, κ) : V ∗ → C, ω 7→ κ−k/2 ∫ δ eΦ(z,t)/κω(z, t). (6.5) Recall that any element ω ∈ V ∗ is a linear combination of elements (Hj1 , . . . ,Hjk) and such an element (Hj1 , . . . ,Hjk) is identified with the differential form ωj1 ∧ · · · ∧ ωjk = dfj1(z, t)/fj1(z, t) ∧ · · · ∧ dfjk(z, t)/fjk(z, t). In (6.5) we integrate over δ such a differential form multiplied by eΦ(z,t)/κ. The element {δ}(z, κ) as a function of z, κ is holomorphic if z is close to z0 and κ 6= 0. Quantum Integrable Model of an Arrangement of Hyperplanes 23 Theorem 6.1. (i) Let κ ∈ R and κ→ +0. Then the function {δ}(z, κ) has an asymptotic expansion {δ}(z, κ) = eΨ(z)/κ ∞∑ m=0 wm(z)κm, (6.6) where (wm(z) ∈ V )m∈Z>0 are functions of z holomorphic at z0 and w0(z) = ±(2π)k/2 ( (−1)k det 16i,j6k ( ∂2Φ ∂ti∂tj ) (z, t(z)) )−1/2 v(z, t(z)). Here v(z, t(z)) is the special vector associated with the critical point (z, t(z)) of the function Φ(z, · ), see Section 2.10. The sign ± depends on the choice of the orientation of δ. (ii) The asymptotic expansion (6.6) gives an asymptotic solution to the Gauss–Manin diffe- rential equations (5.5). (iii) The functions (wm(z))m∈Z>0 take values in Sing V . Part (i) of the theorem is a direct corollary of the method of steepest descent; see, for example, § 11 in [1]. Part (ii) follows from Lemma 5.4 and formula of differentiation of an integral. Part (iii) follows from Stokes’ theorem. Remark 6.2. The definition of δ depends on the choice of local coordinates u1, . . . , uk, but the asymptotic expansion (6.6) does not depend on the choice of δ since the difference of the corresponding integrals is exponentially small. Corollary 6.3. Let z0, t = t(z), Ψ(z) be the same as in Theorem 6.1. Assume that z0 ∈ Cn−∆. In that case the operators Kj(z 0)\|Sing V : Sing V → Sing V , j ∈ J , are all well-defined, see (5.3), and we have Kj ( z0 ) v ( z0, t ( z0 )) = ∂Ψ dzj ( z0 ) v ( z0, t ( z0 )) , j ∈ J, Thus, the special vector v(z0, t(z0)) is an eigenvector of the geometric Hamiltonians Kj(z 0). Proof. The corollary follows from equation (6.3). � Note that ∂Ψ dzj (z0) = aj fj(z0,t(z0)) . Remark 6.4. The Gauss–Manin differential equations (5.5) have singularities over the discrimi- nant ∆ ⊂ Cn. If z0 ∈ ∆, then expansion (6.6) still gives an asymptotic solution to equations (5.5) and that asymptotic solution is regular at z0. 7 Hamiltonians of bad fibers 7.1 Naive geometric Hamiltonians Let us return to the situation of Section 3. Let z0 ∈ ∆ and z ∈ Cn −∆. We have SingFk(A(z0)) ⊂ SingFk(A(z)) ⊂ Fk(A(z)), SingFk(A(z0)) ⊂ Fk(A(z0)) ⊂ Fk(A(z)), (7.1) SingFk(A(z0)) = Fk(A(z0)) ∩ (SingFk(A(z))), see Section 3.4. Recall that Fk(A(z)) was denoted by V . 24 A. Varchenko Consider the map V → V ∗ corresponding to the contravariant form. In Section 5.1 we denoted the images of V and Sing V by W and SingW , respectively. We denote the images of Fk(A(z0)) and SingFk(A(z0)) by W (z0) and SingW (z0), respectively. We have SingW ( z0 ) ⊂ SingW ⊂W, SingW ( z0 ) ⊂W ( z0 ) ⊂W. Recall that ∆ is the union of hyperplanes HC , C ∈ C. Denote C0 = {C ∈ C \| z0 ∈ HC}. Consider the operator-valued functions Kj(z) : V → V , j ∈ J , given by formula (5.3). Denote K0 j (z) = ∑ C∈C0 λCj fC(z) LC , K1 j (z) = Kj(z)−K0 j (z). Each of the summands of K0 j (z) tends to infinity as z tends to z0 in Cn−∆. The operator-valued function K1 j (z) is regular at z0. The operators Kj(z) ∗, L∗C preserve the subspaces SingW ⊂ W ⊂ V ∗ and are symmetric operators on W with respect to the contravariant form on W . The operators Kj(z) ∗ restricted to SingW commute. The point z0 ∈ ∆ defines an edge Xz0 of the arrangement (HC)C∈C, where Xz0 = ∩C∈C0HC . Denote by Tz0 the vector space of constant vectors fields on Cn which are tangent to Xz0 , Tz0 = ξ = ∑ j∈J ξj ∂ ∂zj \| ξj ∈ C, ξ(fC) = 0 for all C ∈ C0  . Lemma 7.1. For any ξ ∈ Tz0, (i) The linear operator Kξ(z) = ∑ j∈J ξjKj(z) : V → V, considered as a function of z, is regular at z0, moreover, Kξ(z) = ∑ j∈J ξjK 1 j (z). (ii) The linear operator Kξ(z 0) preserves the subspace Fk(A(z0)) ⊂ V . (iii) The dual linear operator Kξ(z) ∗ : V ∗ → V ∗, considered as a function of z, is regular at z0, moreover, Kξ(z) ∗ = ∑ j∈J ξjK 1 j (z)∗. (iv) The linear operator Kξ(z 0)∗ preserves the subspace SingW (z0) ⊂ V ∗. Proof. Parts (iii), (iv) follow from parts (i), (ii). Part (i) is clear. Part (ii) follows from a straightforward calculation. � The operators Kξ(z 0)∗ preserve the subspace SingW (z0). The operators Kξ(z 0)∗\|SingW (z0) : SingW (z0)→ SingW (z0), ξ ∈ Tz0 , form a commutative family of linear operators. The operators are symmetric with respect to the contravariant form. These operators will be called naive geometric Hamiltonians on SingW (z0). Quantum Integrable Model of an Arrangement of Hyperplanes 25 7.2 Space Fk(A(z0)) and operators LC Lemma 7.2. (i) The space Fk(A(z0)) lies in the kernel of LC : V → V for any C ∈ C0. (ii) The space W (z0) lies in the kernel of L∗C \|W : W →W for any C ∈ C0. (iii) For any C ∈ C0, the image of L∗C \|W is orthogonal to W (z0) with respect to the contravariant form. Proof. Part (i) follows from a straightforward easy calculation. Part (ii) follows from part (i). Part (iii) follows from part (ii) and the fact that L∗C is symmetric. � 7.3 Conjecture Conjecture 7.3. Let z0 ∈ ∆. Assume that the contravariant form restricted to SingW (z0) is nondegenerate. Let pr : SingW → SingW (z0) be the orthogonal projection with respect to the contravariant form. Then the linear operators prK1 j ( z0 )∗\|SingW (z0) : SingW ( z0 ) → SingW ( z0 ) , j ∈ J, commute and are symmetric with respect to the contravariant form. For z0 ∈ ∆, we define the quantum integrable model assigned to (A(z0), a) to be the collection( SingW ( z0 ) , S(a)\|SingW (z0), prK1 j ( z0 )∗\|SingW (z0) : SingW ( z0 ) → SingW ( z0 ) , where j ∈ J ) . The unital subalgebra of End(SingW (z0)) generated by operators prK1 j ( z0 )∗\|SingW (z0), j ∈ J, will be called the algebra of geometric Hamiltonians of (A(z0), a). Note that the naive geometric Hamiltonians are elements of the algebra of geometric Hamil- tonians, since for any ξ = ∑ j∈J ξj ∂ ∂zj ∈ TXz0 , we have Kξ ( z0 )∗\|SingW (z0) = ∑ j∈J ξjprK1 j ( z0 )∗\|SingW (z0). In the next section we prove the conjecture under Assumption 7.4 of certain positivity and reality conditions, see Theorem 7.5. In Section 9 more results in this direction will be obtained, see Theorems 9.16 and 9.17. For applications to the Gaudin model an equivariant version of the conjecture is needed, see Sections 10 and 11. 7.4 Positive (aj)j∈J , real (gj)j∈J Assumption 7.4. Assume that all weights aj , j ∈ J , are positive and all functions gj = b1j t1 + · · ·+ bkj tk, j ∈ J , have real coefficients bij. The space V has a real structure, V = VR ⊗R C, see Section 2.12. Under Assumption 7.4 all subspaces in (7.1) are real (can be defined by real equations). The contravariant form S(a) is positive definite on VR and is positive definite on the real parts of all of the subspaces in (7.1). Denote by pr : Sing V → SingFk(A(z0)) the orthogonal projection. Theorem 7.5. Assume that Assumption 7.4 is satisfied and z0 ∈ ∆. Then the operators prK1 j ( z0 ) \|SingFk(A(z0)) : SingFk ( A ( z0 )) → SingFk ( A ( z0 )) , j ∈ J, commute and are symmetric with respect to the contravariant form. Theorem 7.5 proves Conjecture 7.3 under Assumption 7.4. 26 A. Varchenko 7.5 Proof of Theorem 7.5 for z0 ∈ ∆ ∩ Rn Assume that z0 ∈ ∆ ∩ Rn. Let r : (C,R, 0) → (Cn,Rn, z0) be a germ of a holomorphic curve such that r(u) ∈ Cn − ∆ for u 6= 0. For u ∈ R>0, the arrangement A(r(u)) is real. Denote U(r(u))R = (Ck − ∪j∈JHj(r(u))) ∩ Rk. Let U(r(u))R = ∪αDα(r(u)) be the decomposition into the union of connected components. We label components so that for any α, the component Dα(r(u)) continuously depends on u > 0. Let A be the set of all α such that Dα(r(u)) is bounded. Let A1 be the set of all α such that Dα(r(u)) is bounded and vanishes as u → +0 (the limit of Dα(r(u)) is not a domain of A(z0)). Let A2 be the set of all α such that Dα(r(u)) is bounded and the limit of Dα(r(u)) as u→ 0 is a domain of A(z0). We have A = A1 ∪A2 and A1 ∩A2 = ∅. All critical points of Φ(r(u), · ) lie in ∪α∈ADα(r(u)). Each domain Dα(r(u)) contains a unique critical point (r(u), t(u)α) and that critical point is nondegenerate. Denote by v(r(u), t(u)α) ∈ Sing V the corresponding special vector. That vector is an eigenvector of the geometric Hamil- tonians, Kj(r(u)) v(r(u), t(u)α) = aj fj(r(u), t(u)α) v(r(u), t(u)α), j ∈ J. If α ∈ A2, then all eigenvalues 1/fj(r(u), t(u)α), j ∈ J , are regular functions at u = 0. If α ∈ A1, then there is an index j ∈ J such that aj/fj(r(u), t(u)α)→∞ as u→ 0. Lemma 7.6. The span 〈v(r(u), t(u)α)〉α∈A2 has a limit as u→ +0. That limit is SingFk(z0) ⊂ Sing V . Similarly, the span 〈v(r(u), t(u)α)〉α∈A1 has a limit as u → +0. That limit is (SingFk(z0))⊥ ⊂ Sing V where ⊥ denotes the orthogonal complement. Proof. The lemma follows from Theorem 2.9 and Corollary 2.10. � Assume now that a curve r(u) = (z1(u), . . . , zn(u)) is linear in u. Then for any j we have K0 j (r(u)) = Nj/u where Nj : Sing V → Sing V is an operator independent of u. Lemma 7.7. The image of Nj is a subspace of (SingFk(z0))⊥. Proof. The lemma follows from Theorem 2.9 and Corollary 2.10. � By formula (6.4) and Lemma 7.7, for any α ∈ A2 we have aj fj(r(0), t(0)α) v(r(0), t(0)α) = K1 j (r(0))v(r(0), t(0)α) + v1, where v1 ∈ (SingFk(z0))⊥. Thus, prK1 j (r(0)) v(r(0), t(0)α) = 1 fj(r(0), t(0)α) v(r(0), t(0)α). (7.2) Thus, all operators prK1 j (r(0)) are diagonal in the basis (v(r(0), t(0)α))α∈A2 of SingFk(A(z0)). Equation (7.2) finishes the proof of the commutativity of operators prK1 j (z0). The symmetry of the operators prK1 j (z0) with respect to the contravariant form follows from the fact that operators prK1 j (z0) are diagonal in the orthogonal basis (v(r(0), t(0)α))α∈A2 . Quantum Integrable Model of an Arrangement of Hyperplanes 27 7.6 Proof of Theorem 7.5 for any z0 ∈ ∆ Consider the arrangement (HC)C∈C in Cn and its arbitrary edge X. The arrangement (HC)C∈C is real, see Remark 3.2. The edge X is the complexification of X ∩ Rn. Denote X̂ = X − ∪C∈C−CXX ∩HC . (7.3) For any z1, z2 ∈ X̂, the subspaces SingFk ( A ( z1 )) ⊂ Sing V, SingFk ( A ( z2 )) ⊂ Sing V coincide. Denote that subspace by SingFk(A(X)) ⊂ Sing V . For z ∈ X̂, the operators prK1 j (z) : SingFk(A(X)) → SingFk(A(X)), j ∈ J , depend on z holomorphically. The operators commute and are symmetric for z ∈ X̂ ∩ Rn, by reasonings in Section 7.5. Hence they commute and are symmetric for all z ∈ X̂. 7.7 Critical points and eigenvectors Theorem 7.8. Assume Assumption 7.4. Let z0 ∈ ∆ and let p be a critical point of Φ(z0, · ) : U(A(z0)) → C. Then the corresponding special vector v(z0, p) ∈ SingFk(z0) (if nonzero) is an eigenvector of the operators prK1 j (z0), j ∈ J , prK1 j ( z0 ) v ( z0, p ) = aj fj(z0, p) v ( z0, p ) . (7.4) Proof. If z0 ∈ ∆ ∩ Rn, then the theorem is just a restatement of formula (7.2). Assume that z0 is an arbitrary point of ∆. Then there exists an edge X of the arrangement (HC)C∈C such that z0 ∈ X̂, see (7.3). For z0 ∈ X̂, all objects in formula (7.4) depend on z0 algebraically. Hence, the fact, that formula (7.4) holds for all critical points if z0 ∈ X̂ ∩ Rn, implies Theorem 7.8 for any z0 ∈ X̂. � 7.8 Hamiltonians, critical points and the canonical element Let Assumption 7.4 be satisfied. Fix z0 ∈ ∆. We have defined the quantum integrable model assigned to (A(z0), a) to be the collection( SingFk ( A ( z0 )) ; S(a)\|SingFk(A(z0)); prK1 j ( z0 ) \|SingFk(A(z0)) : SingFk ( A ( z0 )) → SingFk ( A ( z0 )) , where j ∈ J ) , see Section 7.4. Let p ∈ U(A(z0)) be an isolated critical point of the function Φ(z0, · ) : U(A(z0)) → C. Let Ap,Φ be the local algebra of the critical point and [ ] : C(t)U(A(z0)) → Ap,Φ the canonical projection. Denote by Hess(a) the Hessian of Φ(z0, · ). Let E ∈ C(t)U(A(z0)) ⊗ Fk(A(z0)) be the canonical element associated with A(z0), see Sec- tion 2.10. Denote by [E] the projection of the canonical element to Ap,Φ ⊗ Fk(A(z0)). By Lemma 2.6, we have [E] ∈ Ap,Φ ⊗ SingFk ( A ( z0 )) . Theorem 7.9. We have (i) S(a)([E], [E]) = (−1)k[Hess(a)], (ii) (1⊗ prK1 j (z0))[E] = ([aj/fj(z, ·)]⊗ 1)[E] for j ∈ J . Proof. Part (i) follows from Theorem 2.8. Part (ii) follows from Theorem 7.8. � 28 A. Varchenko 8 Geometric interpretation of the algebra of Hamiltonians 8.1 An abstract setting Let k < n be positive integers and J = {1, . . . , n} as before. Let F be a germ of a holomorphic function at a point p ∈ Ck. Assume that p is an isolated critical point of F with Milnor number µp. Let Ap,F be the local algebra of the critical point and ( , )p the residue bilinear form on Ap,F , see (2.3). Denote by [HessF ] the projection to Ap,F of the germ det16l,m6k(∂ 2F/∂tl∂tm). Let h1, . . . , hµp be a C-basis of Ap,F . Let g1, . . . , gn ∈ Ap,F be a collection of elements such that the unital subalgebra of Ap,F generated by g1, . . . , gn equals Ap,F . Let W be a vector space with a symmetric bilinear form S. Let Mj : W → W , j ∈ J , be a collection of commuting symmetric linear operators, MiMj = MjMi, S(Mju, v) = S(u,Mjv) for all i, j ∈ J and u, v ∈W. Assume that an element w = µp∑ l=1 hl ⊗ wl ∈ Ap,F ⊗W is given such that µp∑ l=1 hl ⊗Mjwl = m∑ l=1 gjhl ⊗ wl, j ∈ J, (8.1) µp∑ l,m=1 S(wl, wm)hlhm = (−1)k[HessF ]. (8.2) Denote by Y ⊂W the vector subspace generated by w1, . . . , wµp . By property (8.1), everyMj , j ∈ J , preserves Y . Denote by AY the unital subalgebra of End(Y ) generated by Mj \|Y , j ∈ J . The subspace Y is an AY -module. Define a linear map α : Ap,F → Y, f 7→ µp∑ l=1 (f, hl)pwl. Theorem 8.1. (i) The map α : Ap,F → Y is an isomorphism of vector spaces. The form S restricted to Y is nondegenerate. (ii) The map gj 7→Mj \|Y , j ∈ J , extends uniquely to an algebra isomorphism β : Ap,F → AY . (iii) The maps α, β give an isomorphism of the regular representation of Ap,F and the AY - module Y , that is Mjα(f) = α(gjf) for any f ∈ Ap,F and j ∈ J . (iv) Define the value w(p) of w at p as the image of w under the natural projection Ap,F ⊗W → Ap,F /mp⊗W = W . Then w(p) = α(HessF )/µp and the value w(p) is nonzero. The vector w(p) is the only (up to proportionality) common eigenvector of the operators Mj \|Y , j ∈ J , and we have Mjw(p) = gj(p)w(p). This theorem is an analog of Theorem 5.5 and Corollary 5.6 in [15]. The proof is analogous to the proofs in [15]. Quantum Integrable Model of an Arrangement of Hyperplanes 29 8.2 Proof of Theorem 8.1 Lemma 8.2. We have Mjα(f) = α(gjf) for any f ∈ Ap,F , j ∈ J . Proof. We have Mjα(f) = µp∑ l=1 (f, hl)pMjwl. By (8.1), that is equal to m∑ l=1 (f, gjhl)pwl = m∑ l=1 (gjf, hl)pwl = α(gjf). � Define a bilinear form ( , )S on Ap,F , (f, g)S = S(α(f), α(g)) = µp∑ l,m=1 S(wl, wm)(f, hl)p(g, hm)p. Lemma 8.3. We have (fg, h)S = (f, gh)S for all f, g, h ∈ Ap,F . Proof. Since gj , j ∈ J , generate Ap,F it is enough to show that (fgj , h)S = (f, gjh)S for all f, h ∈ Ap,F , j ∈ J . We have (fgj , h)S = S(α(fgj), α(h)) = S(Mjα(f), α(g)) = S(α(f),Mjα(g)) = S(α(f), α(gjh)) = (f, gjh)S . � Lemma 8.4. There exists a unique element s ∈ Ap,F such that (f, g)S = (sf, g)p for all f, g ∈ Ap,F . Proof. Consider the linear function Ap,F → C, f 7→ (1, f)S . Since the bilinear form ( , , )p is nondegenerate there exists a unique s ∈ Ap,F such that (1, f)S = (s, f)p for any f ∈ Ap,F . Hence for any f, g ∈ Ap,F we have (f, g)S = (1, fg)S = (s, fg)p = (sf, g)p. � Lemma 8.5. For any f ∈ Ap,F , the trace of the linear operator Lf : Ap,F → Ap,F , h 7→ fh, is given by the formula trLf = (f, [HessF ])p. Proof. We have trL1 = µp = (1, [HessF ])p and trLf = 0 = (f, [HessF ])p for any f ∈ mp. This proves the lemma. � Let h∗1, . . . , h ∗ µp be a C-basis of Ap,F dual to h1, . . . , hµp with respect to the form ( , )p. Then µp∑ l=1 ([HessF ], h∗l )phl = [HessF ]. Indeed, for any f ∈ Ap,F we have µp∑ l=1 (f, h∗l )phl = f . Lemma 8.6. We have µp∑ l=1 hlh ∗ l = [HessF ]. Proof. For any f ∈Ap,F , we have trLf = µp∑ l=1 (h∗l , fhl)p = µp∑ l=1 (h∗l hl, f)p and trLf = (f, [HessF ])p. This proves the lemma. � Lemma 8.7. Let s ∈ Ap,F be the element defined in Lemma 8.4. Then s has the following two properties: (i) the element s is invertible and projects to (−1)k in C = Ap,F /mp, (ii) the form ( , )S is nondegenerate. Proof. To prove the lemma it is enough to show that (−1)k[HessF ] = s[HessF ]. Indeed, on one hand we have (f, g)S = µp∑ l,m=1 S(wl, wm)(f, hl)p(g, hm)p. 30 A. Varchenko On the other hand we have (f, g)S = (sf, g)p = µp∑ l=1 (sf, hl)p(g, h ∗ l )p. Hence, µp∑ l,m=1 S(wl, wm)hl ⊗ hm = µp∑ l=1 shl ⊗ h∗l in Ap,F ⊗Ap,F . Therefore, µp∑ l,m=1 S(wl, wm)hlhm = µp∑ l=1 shlh ∗ l . By Assumption (8.2) and Lemma 8.6 we obtain (−1)k[HessF ] = s[HessF ]. � Let us prove Theorem 8.1. Assume that µp∑ l=1 λlwl = 0 with λl ∈ C. Denote h = µp∑ l=1 λlh ∗ l . Then α(h) = 0 and (f, h)S = S(α(f), α(h)) = 0 for all f ∈ Ap,F . Hence h = 0 since ( , )S is nondegenerate. Therefore, λl = 0 for all l and the vectors v1, . . . , vµT are linearly independent. This proves part (i) of Theorem 8.1. Parts (ii) and (iii) follow from Lemma 8.2. 8.3 Remark on maximal commutative subalgebras Let A be a commutative algebra with unity element 1. Let B be the subalgebra of End(A) generated by all multiplication operators Lf : A→ A, h 7→ fh, where f ∈ A. Lemma 8.8. The subalgebra B is a maximal commutative subalgebra of End(A). Proof. Let T ∈ End(A) be such that [T, Lf ] = 0 for all f ∈ A. Then T = LT (1). � Corollary 8.9. Under assumptions of Section 8.1, the algebra AY is a maximal commutative subalgebra of End(Y ). 8.4 Interpretation of the algebra of Hamiltonians of good fibers Under notations of Section 3 fix a point z ∈ Cn − ∆. Recall that in formula (5.1) we have defined the quantum integrable model assigned to (A(z), a) to be the collection( SingW ; S(a)\|SingW ; K1(z)∗\|SingW , . . . ,Kn(z)∗\|SingW : SingW → SingW ) . Let p ∈ U(A(z)) be an isolated critical point of the master function Φ(z, · ) : U(A(z)) → C. Let Ap,Φ be the local algebra of the critical point and ( , )p the residue bilinear form on Ap,Φ. Let [E] ∈ Ap,Φ⊗SingW be the element corresponding to the canonical element, see Section 5.5. Define a linear map αp : Ap,Φ → SingW, g 7→ (g, [E])p. Denote Yp the image of αp. Theorem 8.10. (i) We have ker αp = 0. (ii) The operators Kj(z) ∗ preserve Yp. Moreover, for any j ∈ J , g ∈ Ap,Φ, we have αp(gaj/[fj(z, ·)]) = Kj(z) ∗αp(g). Quantum Integrable Model of an Arrangement of Hyperplanes 31 (iii) Define the value [E](p) of [E] at p as the image of [E] under the natural projection Ap,Φ⊗ SingW → Ap,Φ/mp ⊗ SingW = SingW . Then the value [E](p) is nonzero. The vector [E](p) is the only (up to proportionality) common eigenvector of the operators Kj(z) ∗\|Yp : Yp → Yp, j ∈ J , and we have Kj(z) ∗[E](p) = aj fj(z, p) [E](p). Proof. By Theorem 5.9 and Remark 5.10 the objects SingW,S(a)\|SingW , Kj(z) ∗\|SingW , [v], [aj/fj(z, ·)], j ∈ J , satisfy the assumptions of Theorem 8.1. Now Theorem 8.10 follows from Theorem 8.1. � Theorem 8.11. The linear map αp identifies the contravariant form on Yp and the residue form ( , )p on Ap,Φ multiplied by (−1)k, S(a)(αp(f), αp(g)) = (−1)k(f, g)p (8.3) for any f, g ∈ Ap,Φ. Proof. If the Milnor number of p is one, then the theorem follows from Lemma 8.7. If the Milnor number is greater than one, the theorem follows by continuity from the case of the Milnor number equal to one, since all objects involved depend continuously on the weights a and parameters z. Note that the theorem says that the element s of Lemma 8.7 in our situation equals (−1)k. � Remark 8.12. In formula (8.3), each of αp(f), αp(g) is given by the Grothendieck residue, so each of αp(f), αp(g) is a k-dimensional integral. The quantity (f, g)p is also a k-dimensional in- tegral. Thus formula (8.3) is an equality relating a bilinear expression in k-dimensional integrals to an individual k-dimensional integral. Denote by AYp the unital subalgebra of End(Yp) generated by Kj(z) ∗\|Yp , j ∈ J . Corollary 8.13. (i) The map [aj/fj(z, ·)] 7→ Kj(z) ∗\|Yp, j ∈ J , extends uniquely to an algebra isomorphism βp : Ap,Φ → AYp. (ii) The maps αp, βp give an isomorphism of the regular representation of Ap,Φ and the AYp- module Yp, that is βp(h)αp(g) = αp(hg) for any h, g ∈ Ap,Φ. (iii) The algebra AYp is a maximal commutative subalgebra of End(Yp). (iv) All elements of the algebra AYp are symmetric operators with respect to the contravariant form S(a). Theorem 8.14. Let p1, . . . , pd be a list of all distinct isolated critical points of Φ(z, · ). Let Yps = αps(Aps,Φ) ⊂ SingW , s = 1, . . . , d, be the corresponding subspaces. Then the sum of these subspaces is direct. The subspaces are orthogonal. Proof. It follows from Theorem 8.10 that for any s = 1, . . . , d and j ∈ J the operator Kj(z) ∗− aj/fj(z, ps) restricted to Yps is nilpotent. We also know that the numbers aj/fj(z, ps) separate the points p1, . . . , pd. These observations imply Theorem 8.14. � Corollary 8.15. The sum of Milnor numbers of the critical points p1, . . . , pd is not greater than the rank of the contravariant form S(a)\|SingW . 32 A. Varchenko Denote Y = ⊕ds=1Yps . Denote by AY the unital subalgebra of End(Y ) generated by Kj(z) ∗\|Y , j ∈ J . Consider the isomorphisms α = ⊕ds=1αps : ⊕ds=1Aps,Φ → ⊕ds=1Yps , β = ⊕ds=1βs : ⊕ds=1Aps,Φ → ⊕ds=1AYps . Corollary 8.16. We have (i) AY = ⊕ds=1Aps,Yps ; (ii) AY is a maximal commutative subalgebra of End(Y ). (iii) The isomorphisms α, β identify the regular representation of the algebra ⊕ds=1Aps,Φ and the AY -module Y . The isomorphism α identifies the contravariant form on Y and the residue form ( , ) = ⊕ds=1( , )ps on ⊕ds=1Aps,Φ multiplied by (−1)k. (iv) In particular, if the dimension of SingW equals the sum of Milnor numbers d∑ s=1 µs, then the module SingW over the unital subalgebra of End(SingW ) generated by geometric Hamilto- nians Kj(z) ∗\|SingW : SingW → SingW , j ∈ J , is isomorphic to the regular representation of the algebra ⊕ds=1Aps,Φ. (v) If for z ∈ Cn −∆ the arrangement (A(z), a) is unbalanced, then the module SingW over the unital subalgebra of End(SingW ) generated by geometric Hamiltonians Kj(z) ∗\|SingW , j ∈ J , is isomorphic to the regular representation of the algebra ⊕ds=1Aps,Φ. Corollary 8.17. If for z ∈ Cn−∆ the arrangement (A(z), a) is unbalanced, then the contrava- riant form is nondegenerate on SingFk(A(z)). Proof. Indeed in this case the sum of Milnor numbers of critical points of the master function equals \|χ(U(A(z)))\| and equals dim SingFk(A(z)). � 8.5 Interpretation of the algebra of Hamiltonians of bad fibers if Assumption 7.4 is satisfied Let Assumption 7.4 be satisfied. Fix z0 ∈ ∆. We have defined the quantum integrable model assigned to (A(z0), a) to be the collection( SingFk ( A ( z0 )) ; S(a)\|SingFk(A(z0)); K̃1 ( z0 ) , . . . , K̃n ( z0 ) : SingFk ( A ( z0 )) → SingFk ( A ( z0 ))) , where K̃j(z 0) = prK1 j (z0)\|SingFk(A(z0)), see Theorem 7.5. Let p ∈ U(A(z0)) be an isolated critical point of the master function Φ(z0, · ) : U(A(z0))→ C. Let Ap,Φ be the local algebra of the critical point and ( , )p the residue bilinear form on Ap,Φ. Let [E] ∈ Ap,Φ ⊗ SingFk(A(z0)) be the canonical element corresponding to the arrangements A(z0), see Section 7.8. Define a linear map αp : Ap,Φ → SingFk ( A ( z0 )) , g 7→ (g, [E])p. Denote Yp the image of αp. Theorem 8.18. Let Assumption 7.4 be satisfied. Then (i) We have ker αp = 0. The isomorphism αp identifies the contravariant form on Yp and the residue form ( , )p on Ap,Φ multiplied by (−1)k. Quantum Integrable Model of an Arrangement of Hyperplanes 33 (ii) For any j ∈ J , the operator K̃j(z 0) preserves Yp. Moreover, for any j ∈ J and g ∈ Ap,Φ, we have αp(gaj/[fj(z 0, ·)]) = K̃j(z 0)αp(g). (iii) Define the value [E](p) of [E] at p as the image of [E] under the natural projection Ap,Φ⊗ SingFk(A(z0)) → Ap,Φ/mp ⊗ SingFk(A(z0)) = SingFk(A(z0)). Then the value [E](p) is nonzero. The vector [E](p) is the only (up to proportionality) common eigenvector of the operators K̃j(z 0)\|Yp : Yp → Yp, j ∈ J , and we have K̃j(z 0)[E](p) = aj fj(z0, p) [E](p). Denote by AYp the unital subalgebra of End(Yp) generated by K̃j(z 0)\|Yp , j ∈ J . Corollary 8.19. Let Assumption 7.4 be satisfied. Then (i) The map [aj/fj(z 0, ·)] 7→ K̃j(z 0)\|Yp, j ∈ J , extends uniquely to an algebra isomorphism βp : Ap,Φ → AYp. (ii) The maps αp, βp give an isomorphism of the regular representation of Ap,Φ and the AYp- module Yp, that is βp(h)αp(g) = αp(hg) for any h, g ∈ Ap,Φ. (iii) The algebra AYp is a maximal commutative subalgebra of End(Yp). Recall that under Assumption 7.4 all critical points of Φ(z0, · ) are isolated, the sum of their Milnor numbers equals dim SingFk(A(z0)) and the form S(a)\|SingFk(A(z0)) is nondegenerate. Theorem 8.20. Let Assumption 7.4 be satisfied. Let p1, . . . , pd be a list of all distinct critical points of Φ(z0, · ). Let Yps = αps(Aps,Φ) ⊂ SingFk(A(z0)), s = 1, . . . , d, be the corresponding subspaces. Then the sum of these subspaces is direct, orthogonal and equals SingFk(A(z0)). Denote by AA(z0),a the unital subalgebra of End(SingFk(A(z0))) generated by K̃j(z 0), j ∈ J . The algebra AA(z0),a is called the algebra of geometric Hamiltonian of the arrangement (A(z0), a), see Section 7.3. Consider the isomorphisms α = ⊕ds=1αps : ⊕ds=1Aps,Φ → SingFk ( A ( z0 )) = ⊕ds=1Yps , β = ⊕ds=1βs : ⊕ds=1Aps,Φ → ⊕ds=1AYps . Corollary 8.21. Let Assumption 7.4 be satisfied. Then (i) AA(z0),a = ⊕ds=1AYps . (ii) AA(z0),a is a maximal commutative subalgebra of End(SingFk(A(z0))). (iii) The isomorphisms α, β identify the regular representation of the algebra ⊕ds=1Aps,Φ and the AA(z0),a-module SingFk(A(z0)). The isomorphism α identifies the contravariant form on SingFk(A(z0)) and the residue form ( , ) = ⊕ds=1( , )ps on ⊕ds=1Aps,Φ multiplied by (−1)k. 9 More on Hamiltonians of bad fibers 9.1 An abstract setting Let k < n be positive integers and J = {1, . . . , n} as before. Let B ⊂ Ck be a ball with center at a point p. Let Fu be a holomorphic function on B depen- ding holomorphically on a complex parameter u at u = 0. Assume that F0 = Fu=0 has a single 34 A. Varchenko critical point at p with Milnor number µ. Let C(t)B be the algebra of holomorphic functions on B, Iu ⊂ C(t)B the ideal generated by ∂Fu/∂ti, i = 1, . . . , k, and Au = C(t)B/Iu. Assume that dimCAu does not depend on u for u in a neighborhood of 0. Let [ ]u : C(t)B → Au be the cano- nical projection, ( , )u the residue bilinear form on Au and HessFu = det16l,m6k(∂ 2Fu/∂tl∂tm). Let h1, . . . , hµ ∈ C(t)B be a collection of elements such that for any u the elements [h1]u, . . . , [hµ]u form a C-basis of Au. Let g1,u, . . . , gn,u ∈ C(t)B be elements depending on u holomorphically at u = 0 and such that for any u (close to 0) the unital subalgebra of Au generated by [g1,u]u, . . . , [gn,u]u equals Au. Let W be a vector space with a symmetric bilinear form S. For u 6= 0, let Mj,u : W → W , j ∈ J , be a collection of commuting symmetric linear operators, Mi,uMj,u = Mj,uMi,u, S(Mj,ux, y) = S(x,Mj,uy) for all i, j ∈ J and x, y ∈W. We assume that every Mj,u depends on u meromorphically (for u close to 0) and has at most simple pole at u = 0, Mj,u = M (−1) j u +M (0) j +M (1) j;1 u+ · · · , M (i) j ∈ End(W ). (9.1) Let w1,u, . . . , wµ,u ∈W be a collection of vectors depending on u holomorphically at u = 0. Consider the element [w]u = µ∑ l=1 [hl]u ⊗ wl,u ∈ Au ⊗W. Assume that for every nonzero u (close to 0) we have µ∑ l=1 [hl]u ⊗Mj,uwl,u = µ∑ l=1 [gj,u]u[hl]u ⊗ wl,u, j ∈ J, (9.2) and for every u (close to 0) we have µp∑ l,m=1 S(wl,u, wm,u)[hl]u[hm]u = (−1)k[HessFu]u. For any u, denote by Yu ⊂ W the vector subspace generated by w1,u, . . . , wµ,u. By proper- ty (9.2), for any nonzero u (close to 0) every Mj,u, j ∈ J , preserves Yu. For any nonzero u (close to 0) denote by AYu the unital subalgebra of End(Yu) generated by Mj,u\|Yu , j ∈ J . The subspace Yu is an AYu-module. For any u (u = 0 included) define a linear map αu : Au → Yu, [f ]u 7→ µ∑ l=1 ([f ]u, [hl]u)uwl,u. Theorem 9.1. For any u (in particular, for u = 0) the map αu : Au → Yu is an isomorphism of vector spaces. The form S restricted to Yu is nondegenerate. Proof. Define a bilinear form ( , )S,u on Au, ([f ]u, [g]u)S,u = S(αu([f ]u), αu([g]u)) = µ∑ l,m=1 S(wl,u, wm,l)([f ]u, [hl]u)u([g]u, [hm]u)u. Quantum Integrable Model of an Arrangement of Hyperplanes 35 Lemma 9.2. For any u, we have ([f ]u[g]u, [h]u)S,u=([f ]u, [g]u[h]u)S,u for all [f ]u, [g]u, [h]u ∈ Au. Proof. For u 6= 0, the statement follows from Lemma 8.3. For u = 0, the statement follows by continuity. � The next two lemmas are similar to the corresponding analogs in Section 8.2. Lemma 9.3. There exists a unique element [s]0 ∈ A0 such that ([f ]0, [g]0)S,0 = ([s]0[f ]0, [g]0)0 for all [f ]0, [g]0 ∈ A0. Lemma 9.4. Let [s]0 ∈ A0 be the element defined in Lemma 9.3. Then [s]0 has the following two properties: (i) the element [s]0 is invertible and projects to (−1)k in C = A0/m0, where m0 ⊂ A0 is the maximal ideal, (ii) the form ( , )S,0 is nondegenerate. Lemma 9.4 implies Theorem 9.1, cf. the end of Section 8.2. � Define the value [w]0(p) of [w]0 at p as the image of [w]0 under the natural projection A0 ⊗W → A0/m0 ⊗W = W . Corollary 9.5. The value [w]0(p) is nonzero. Corollary 9.6. The space Y0 is of dimension µ and w1,0, . . . , wl,0 is its basis. For any [g]0 ∈ A0, denote by L[g]0 ∈ End(A0) the linear operator on A0 of multiplication by [g]0. For any j ∈ J , define a linear map L̄j,0 : Y0 → Y0 by the formula L̄j,0 = α0L[gj,0]0(α0)−1. Denote by AY0 the unital subalgebra of End(Y0) generated by L̄j,0, j ∈ J . Clearly, AY0 is commutative. The subspace Y0 is an AY0-module. Theorem 9.7. (i) The map [gj,0]0 7→ L̄j,0, j ∈ J , extends uniquely to an algebra isomorphism β0 : A0 → AY0. (ii) the algebra AYu tends to the algebra AY0 as u → 0. More precisely, for any j ∈ J and l = 1, . . . , µ, we have Mj,uwl,u → L̄j,0wl,0 as u→ 0. Proof. Part (i) is clear. Part (ii) follows from (9.2). � Let Ỹ ⊂W be a vector subspace such that (a) Y0 ⊂ Ỹ ; (b) the bilinear form S restricted on Ỹ is nondegenerate; (c) for any j ∈ J , the subspace Ỹ lies in the kernel of M (−1) j . For example, we can choose Ỹ = Y0. Let prỸ : W → Ỹ be the orthogonal projection. Theorem 9.8. (i) For j ∈ J , let M (0) j be the constant coefficient of the Laurent expansion of Mj,u, see (9.1). Then L̄j,0 = prỸM (0) j \|Y0 . (9.3) In particular, that means that the operators prỸM (0) j \|Y0 do not depend on the choice of Ỹ . 36 A. Varchenko (ii) Let (ξj)j∈J ⊂ C be numbers such that ∑ j∈J ξjM (−1) j = 0, then∑ j∈J ξjL̄j,0 = ∑ j∈J ξjM (0) j \|Y0 . Proof. For j ∈ J , let wl,u = wl,0 +w (1) l u+ · · · be the Taylor expansion of wl,u. By part (ii) of Theorem 9.7 we have wj,0 ∈ kerM (−1) j and L̄j,0wl,0 = M (0) j wl,0 +M (−1) j w (1) l,0 . (9.4) The operator M (−1) j is symmetric since Mj,u is symmetric. The image of M (−1) j is orthogonal to the kernel of M (−1) j . Hence, formula (9.3) follows from formula (9.4). Part (ii) of the theorem also follows from formula (9.4). � Corollary 9.9. For any i, j ∈ J , the operators prỸM (0) i \|Y0, prỸM (0) j \|Y0 are symmetric and commute. Corollary 9.10. The unital subalgebra of End(Y0) generated by the operators prỸM (0) j \|Y0, j ∈ J , is a maximal commutative subalgebra. Proof. The proof follows from remarks in Section 8.3. � Corollary 9.11. The vector [w]0(p) is the only (up to proportionality) common eigenvector of the operators prỸM (0) j \|Y0, j ∈ J , and we have prỸM (0) j [w]0(p) = gj,0(p)[w]0(p). Assume that the parameter u is changed holomorphically, u = c1v+ c2v 2 + · · · where ci ∈ C, c1 6= 0, and v is a new parameter. Let Mj,u(v) = M̃ (−1) j v + M̃ (0) j + M̃ (1) j;1 v + · · · , M̃ (i) j ∈ End(W ), be the new Laurent expansion. Lemma 9.12. For any j ∈ J , we have prỸM (0) i \|Y0 = prỸ M̃ (0) i \|Y0 and the algebra AY0 ⊂ End(Y0) does not change under the reparametrization of u. Proof. One proof of the lemma follows from Theorem 9.7. Another proof follows from the fact that M̃ (−1) i = M (−1) i /c1, M̃ (0) i = M (0) i − c2M (−1) i /c2 1. � 9.2 Hamiltonians of bad fibers Let us return to the situation of Sections 3 and 7 and recall the previous constructions. Let z0 ∈ ∆. Let V → V ∗ be the map associated with the contravariant form. Let W , SingW , W (z0), SingW (z0) be the images of V , Sing V , Fk(A(z0)), SingFk(A(z0)), respectively. The contravariant form on V induces a nondegenerate symmetric bilinear form on W also denoted by S(a). For z ∈ Cn−∆, we have linear operators Kj(z) : V → V , j ∈ J , where Kj(z) = ∑ C∈C λCj fC(z)LC , see (5.3). For C0 = {C ∈ C \| z0 ∈ HC}, we define K0 j (z) = ∑ C∈C0 λCj fC(z) LC , K1 j (z) = Kj(z)−K0 j (z). Quantum Integrable Model of an Arrangement of Hyperplanes 37 The dual operators Kj(z) ∗ : V ∗ → V ∗ preserve the subspaces SingW ⊂ W ⊂ V ∗, commute on the subspace SingW and are symmetric on W with respect to the contravariant form. The operators L∗C : V ∗ → V ∗, C ∈ C, preserve the subspaces SingW ⊂ W ⊂ V ∗. The space W (z0) lies in the kernel of L∗C \|W : W →W for any C ∈ C0. Let Tz0 = {ξ = ∑ j∈J ξj ∂ ∂zj \| ξj ∈ C, ξ(fC) = 0 for all C ∈ C0}. Let p ∈ U(A(z0)) be an isolated critical point of the master function Φ(z0, · ) : U(A(z0))→ C. Let Ap,Φ be the local algebra of the critical point and ( , )p the residue bilinear form on Ap,Φ. Let [ ] : C(t)U(A(z0)) → Ap,Φ be the canonical projection and [E] ∈ Ap,Φ ⊗ SingW (z0) the element corresponding to the canonical element. Define a linear map αp : Ap,Φ → SingW ( z0 ) , g 7→ (g, [E])p. Denote Yp the image of αp. Theorem 9.13. (i) We have ker αp = 0. The isomorphism αp identifies the contravariant form on Yp and the residue form ( , )p on Ap,Φ multiplied by (−1)k. In particular, the contravariant form on Yp is nondegenerate. (ii) Let Ỹ ⊂ SingW be a vector subspace such that (a) Yp ⊂ Ỹ ; (b) the contravariant form restricted on Ỹ is nondegenerate; (c) for any j ∈ J , the subspace Ỹ lies in the kernel of L∗C , C ∈ C0. Let prỸ : SingW → Ỹ be the orthogonal projection. Then for any j ∈ J the operator prỸK 1 j (z0)∗\|Yp maps Yp to Yp and does not depend on the choice of Ỹ . The operators prỸK 1 j (z0)∗\|Yp : Yp → Yp, j ∈ J , commute and are symmetric with respect to the con- travariant form on Yp. (iii) The unital subalgebra AYp ⊂ End(Yp) generated by prỸK 1 j (z0)∗\|Yp, j ∈ J , is a maximal commutative subalgebra. (iv) The naive geometric Hamiltonians Kξ(z 0)∗, ξ ∈ Tz0, preserve the subspace Yp and the operators Kξ(z 0)∗\|Yp are elements of the subalgebra AYp. (v) The value [E](p) of [E] at p is nonzero. The vector [E](p) is the only (up to proportionality) common eigenvector of the operators prỸK 1 j (z0)∗\|Yp, j ∈ J , and we have prỸK 1 j ( z0 )∗ [E](p) = aj fj(z0, p) [E](p). (vi) For any j ∈ J , g ∈ Ap,Φ, we have αp(gaj/[fj(z 0, ·)]) = prỸK 1 j (z0)∗αp(g). The map [aj/fj(z 0, )] 7→ prỸK 1 j (z0)∗\|Yp, j ∈ J , extends uniquely to an algebra isomorphism βp : Ap,Φ → AYp. (vii) The isomorphisms αp, βp identify the regular representation of the algebra Ap,Φ and the AYp-module Yp. All statements of the theorem (but the second statement of part (i)) follow from the cor- responding statements of Section 9.1. The second statement of part (i) has the same proof as Theorem 8.11. 38 A. Varchenko Theorem 9.14. Let p1, . . . , pd be a list of all distinct isolated critical points of Φ(z0, · ). Let Yps = αps(Aps,Φ) ⊂ SingW (z0), s = 1, . . . , d, be the corresponding subspaces. Then the sum of these subspaces is direct and orthogonal with respect to the contravariant form. Corollary 9.15. The sum of Milnor numbers of the critical points p1, . . . , pd is not greater than the rank of the contravariant form S(a)\|SingFk(A(z0)). Denote Y = ⊕ds=1Yps . Let Ỹ ⊂ SingW be a vector subspace such that (a) Y ⊂ Ỹ ; (b) the contravariant form restricted on Ỹ is nondegenerate; (c) for any j ∈ J , the subspace Ỹ lies in the kernel of L∗C , C ∈ C0. For example, we can choose Ỹ = Y . Let prỸ : SingW → Ỹ be the orthogonal projection. Theorem 9.16. (i) For any j ∈ J the operator prỸK 1 j (z0)∗ maps Y to Y and does not depend on the choice of Ỹ . The operators prỸK 1 j (z0)∗\|Y : Y → Y , j ∈ J , commute, preserve each of the subspaces Yps and are symmetric with respect to the contravariant form on Y . Denote by AY be the unital subalgebra of End(Y ) generated by prỸK 1 j (z0)∗\|Y , j ∈ J . (ii) The naive geometric Hamiltonians Kξ(z 0)∗, ξ ∈ Tz0, preserve the subspace Y and the operators Kξ(z 0)∗\|Y are elements of the subalgebra AY . (iii) Consider the isomorphisms α = ⊕ds=1αps : ⊕ds=1Aps,Φ → ⊕ds=1Yps , β = ⊕ds=1βs : ⊕ds=1Aps,Φ → ⊕ds=1AYps . Then (a) AY = ⊕ds=1AYps ; (b) AY is a maximal commutative subalgebra of End(Y ); (c) the isomorphisms α, β identify the regular representation of the algebra ⊕ds=1Aps,Φ and the AY -module Y ; the isomorphism α identifies the contravariant form on Y and the residue form ( , ) = ⊕ds=1( , )ps on ⊕ds=1Aps,Φ multiplied by (−1)k; (d) in particular, if the rank of the contravariant form S(a)\|SingFk(z0) equals the sum of Milnor numbers of the points p1, . . . , pd, then Y = SingW (z0) and the module SingW (z0) over the unital subalgebra of End(SingW (z0)) generated by geometric Hamiltonians prỸK 1 j (z0)∗\|SingW (z0), j ∈ J , is isomorphic to the regular representa- tion of the algebra ⊕ds=1Aps,Φ; The theorem follows from the corresponding statements of Section 9.1. Theorem 9.17. Assume that the arrangement (A(z0), a) is unbalanced. (i) Then the contravariant form is nondegenerate on SingW (z0). (ii) Let prSingW (z0) : SingW → SingW (z0) be the orthogonal projection, ASingW (z0) be the uni- tal subalgebra of End(SingW (z0)) generated by the operators prSingW (z0)K 1 j (z0)∗\|SingW (z0), j ∈ J . Then ASingW (z0) is commutative and its elements are symmetric with respect to the contravariant form on SingW (z0). Quantum Integrable Model of an Arrangement of Hyperplanes 39 (iii) Let p1, . . ., pd be a list of all distinct isolated critical points of Φ(z0, · ). Then the ASingW (z0)- module SingW (z0) is isomorphic to the regular representation of the algebra ⊕ds=1Aps,Φ. Proof. If (A(z0), a) is unbalanced, then the sum of Milnor numbers of the master function equals \|χ(U(A(z0)))\| and equals dim SingFk(A(z0)). This implies part (i) of the theorem. Parts (ii) and (iii) follow from Theorem 9.16. � 9.3 Remark on critical points of real arrangements Assume that (gj)j∈J are real, see Remark 3.2. Assume that z0 ∈ Rn ⊂ Cn. Assume that the contravariant form is positive definite on SingW (z0). Let prSingW (z0) : SingW → SingW (z0) be the orthogonal projection. Assume that the operators prSingW (z0)K 1 j ( z0 )∗\|SingW (z0) : SingW ( z0 ) → SingW ( z0 ) , j ∈ J, commute and are symmetric with respect to the contravariant form. Theorem 9.18. Under these assumptions, any critical point p of the master function Φ(z0, ·) : U(A(z0)) → U(A(z0)), is nondegenerate and the nonzero value [E](p) at p of the canonical element lies in the real part SingW (z0)R of SingW (z0) (up to multiplication by a nonzero complex number). Proof. On one hand, under assumptions of the theorem all the linear operators preserve SingW (z0)R and can be diagonalized simultaneously. That means that any element of the alge- bra of geometric Hamiltonians ASingW (z0) (generated by operators prSingW (z0)K 1 j (z0)∗\|SingW (z0), j ∈ J) is diagonalizable. On the other hand, if the Milnor number of p is greater than one, then the local algebra Ap,Φ has nilpotent elements and, by Theorem 9.13, the algebra ASingW (z0) has nondiagonalizable elements. The second part of the theorem is clear. � Theorem 9.18 is in the spirit of the main theorem of [12] and Conjecture 5.1 in [11]. The main theorem of [12] says that certain Schubert cycles intersect transversally and all intersection point are real. These two statements correspond to the two statements of Theorem 9.18. 10 Arrangements with symmetries 10.1 A family of prediscriminantal arrangements In this section we consider a special family of parallelly translated hyperplanes, see Section 3. The members of that family will be called prediscriminantal arrangements. Data 10.1. Let h∗ be a complex vector space of dimension r with a collection of vectors α1, . . . , αr,Λ1, . . . ,ΛN ∈ h∗ and a symmetric bilinear form ( , ). We assume that (αi, αi) 6= 0 for every i = 1, . . . , r. Let k = (k1, . . . , kr) be a collection of nonnegative integers. We denote k = ∑ i ki and assume that k > 0. We assume that for every b = 1, . . . , N there exists i such that (αi,Λb) 6= 0 and ki > 0. Consider the expressions f(i),l,l′ = t (i) l − t (i) l′ + z(i),l,l′ such that i = 1, . . . , r and 1 6 l < l′ 6 ki; (10.1) f(i,i′),l,l′ = t (i) l − t (i′) l′ + z(i,i′),l,l′ such that 1 6 i < i′ 6 r, 1 6 l 6 ki, 1 6 l ′ 6 ki′ and (αi, αi′) 6= 0; f(i,b),l = −t(i)l + z(i,b),l such that 1 6 i 6 r, 1 6 l 6 ki, 1 6 b 6 N and (αi,Λb) 6= 0. 40 A. Varchenko Let J denote the set of all low indices of the letters f in these expressions. So J is the union of three nonintersecting subsets J1, J2, J3 where J1 consists of triples {(i), l, l′} from the first line of (10.1), J2 consists of four-tuples {(i, i′), l, l′} from the second line, J3 consists of triples {(i, b), l} from the third line. Let n be the number of elements in J . Consider Ck with coordinates t = ( t (1) 1 , . . . , t (1) k1 , . . . , t (r) 1 , . . . , t (r) kr ) . Consider Cn with coordinates z = (zj)j∈J and Cn × Ck with coordinates z, t. For any j ∈ J the expression fj can be considered as a linear function on Cn × Ck. We have fj = zj + gj where gj = t (i) l − t (i) l′ if j = {(i), l, l′}, where gj = t (i) l − t (i′) l′ if j = {(i, i′), l, l′} and gj = −t(i)l if j = {(i, b), l}. The functions gj , j ∈ J , can be considered as linear functions on Ck. For j ∈ J , the equation fj(z, t) = 0 defines a hyperplane H̃j ⊂ Cn × Ck and we get an arrangement Ã = {H̃j \| j ∈ J} in Cn × Ck. We assign (nonzero) weights aj to hyperplanes of C̃ by putting a(i),l,l′ = (αi, αi), a(i,i′),l,l′ = (αi, αi′), a(i,b),l = −(αi,Λb). (10.2) The weighted arrangement C̃ is an example of a family of parallelly translated hyperplanes considered in Sections 3–9. 10.2 Discriminantal arrangements Let X ⊂ Cn be the subset defined by the following equations: z(i),l,l′ = 0, i = 1, . . . , r and 1 6 l < l′ 6 ki; z(i,i′),l,l′ = 0, 1 6 i < i′ 6 r, 1 6 l 6 ki, 1 6 l′ 6 ki′ and (αi, αi′) 6= 0; z(i,b),l = z(i′,b),l′ , 1 6 i 6 r, 1 6 l 6 ki, 1 6 i′ 6 r, 1 6 l′ 6 ki′ , 1 6 b 6 N, (αi,Λb) 6= 0, (αi′ ,Λb) 6= 0. The subset X is an N -dimensional affine space. We will use the following coordinates x1, . . . , xN on X defined by the equations xb = z(i,b),l, where 1 6 b 6 N , 1 6 i 6 r, 1 6 l 6 ki and (αi,Λb) 6= 0. We will be interested in the open subset U(X) ⊂ X, U(X) = {z ∈ X \| x1(z), . . . , xN (z) are all distinct}. Let us consider the arrangement A(z) = (Hj(z))j∈J in the fiber Ck of the projection Cn×Ck → Cn over a point z ∈ U(X) with coordinates x1(z), . . . , xN (z). Its hyperplanes are defined by the equations: t (i) l − t (i) l′ = 0, i = 1, . . . , r and 1 6 l < l′ 6 ki; t (i) l − t (i′) l′ = 0, 1 6 i < i′ 6 r, 1 6 l 6 ki, 1 6 l′ 6 ki′ and (αi, αi′) 6= 0; t (i) l − xb(z) = 0, 1 6 i 6 r, 1 6 l 6 ki, 1 6 b 6 N and (αi,Λb) 6= 0. The weights of these hyperplanes are defined by formula (10.2). This weighted arrangement is called discriminantal, see [23, 25]. Quantum Integrable Model of an Arrangement of Hyperplanes 41 10.3 Symmetries of the family of prediscriminantal arrangements The product of symmetric groups Sk = Sk1 × · · ·×Skr acts on Ck by permuting coordinates t (i) l with the same upper index. More precisely, a point p ∈ Ck with coordinates (t (1) 1 (p), . . . , t (1) k1 (p), . . . , t (r) 1 (p), . . . , t (r) kr (p)) is mapped by an element σ = (σ1, . . . , σr) to the point with coordinates (t (1) σ−1 a (1) (p), . . . , t (1) σ−1 1 (k1) (p), . . . , t (r) σ−1 r (1) (p), . . . , t (r) σ−1 r (kr) (p)). The group Sk acts also on Cn. Namely, an element σ = (σ1, . . . , σr) sends a coordina- te z(i),l,l′ to the coordinate z(i),σi(l),σi(l′) if σi(l) < σi(l ′) and to −z(i),σi(l),σi(l′) if σi(l) > σi(l ′). An element σ sends a coordinate z(i,i′),l,l′ to z(i,i′),σi(l),σi′ (l ′). An element σ sends a coordinate z(i,b),l to the coordinate z(i,b),σi(l). Clearly every point of X ⊂ Cn is a fixed point of the Sk-action. The actions of Sk on Cn and Ck induce an action on Cn × Ck. The Sk-action on Cn × Ck preserves the arrangement Ã and sends fibers of the projection Cn × Ck → Cn to fibers. The Sk-action on Ã corresponds to an Sk-action on the set J = J1 ∪ J2 ∪ J3. That action preserves the summands. For σ = (σ1, . . . , σr) ∈ Sk we have {(i), l, l′} 7→ {(i),min(σi(l), σi(l ′)),max(σi(l), σi(l ′))}, {(i, i′), l, l′} 7→ {(i, i′), σi(l), σi′(l′)}, {(i, b), l} 7→ {(i, b), σi(l)}, where {(i), l, l′} ∈ J1, {(i, i′), l, l′} ∈ J2, {(i, b), l} ∈ J3. Consider the discriminant ∆ = ∪C∈CHC ⊂ Cn, see Section 3.2. Here C is the set of all circuits of the matroid of the collection (gj)j∈J . Clearly the Sk-action on Cn preserves the discriminant and permutes the hyperplanes {HC \| C ∈ C}. The action of Sk on the hyperplanes of the discriminant corresponds to the following action on C. If C = {j1, . . . , jl} ⊂ J is a circuit and σ ∈ Sk, then σ(C) is the circuit {σ(j1), . . . , σ(jl)}. 10.4 The Sk-action on geometric Hamiltonians We use notations of Section 3.3 and for z∈Cn−∆ denote V = Fk(A(z)), Sing V = SingFk(A(z)). The triple (V,Sing V, S(a)) does not depend on z ∈ Cn −∆ as explained in Section 3.3. Fix an order on the set J . Recall that the standard basis of V ∗ = Ak(A(z)), associated with an order on J , is formed by elements (Hj1 , . . . ,Hjk) where {j1 < · · · < jk} runs through the set of all independent ordered k-element subsets of J . The (dual) standard basis of V is formed by the corresponding vectors F (Hj1(z), . . . ,Hjk(z)). We have F (Hj1(z), . . . ,Hjk(z)) = (−1)\|µ\|F (Hjµ(1)(z), . . . ,Hjµ(k)(z)) for any µ ∈ Sk, see Section 2.11. The Sk-action on J induces an action on V and V ∗. For σ ∈ Sk, we have σ : F (Hj1(z), . . . ,Hjk(z)) 7→ F (Hσ(j1)(z), . . . ,Hσ(jk)(z)) and (Hj1(z), . . . ,Hjk(z)) 7→ (Hσ(j1)(z), . . . ,Hσ(jk)(z)). The Sk-action on V preserves the sub- space Sing V and preserves the contravariant form S(a) on V . Let z0 ∈ U(X). Then z0 is Sk-invariant and the group Sk acts on the fiber Ck over z0 and on the weighted arrangement (A(z0), a) in that fiber. The subspaces Fk(A(z0)), SingFk(A(z0)) of V are Sk-invariant. Let V → V ∗ be the map associated with the contravariant form. Let W , SingW , W (z0), SingW (z0) be the images of V , Sing V , Fk(A(z0)), SingFk(A(z0)), respectively. All these subspaces are Sk-invariant. An Sk-action on a vector space defines an Sk-action on linear operators on that space. For σ ∈ Sk and a linear operator L we define σ(L) = σLσ−1. In Section 5.2 we have defined operators LC : V → V , C ∈ C. Clearly for any σ ∈ Sk and any C ∈ C we have σ(LC) = Lσ(C). 42 A. Varchenko In Section 5.2 we have considered differential 1-forms on Cn×Ck which were denoted by ωj , j ∈ J , and ωC , C ∈ C. The Sk-action on Cn × Ck preserves this set of differential 1-forms. Namely for any j ∈ J , C ∈ C, σ ∈ Sk, we have σ : ωj 7→ ωσ(j), ωC 7→ ωσ(C). Lemma 10.2. The following objects are Sk-invariant:∑ j∈J ajωj , ∑ C∈C ωC ⊗ LC , ∑ independent {j1<···<jk}⊂J ωj1 ∧ · · · ∧ ωjk ⊗ F (Hj1 , . . . ,Hjk). By the definition of Kj(z) : V → V , j ∈ J , we have∑ C∈C ωC ⊗ LC = ∑ j∈J dzj ⊗Kj(z), see formula (5.4). The functions Kj(z) are End(V )-valued meromorphic functions on Cn. Since Sk acts on Cn and End(V ) it also acts on End(V )-valued functions on Cn, σ : F (q) 7→ σF (σ−1(q))σ−1 for q ∈ Cn. Lemma 10.2 allows us to describe the Sk-action on functions Kj(z). Corollary 10.3. An element σ = (σ1, . . . , σr) ∈ Sk acts on functions Kj, j ∈ J , by the formulas: K(i),l,l′(q) 7→ K(i),σi(l),σi(l′) ( σ−1(q) ) , if σi(l) < σi(l ′), K(i),l,l′(q) 7→ −K(i),σi(l′),σi(l) ( σ−1(q) ) , if σi(l) > σi(l ′), K(i,i′),l,l′(q) 7→ K(i),σi(l),σi′ (l ′) ( σ−1(q) ) , K(i,b),l(q) 7→ K(i,b),σi(l) ( σ−1(q) ) . Let z0 ∈ U(X). Recall that C0 = {C ∈ C \| z0 ∈ HC}, K0 j (z) = ∑ C∈C0 λCj fC(z) LC , K1 j (z) = Kj(z)−K0 j (z). Corollary 10.4. An element σ ∈ Sk acts on operators K1 j (z0) : V → V , j ∈ J , by the formula σ ( K1 j ( z0 )) = ±K1 σ(j) ( z0 ) , where the minus sign is chosen only if j = {(i), l, l′} and σi(l) > σ(l′). 10.5 Functions K∂xb (z) Let z0 ∈ U(X). Recall that Tz0 = {ξ = ∑ j∈J ξj ∂ ∂zj \| ξj ∈ C, ξ(fC) = 0 for all C ∈ C0}, see Section 7.1. Define the following constant vector fields on Cn, ∂xb = ∑ 16i6r, 16l6ki, (αi,Λb)6=0 ∂ ∂z(i,b),l , b = 1, . . . , N. Lemma 10.5. The vector fields ∂xb, b = 1, . . . , N , are elements of Tz0. The End(V )-valued functions K∂xb (z) = ∑ 16i6r, 16l6ki, (αi,Λb)6=0 K(i,b),l(z), b = 1, . . . , N, are Sk-invariant. Quantum Integrable Model of an Arrangement of Hyperplanes 43 By Lemma 7.1, the functions K∂yb (z) are regular at z0 and their dual operators preserve the subspace SingW (z0) ⊂ V ∗. Corollary 10.6. For z0 ∈ U(X), the operators K∂xb ( z0 )∗\|SingW (z0) : SingW ( z0 ) → SingW ( z0 ) , b = 1, . . . , N, are Sk-invariant. The operators K∂xb (z0)∗\|SingW (z0) are naive geometric Hamiltonians on SingW (z0) in the sense of Section 7.1. They commute and they are symmetric operators with respect to the contravariant form. 10.6 Naive geometric Hamiltonians on Sing W−(z0) The space W has the canonical direct sum decomposition into isotypical components correspon- ding to irreducible representations of Sk. One of the isotypical components is the component W− = { x ∈W \|σ(x) = (−1)\|σ\|x for any σ ∈ Sk } , corresponding to the alternating representation. If L : W → W is an Sk-invariant linear operator, then L preserves the canonical decomposition and, in particular, it preserves the subspace W−. Let z0 ∈ U(X). Then the subspace SingW (z0) ⊂ W is an Sk-submodule. We define SingW−(z0) = W− ∩ SingW (z0). For any b = 1, . . . , N , the operator K∂xb (z0)∗ preserves SingW−(z0). The operators K∂xb ( z0 )∗\|SingW−(z0) : SingW− ( z0 ) → SingW− ( z0 ) will be called naive geometric Hamiltonians on SingW−(z0). 10.7 Sk-symmetries of the canonical element Let z0 ∈ U(X). Let p ∈ U(A(z0)) be an isolated critical point of the master function Φ(z0, · ) : U(A(z0)) → C. Let O(p) = {σ(p) \| σ ∈ Sk} be the Sk-orbit of p. The orbit consists of k1! · · · kr! points. Let Aσ(p),Φ be the local algebra of the critical point σ(p) and ( , )σ(p) the residue bilinear form on Aσ(p),Φ. Let [ ]σ(p) : C(t)U(A(z0)) → Aσ(p),Φ be the canonical projection and [E]σ(p) ∈ Aσ(p),Φ ⊗ SingW (z0) the projection of the the canonical element. The group Sk acts on functions on U(A(z0)). If g ∈ C(t)U(A(z0)) and σ ∈ Sk, then σ(g)(q) = g(σ−1(q)) for q ∈ U(A(z0)). Lemma 10.7. An element σ ∈ Sk acts on functions fj, j ∈ J , by the formula σ(fj) = ±fσ(j), where the minus sign is chosen only if j = {(i), l, l′} and σi(l) > σ(l′). The Sk-action on C(t)U(A(z0)) induces an isomorphism σ : Ap,Φ → Aσ(p),Φ, [g]p 7→ [σ(g)]σ(p). Let us compare the residue bilinear forms on Ap,Φ and Aσ(p),Φ and projections of the canonical element to Ap,Φ and Aσ(p),Φ. Lemma 10.8. For f, g ∈ Ap,Φ, we have (σ(f), σ(g))σ(p) = (−1)σ(f, g)p and σ([E]p) = (−1)σ[E]σ(p), where (−1)σ = r∏ i=1 (−1)σi. 44 A. Varchenko For σ ∈ Sk, let ασ(p) : Aσ(p),Φ → SingW (z0) be the linear monomorphisms constructed in Section 9.2. Let Yσ(p) be the image of ασ(p). If µ is the Milnor number of p, then dimYσ(p) = µ. Lemma 10.9. For f ∈ Ap,Φ, we have σ(αp(f)) = ασ(p)(σ(f)). Corollary 10.10. We have σ(Yp) = Yσ(p). By Theorem 9.14, the subspaces Yσ(p) are all orthogonal and the contravariant form on YO(p) = ⊕σ∈Sk Yσ(p) is nondegenerate. The group Sk acts on YO(p). Denote Y −O(p) = { v ∈ YO(p) \| σ(v) = (−1)σv for all σ ∈ Sk } . Corollary 10.11. We have Y −O(p) = { ∑ σ∈Sk (−1)σσ(v) \| v ∈ Yp} and dimY −O(p) = µ. Let Ant = ∑ σ∈Sk (−1)σσ : SingW (z0)→ SingW−(z0) be the anti-symmetrization operator. Denote [E]−p = (1⊗Ant)[E]p. Define a linear monomorphism α−p : Ap,Φ → SingW− ( z0 ) , f → (f, [E]−p )p . Lemma 10.12. We have Y −O(p) = {α−p (f) \| f ∈ Ap,Φ}. The linear map α−p identifies the contravariant form on Y −O(p) and the residue bilinear form on Ap,Φ multiplied by (−1)kk1! · · · kr!. In particular, the contravariant form on Y −O(p) is nonde- generate. We also have S(a)([E])−p , [E]−p ) = (−1)kk1! · · · kr! [ Hess(a)Φ ( z0, · )] p . (10.3) Let Ỹ ⊂ SingW be a vector subspace such that (a) YO(p) ⊂ Ỹ ; (b) the contravariant form restricted on Ỹ is nondegenerate; (c) for any j ∈ J , the subspace Ỹ lies in the kernel of L∗C , C ∈ C0; (d) the subspace Ỹ is Sk-invariant. For example, we can choose Ỹ = YO(p). Let prỸ : SingW → Ỹ be the orthogonal projection with respect to the contravariant form. For every j ∈ J and σ ∈ Sk, the map prỸK 1 j (z0)∗ : SingW → SingW preserves Yσ(p) and prỸK 1 j (z0)∗\|Ỹ does not depend on the choice of Ỹ , see Theorem 9.16. We also have σ : prỸK 1 j (z0)∗ 7→ ±prỸK 1 σ(j)(z 0)∗ where the minus sign is chosen only if j = {(i), l, l′} and σi(l) > σ(l′). Important definition. Denote by PSk the algebra of polynomials with complex coefficients in n variables aj/fj , j ∈ J , such that for any F (aj/fj , j ∈ J) ∈ PSk the function F (aj/fj(z 0, ·), j ∈ J) ∈ U(A(z0)) is Sk-invariant. Lemma 10.13. The natural homomorphism PSk → Ap,Φ is an epimorphism. Quantum Integrable Model of an Arrangement of Hyperplanes 45 Let F (aj/fj(z 0, ·), j ∈ J) ∈ PSk . Replace in F each variable aj/fj with the operator prỸK 1 j (z0)∗. Denote the resulting operator on SingW by F (prỸK 1 j (z0)∗, j ∈ J). This operator preserves Yσ(p) for any σ ∈ Sk and its restriction to Yσ(p) does not depend on the choice of Ỹ . The operator F (prỸK 1 j (z0)∗, j ∈ J) is Sk-invariant by Corollary 10.4. Hence, F (prỸK 1 j (z0)∗, j ∈ J) preserves Y −O(p). Theorem 10.14. (i) For F ∈ PSk, the operators F ( prỸK 1 j ( z0 )∗ , j ∈ J ) \|Y − O(p) : Y −O(p) → Y −O(p) commute and are symmetric with respect to the contravariant form. (ii) The map F 7→ F (prỸK 1 j (z0)∗, j ∈ J)\|Y − O(p) induces an algebra monomorphism β−p : Ap,Φ → End(Y −O(p)). (iii) The image of this monomorphism, denoted by AY − O(p) , is a maximal commutative subalgebra of End(Y −O(p)). (iv) The naive geometric Hamiltonians K∂xb (z0)∗\|Y − O(p) , b = 1, . . . , N , are elements of AY − O(p) . (v) The maps α−p , β−p define an isomorphism of the regular representation of Ap,Φ and the AY − O(p) -module Y −O(p). The linear map α−p identifies the contravariant form on Y −O(p) and the residue bilinear form on Ap,Φ multiplied by (−1)kk1! · · · kr!. (vi) Define the value of [E]−p at p as the image of the natural projection of [E]−p to Ap,Φ/mp ⊗ SingW−(z0) = SingW−(z0). Then the value [E]−p (p) is nonzero and lies in Y −O(p). The vector [E]−p (p) is the only (up to proportionality) common eigenvector of the operators F (prỸK 1 j (z0)∗, j ∈ J)\|Y − O(p) , F ∈ PSk, we have F ( prỸK 1 j ( z0 )∗ , j ∈ J ) [E]−p (p) = F ( aj/f ( z0, p ) , j ∈ J ) [E]−p (p). Theorem 10.15. Let p1, . . . , pd be a list of isolated critical points of Φ(z0, · ) such that the orbits O(p1), . . . , O(pd) do not intersect. Let Y −ps = α−ps(Aps,Φ) ⊂ SingW−(z0), s = 1, . . . , d, be the corresponding subspaces. Then the sum of these subspaces is direct and orthogonal with respect to the contravariant form. Corollary 10.16. The sum of Milnor numbers of the critical points p1, . . . , pd is not greater than the rank of the contravariant form S(a)\|SingW−(z0). Let Ỹ ⊂W be a vector subspace such that (a) ⊕ds=1YO(ps) ⊂ Ỹ ; (b) the contravariant form restricted on Ỹ is nondegenerate; (c) for any j ∈ J , the subspace Ỹ lies in the kernel of L∗C , C ∈ C0; (d) the subspace Ỹ is Sk-invariant. For example, we can choose Ỹ = ⊕ds=1YO(ps). Let prỸ : SingW → Ỹ be the orthogonal projection. Denote Y − = ⊕ds=1Y − O(ps) . 46 A. Varchenko Theorem 10.17. For F ∈ PSk, the operators F ( prỸK 1 j ( z0 )∗ , j ∈ J ) \|Y − : Y − → Y − do not depend on the choice of Ỹ . They commute and are symmetric with respect to the con- travariant form. Denote by AY − the unital subalgebra of End(Y −) generated by F (prỸK 1 j (z0)∗, j ∈ J)\|Y −, F ∈ PSk. The naive geometric Hamiltonians K∂xb (z0)∗\|Y −, b = 1, . . . , N , are elements of the algebra AY −. Consider the isomorphisms α− = ⊕ds=1α − ps : ⊕ds=1Aps,Φ → ⊕ds=1Y − O(ps) , β− = ⊕ds=1β − ps : ⊕ds=1Aps,Φ → ⊕ds=1AY − O(ps) . Then (a) AY − = ⊕ds=1AY − O(ps) ; (b) AY − is a maximal commutative subalgebra of End(Y −); (c) the isomorphisms α−, β− identify the regular representation of the algebra ⊕ds=1Aps,Φ and the AY −-module Y −; the linear map α− identifies the contravariant form on Y − and the residue bilinear form ( , ) = ⊕ds=1( , )ps on ⊕ds=1Aps,Φ multiplied by (−1)kk1! · · · kr!; (d) in particular, if the rank of the contravariant form S(a)\|SingW−(z0) equals the sum of Milnor numbers of the points p1, . . . , pd, then the module SingW−(z0) = Y − over the unital subalgebra of End(SingW−(z0)) generated by geometric Hamiltonians F (prỸK 1 j (z0)∗, j ∈ J)\|Y −, F ∈ PSk, is isomorphic to the regular representation of the algebra ⊕ds=1Aps,Φ. Corollary 10.18. If z0 ∈ U(X) and the arrangement (A(z0), a) is unbalanced, then the opera- tors F (prỸK 1 j (z0)∗, j ∈ J)\|SingW−(z0) : SingW−(z0) → SingW−(z0), F ∈ PSk, commute and are symmetric with respect to the contravariant form. Proof. In this case Y − = SingW−(z0) and the corollary follows from Theorem 10.17. � 11 Applications to the Bethe ansatz of the Gaudin model 11.1 Gaudin model Let g be a simple Lie algebra over C with Cartan matrix (ai,j) r i,j=1. Let h ⊂ g be a Cartan subalgebra. Fix simple roots α1, . . . , αr in h∗ and a nondegenerate g-invariant bilinear form ( , ) on g. The form identifies g and g∗ and defines a bilinear form on g∗. Let H1, . . . ,Hr ∈ h be the corresponding coroots, 〈λ,Hi〉 = 2(λ, αi)/(αi, αi) for λ ∈ h∗. In particular, 〈αj , Hi〉 = ai,j . Let E1, . . . , Er ∈ n+, H1, . . . ,Hr ∈ h, F1, . . . , Fr ∈ n− be the Chevalley generators of g, [Ei, Fj ] = δi,jHi, i, j = 1, . . . , r, [h, h′] = 0, h, h′ ∈ h, [h,Ei] = 〈αi, h〉Ei, h ∈ h, i = 1, . . . , r, [h, Fi] = −〈αi, h〉Fi, h ∈ h, i = 1, . . . , r, and (adEi) 1−ai,jEj = 0, (adFi) 1−ai,jFj = 0, for all i 6= j. Let (Ji)i∈I be an orthonormal basis of g, Ω = ∑ i∈I Ji ⊗ Ji ∈ g⊗ g the Casimir element. Quantum Integrable Model of an Arrangement of Hyperplanes 47 For a g-module V and µ ∈ h∗ denote by V [µ] the weight subspace of V of weight µ and by Sing V [µ] the subspace of singular vectors of weight µ, Sing V [µ] = {v ∈ V \| n+v = 0, hv = 〈µ, h〉v}. Let N > 1 be an integer and Λ = (Λ1, . . . ,ΛN ), Λb ∈ h∗, a set of weights. For µ ∈ h∗ let Vµ be the irreducible g-module with highest weight µ. Denote VΛ = VΛ1 ⊗ · · · ⊗ VΛN . For X ∈ End(VΛi), denote by X(i) = 1⊗ · · · ⊗ 1⊗X ⊗ 1⊗ · · · ⊗ 1 ∈ End(VΛ) the operator acting nontrivially on the i-th factor only. For X = ∑ mXm⊗Ym ∈ End(VΛi⊗VΛj ), we set X(i,j) = ∑ mX (i) m ⊗ Y (j) m ∈ End(VΛ). Let x0 = (x0 1, . . . , x 0 N ) be a point of CN with distinct coordinates. Introduce linear operators K1(x0), . . . ,KN (x0) on VΛ by the formula Kb(x 0) = ∑ c 6=b Ω(b,c) x0 b − x0 c , b = 1, . . . , N. The operators are called the Gaudin Hamiltonians. The Hamiltonians commute, [Kb(x 0),Kc(x 0)] = 0 for all b, c. The Hamiltonians commute with the g-action on VΛ. Hence they preserve the subspaces Sing VΛ[µ] ⊂ VΛ of singular vectors of a given weight µ. Let τ : g → g be the anti-involution sending Ei, Hi, Fi, to Fi, Hi, Ei, respectively, for all i. Let W be a highest weight g-module with a highest weight vector w. The Shapovalov form S on W is the unique symmetric bilinear form such that S(w,w) = 1, S(gu, v) = S(u, τ(g)v) for all u, v ∈W and g ∈ g. Fix highest weight vectors v1, . . . , vN of VΛ1 , . . . , VΛN , respectively. Define a symmetric bi- linear form on the tensor product VΛ by the formula SΛ = S1 ⊗ · · · ⊗ SN , where Sb is the Shapovalov form on VΛb . The form SΛ is called the tensor Shapovalov form. The Gaudin Hamiltonians are symmetric with respect to the tensor Shapovalov form, SΛ(Kb(y)u, v) = SΛ(u,Kb(y)v) for any u, v ∈ VΛ and b = 1, . . . , N , see [19]. For any µ ∈ h∗, the Gaudin model on Sing VΛ[µ] ⊂ VΛ is the collection( Sing VΛ[µ]; SΛ\|Sing VΛ[µ]; Kb(y)\|Sing VΛ[µ] : Sing VΛ[µ]→ Sing VΛ[µ], b = 1, . . . , N ) , see [7, 8]. The main problem for the Gaudin model is to find common eigenvectors and eigen- values of the Gaudin Hamiltonians. 48 A. Varchenko 11.2 Master function and weight function, [23] The eigenvectors of the Gaudin Hamiltonians are constructed by the Bethe ansatz method. We remind that construction in this section. Fix a collection of nonnegative integers k = (k1, . . . , kr). Denote k = k1 + · · ·+ kr, Λ∞ = N∑ b=1 Λb − r∑ i=1 kiαi. Consider Ck with coordinates t = ( t (1) 1 , . . . , t (1) k1 , . . . , t (r) 1 , . . . , t (r) kr ) . Define the master function Φ(x0, t,Λ,k) = r∑ i=1 ∑ 16j<j′6ki (αi, αi) log ( t (i) j − t (i) j′ ) (11.1) + ∑ 16i<i′6r ki∑ j=1 ki′∑ j′=1 (αi, αi′) log ( t (i) j − t (i′) j′ ) − r∑ i=1 ki∑ j=1 N∑ b=1 (Λb, αi) log ( t (i) j − x 0 b ) . We consider Φ as a function of t depending on parameters x0. Denote by U the set of all points p ∈ Ck such that for any log h entering (11.1) (with a nonzero coefficient) we have h(p) 6= 0. The set U is the complement in Ck to the union of hyperplanes. The coefficients of the logarithms in (11.1) define weights of the hyperplanes. This weighted arrangement is called discriminantal. Discriminantal arrangements were considered in Section 10.2. Let us construct the weight function ω : Ck → VΛ[Λ∞] introduced in [23], cf. [20]. Let b = (b1, . . . , bN ) be a sequence of nonnegative integers with N∑ i=1 bi = k. The set of all such sequences will be denoted by B. For b ∈ B, let {c1 1, . . . , c 1 b1 , . . . , cN1 , . . . , c N bN } be a set of letters. Let Σ(b) be the set of all bijections σ from the set {c1 1, . . . , c 1 b1 , . . . , cN1 , . . . , c N bN } to the set of variables {t(1) 1 , . . . , t (1) k1 , . . . , t (r) 1 , . . . , t (r) kr }. Denote d(t (i) j ) = i, and dσ,b = (d1 1, . . . , d 1 b1 , . . . , dN1 , . . . , d N bN ), where dml = d(σ(cml )). To each b ∈ B and σ ∈ Σ(b) we assign the vector Fdσ,bv = Fd11 . . . Fd1b1 v1 ⊗ · · · ⊗ FdN1 · · ·FdNbN vN ∈ VΛ[Λ∞]. and the rational function ωσ,b = ω1 σ,b ( x0 1 ) · · ·ωNσ,b ( x0 N ) , with ωeσ,b ( x0 e ) = 1 (σ(ce1)− σ(ce2)) · · · (σ(cebe−1)− σ(cebe))(σ(cebe)− x 0 e) and ωeσ,b(x 0 e) = 1 if be = 0. Then the weight function is given by the formula ω ( x0, t,k ) = ∑ b∈B ∑ σ∈Σ(b) ωσ,b ( x0, t ) Fdσ,bv. (11.2) The weight function is a function of t depending on parameters x0. Lemma 11.1 (Lemma 2.1 in [16]). The weight function is regular on U . Quantum Integrable Model of an Arrangement of Hyperplanes 49 11.3 Bethe vectors Theorem 11.1 ([2, 3, 19]). Let g be a simple Lie algebra. Let Λ = (Λ1, . . . ,ΛN ), Λb ∈ h∗, be a collection of weights, k = (k1, . . . , kr) a collection of nonnegative integers. Assume that x0 ∈ CN has distinct coordinates. Assume that p ∈ Ck is a critical point of the master function Φ(x0, ·,Λ,k) : U → U . Then the vector ω(x0, p,k) (if nonzero) belongs to Sing VΛ[Λ∞] and is an eigenvector of the Gaudin Hamiltonians K1(x0), . . . ,KN (x0). The theorem also follows directly from Theorem 6.16.2 in [23], cf. Theorem 7.2.5 in [23], see also Theorem 4.2.2 in [6]. The vector ω(x0, p,k) is called the Bethe vector corresponding to the critical point p. Theorem 11.2. If p is an isolated critical point of the master function, then SΛ ( ω ( x0, p,k ) , ω ( x0, p,k )) = det  ∂2Φ ∂t (i) j ∂t (i′) j′ (x0, p,Λ,k ) . In particular, if the critical point is nondegenerate, then the Bethe vector is nonzero. This theorem is proved for g = slr+1 in [16] and for any simple Lie algebra in [27]. 11.4 Identification of Gaudin and naive geometric Hamiltonians Let us identify constructions and statements of Theorems 11.1, 11.2 and of Theorem 10.14. First of all let us define the discriminantal arrangement associated with the Gaudin model in Theorems 11.2, 11.1. The Gaudin model is determined by a simple Lie algebra g, a nondegenerate g-invariant bilinear form ( , ), simple roots α1, . . . , αr, highest weights Λ1, . . . ,ΛN , distinct complex numbers x0 1, . . . , x 0 N and a vector k = (k1, . . . , kr) of nonnegative integers. Let us take these h∗, α1, . . . , αr, Λ1, . . . ,ΛN , ( , ), k = (k1, . . . , kr) as Data 10.1 to define a family of prediscriminantal arrangements. Let us choose a point z0 ∈ U(X) by conditions xb(z 0) = x0 b for b = 1, . . . , N . The corresponding weighted discriminantal arrangement (A(z0), a) of Section 10.2 is exactly the weighted arrangement defined by the master function in (11.1). In particular, we have Φ(x0, · ,Λ,k) = Φ(z0, · ) where Φ(z0, · ) is the master function of the arrangement (A(z0), a). In [23] an isomorphism γ : W−(z0)→ VΛ[Λ∞] was constructed with the following properties (i)–(iv). (i) The isomorphism γ identifies SingW−(z0) with Sing VΛ[Λ∞]. (ii) The isomorphism γ identifies a naive geometric Hamiltonian K∗∂xb \|SingW−(z0) with the Gaudin Hamiltonian Kb(x 0)\|Sing VΛ[Λ∞] up to addition of a scalar operator. More precisely, for any b = 1, . . . , N , there is a number cb such that γ\|SingW−(z0) ( K∗∂xb + cb ) \|SingW−(z0) = Kb ( x0 ) \|Sing VΛ[Λ∞]γ\|SingW−(z0). (iii) The isomorphism γ identifies the contravariant form on W−(z0) (multiplied by (−1)kk1! · · · kr!) with the tensor Shapovalov form on VΛ[Λ∞], SΛ(γ(x), γ(y)) = (−1)kk1! · · · kr!S(a)(x, y) for any x, y ∈ VΛ[Λ∞]. 50 A. Varchenko (iv) Let E ∈ C(t)U(A(z0)) ⊗W (z0) be the canonical element of the arrangement A(z0). Let Ant = ∑ σ∈Sk (−1)σσ : W (z0) → W−(z0) be the anti-symmetrization operator. Consider the element (1⊗ Ant)E ∈ C(t)U(A(z0)) ⊗W−(z0). Let ω(x0, ·,k) ∈ C(t)U(A(z0)) ⊗ VΛ[Λ∞] be the weight function defined by (11.2). Then (1⊗ γ)(1⊗Ant)E = k1! · · · kr!ω. See Theorems 5.13, 6.16.2, 6.6 and 7.2.5 in [23], see also Section 5.5 in [27]. Now Theorem 11.2 follows from items (iv) and (vi) of Theorem 10.14 and Theorem 11.1 follows from Lemma 10.3 and Theorem 2.8. From statements (i)–(iv) and Theorem 10.14 we get the following improvement of Theo- rem 11.2. Theorem 11.3. For any simple Lie algebra g, if p is an isolated critical point of the master function Φ(x0, ·,Λ,k), then the corresponding Bethe vector ω(x0, p,k) is nonzero. 11.5 Bethe algebra The subalgebra of End(Sing VΛ[µ]) generated by the Gaudin Hamiltonians can be extended to a larger commutative subalgebra called the Bethe algebra. A construction of the Bethe algebra for any simple Lie algebra g is given in [4]. As a result of that construction, for any x0 one obtains a commutative subalgebra B(x0) ⊂ (Ug)⊗N which commutes with the diagonal subalgebra Ug ⊂ (Ug)⊗N . To define the Bethe algebra of VΛ or of Sing VΛ[µ] one considers the image of B(x0) in End(VΛ) or in End(Sing VΛ[µ]). The Gaudin Hamiltonians Kb(x 0) are elements of the Bethe algebra of VΛ or of Sing VΛ[Λ∞]. A more straightforward construction of the Bethe algebra is known for the Gaudin model of glr+1, see [24]. Below we give its description. Let eij , i, j = 1, . . . ,r+1, be the standard generators of glr+1 satisfying the relations [eij , esk] = δjseik − δikesj . Let h ⊂ glr+1 be the Cartan subalgebra generated by eii, i = 1, . . . ,r+ 1. Let h∗ be the dual space. Let εi, i = 1, . . . ,r+ 1, be the basis of h∗ dual to the basis eii, i = 1, . . . ,r+ 1, of h. Let α1, . . . , αr ∈ h∗ be simple roots, αi = εi− εi+1. Let ( , ) be the standard scalar product on h∗ such that the basis εi, i = 1, . . . ,r + 1, is orthonormal. Let glr+1[s] = glr+1⊗C[s] be the Lie algebra of glr+1-valued polynomials with the pointwise commutator. For g ∈ glr+1, we set g(u) = ∞∑ i=0 (g ⊗ si)u−i−1. We identify glr+1 with the subal- gebra glr+1 ⊗ 1 of constant polynomials in glr+1[s]. Hence, any glr+1[s]-module has a canonical structure of a glr+1-module. For each a ∈ C, there exists an automorphism ρa of glr+1[s], ρa : g(u) 7→ g(u − a). Given a glr+1[s]-module W , we denote by W (a) the pull-back of W through the automorphism ρa. As glr+1-modules, W and W (a) are isomorphic by the identity map. We have the evaluation homomorphism, glr+1[s]→ glr+1, g(u) 7→ gu−1. Its restriction to the subalgebra glr+1 ⊂ glr+1[s] is the identity map. For any glr+1-module W , we denote by the same letter the glr+1[s]-module, obtained by pulling W back through the evaluation homomorphism. Given an algebra A and an (r + 1)× (r + 1)-matrix C = (cij) with entries in A, we define its row determinant to be rdetC = ∑ σ∈Σr+1 (−1)σc1σ(1)c2σ(2) · · · cr+1σ(r+1). Quantum Integrable Model of an Arrangement of Hyperplanes 51 Define the universal differential operator DB by the formula DB = rdet  ∂u − e11(u) −e21(u) . . . −er+1 1(u) −e12(u) ∂u − e22(u) . . . −er+1 2(u) . . . . . . . . . . . . −e1 r+1(u) −e2 r+1(u) . . . ∂u − er+1 r+1(u)  . We have DB = ∂r+1 u + r+1∑ i=1 Bi∂ r+1−i u , Bi = ∞∑ j=i Biju −j , Bij ∈ Uglr+1[s]. The unital subalgebra of Uglr+1[s] generated by Bij , i = 1, . . . ,r + 1, j > i, is called the Bethe algebra and denoted by B. By [24], cf. [10], the algebra B is commutative, and B commutes with the subalgebra Uglr+1 ⊂ Uglr+1[s]. As a subalgebra of Uglr+1[s], the algebraB acts on any glr+1[s]-moduleW . SinceB commutes with Uglr+1, it preserves the glr+1 weight subspaces of W and the subspace SingW of glr+1- singular vectors. If W is a B-module, then the image of B in End(W ) is called the Bethe algebra of W . Let VΛ = ⊗nb=1VΛb be a tensor product of irreducible highest weight glr+1-modules. For given x0 = (x0 1, . . . , x 0 N ), consider VΛ as the glr+1[s]-module ⊗nb=1VΛb(x 0 b). This glr+1[s]-module structure on VΛ provides VΛ with a Bethe algebra, a commutative subalgebra of End(VΛ). It is known that this Bethe algebra of VΛ contains the Gaudin Hamiltonians Kb(x 0), b = 1, . . . , N . In fact, the Gaudin Hamiltonians are suitably normalized residues of the generating function B2(u), see Appendix B in [10]. Theorem 11.4 ([12]). Consider VΛ as the glr+1[s]-module ⊗nb=1VΛb(x 0 b). Then any ele- ment B ∈ B acts on VΛ as a symmetric operator with respect to the tensor Shapovalov form, SΛ(Bu, v) = SΛ(u,Bv) for any u, v ∈ VΛ. 11.6 glr+1 Bethe algebra and critical points of the master function In [15] the following generalization of Theorem 11.1 for g = glr+1 was obtained. Let g = glr+1. A sequence of integers Λ = (λ1, . . . , λr+1) such that λ1 > λ2 > · · · > λr+1 > 0 is called a partition with at most r + 1 parts. We identify partitions Λ with vectors λ1ε1 + · · ·+ λr+1εr+1 of h∗. Let Λ = (Λ1, . . .,ΛN ) be a collection of partitions, where Λb = (λb,1, . . ., λb,r+1) and λb,r+1 = 0. Let k = (k1, . . . , kr) be nonnegative integers such that Λ∞ = N∑ b=1 Λb − r∑ i=1 kiαi is a partition. We consider the glr+1 Gaudin model with parameters x0 = (x0 1, . . . , x 0 N ) on Sing VΛ[Λ∞] where VΛ = VΛ1 ⊗ · · · ⊗ VΛN . Consider the master function Φ(x0, t,Λ,k) defined by (11.1) and the weight function ω(x0, t,k) defined by (11.2). Let u be a variable. Define polynomials T1, . . . , Tr ∈ C[u], Q1, . . . , Qr ∈ C[u, t], Ti(u) = N∏ b=1 (u− xb)(Λb,αi), Qi(u, t) = ki∏ j=1 ( u− t(i)j ) , 52 A. Varchenko and the differential operator DΦ = ( ∂u − log′ ( T1 . . . Tr Q1 )) × ( ∂u − log′ ( Q1T2 · · ·Tr Q2 )) · · · ( ∂u − log′ ( Qr−1Tr Qr )) (∂u − log′(Qr)), where ∂u = d/du and log′ f denotes (df/du)/f . We have DΦ = ∂N+1 u + N+1∑ i=1 Gi∂ N+1−i u , Gi = ∞∑ j=i Giju −j , where Gij ∈ C[t]. Let p ∈ U be an isolated critical point of the master function Φ(x, , · ,Λ,k) with Milnor number µ. Let Ap,Φ be its local algebra. For f ∈ C(t)U denote by [f ] the image of f in Ap,Φ. Denote [DΦ] = ∂N+1 u + N+1∑ i=1 [Gi]∂ N+1−i u , where [Gi] = ∞∑ j=i [Gij ]u −j . Let [ω] ∈ Ap,Φ⊗VΛ[Λ∞] be the element induced by the weight function. The element [ω] belongs to Ap,Φ ⊗ Sing VΛ[Λ∞], see [23], and we have SΛ([ω], [ω]) = [HesstΦ] , see [16, 27]. Theorem 11.5 ([10]). For any i = 1, . . . , r + 1, j > i, we have (1⊗Bij)[ω] = ([Gij ]p ⊗ 1)[ω] in Ap,Φ ⊗ Sing VΛ[Λ∞]. This statement is the Bethe ansatz method to construct eigenvectors of the Bethe algebra in the glr+1 Gaudin model starting with a critical point of the master function. Let g1, . . . , gµ be a basis of Ap,Φ considered as a C-vector space. Write [ω]p = ∑ i gi⊗wi, with wi ∈ Sing VΛ[Λ∞]. Denote by Yp ⊂ Sing VΛ[Λ∞] the vector subspace spanned by w1, . . . , wµ. Let ( , )p be the bilinear form on Ap,Φ. Define a linear map αp : Ap,Φ → Yp, f 7→ (f, [ω]p)p = µ∑ i=1 (f, gi)pwi. Theorem 11.6 ([15]). (i) The subspace Yp ⊂ Sing VΛ[Λ∞] is a B-submodule. Let AYp ⊂ End (Yp) be the Bethe algebra of Yp. Denote by B̄ij the image in AYp of generators Bij ∈ B. (ii) The map αp : Ap,Φ → Yp is an isomorphism of vector spaces. (iii) The map [Gij ]p 7→ B̄ij extends uniquely to an algebra isomorphism βp : Ap,Φ → AYp. (iv) The isomorphisms αp and βp identify the regular representation of Ap,Φ and the B-modu- le Yp, that is, for any f, g ∈ Ap,Φ we have αp(fg) = βp(f)αp(g). (v) The value of the weight function at p is a nonzero vector of Sing VΛ[Λ∞]. (vi) Let p1, . . . , pd be a list of isolated critical points of Φ(x0, · ,Λ,k) such that the orbits O(p1), . . . , O(pd) do not intersect. Let Yps = αps(Aps,Φ) ⊂ Sing VΛ[Λ∞], s = 1, . . . , d, be the corresponding subspaces. Then the sum of these subspaces is direct and orthogonal. Quantum Integrable Model of an Arrangement of Hyperplanes 53 (vii) Denote Y = ⊕ds=1Yps. Denote AY the Bethe algebra of Y. Consider the isomorphisms α = ⊕ds=1αps : ⊕ds=1Aps,Φ → ⊕ds=1Yps , β = ⊕ds=1βps : ⊕ds=1Aps,Φ → ⊕ds=1AYps . Then (a) AY = ⊕ds=1AYps ; (b) AY is a maximal commutative subalgebra of End(Y); (c) the isomorphisms α, β identify the regular representation of the algebra ⊕ds=1Aps,Φ and the AY-module Y. Let us identify constructions and statements of Theorem 11.6 and of Theorems 10.14, 10.17. Consider the discriminantal arrangement A(z0), z0 ∈ U(X), defined in Section 11.4 and the isomorphism γ\|SingW−(z0) : SingW−(z0) → Sing VΛ[Λ∞] of that section. According to statements (i)–(iv) of Section 11.4, we have γ(Y −O(ps) ) = Yps , where Y −O(ps) ⊂ SingW−(z0) is the subspace in Theorems 10.14, 10.17 and Yps ⊂ Sing VΛ[Λ∞] is the subspace in Theorem 11.6. Moreover, γ identifies the algebra AY − O(ps) of geometric Hamiltonians on Y −O(ps) with the Bethe algebra of Yps . Additional information that is given by Theorems 10.14, 10.17 is the following theorem. Theorem 11.7. The monomorphism αp of Theorem 11.6 identifies the tensor Shapovalov form on Yp and the residue bilinear form on Ap,Φ. Proof. The theorem is a corollary of properties (iii), (iv) in Section 11.4 and Lemma 10.12. � 11.7 Expectations One may expect that for the Gaudin models of any simple Lie algebras the associated Bethe algebra defined in [4] coincides with the algebra of geometric Hamiltonians associated with the arrangement of the corresponding master function. That topic will be discussed in a forthcoming paper. Below we consider three examples in which we have a well-defined algebra of geometric Hamiltonians on Sing VΛ[Λ∞]. 11.7.1 Example Consider the Gaudin model corresponding to the following data: the Lie algebra glr+1, the collection of dominant integral weights Λ = (Λ1, . . . ,ΛN ) with Λb = (1, 0, . . . , 0) for all b, a vector of nonnegative integers k = (k1, . . . , kr) such that Λ∞ is a partition, a collection of generic distinct complex numbers x0 = (x0 1, . . . , x 0 N ). By [16], for generic x0 the associated master function has a collection of critical points p1, . . . , pd such that the Sk-orbits of these points do not intersect and the sum of Milnor numbers of these points equals the dimension of Sing VΛ[Λ∞]. In this case Theorem 10.17 defines a ma- ximal commutative subalgebra ASing (W−(z0)) ⊂ End(Sing (W−(z0))) containing naive geometric Hamiltonians. The isomorphism γ sends ASing (W−(z0)) to a maximal commutative subalgebra of End(Sing VΛ[Λ∞]) containing the Gaudin Hamiltonians. By [12, 13], the Bethe algebra of Sing VΛ[Λ∞] is a maximal commutative subalgebra of End(Sing VΛ[Λ∞]) and the Bethe algebra of Sing VΛ[Λ∞] is generated by the Gaudin Hamiltonians. Hence, in this case, γ establishes an isomorphism of the algebra of geometric Hamiltonians ASing (W−(z0)) and the Bethe algebra of Sing VΛ[Λ∞]. 54 A. Varchenko 11.7.2 Example Consider the Gaudin model corresponding to the following data: the Lie algebra gl2, a collection of weights Λ = (Λ1, . . . ,ΛN ) with Λb = λbα1 such that λb ∈ R<0 for all b, a nonnegative integer k = (k1), a collection of distinct complex numbers x0 = (x0 1, . . . , x 0 N ). Let (A(z0), a) be the associated discriminantal arrangement defined in Section 11.4. By our assumptions, the weights a of the discriminant arrangement are all positive. Corollary 10.18 defines in this case a maximal commutative subalgebra of End(SingW−(z0))) and the isomor- phism γ sends this subalgebra to a maximal commutative subalgebra of End(Sing VΛ[Λ∞]), which contains Gaudin Hamiltonians. One may expect that this commutative subalgebra of End(Sing VΛ[Λ∞]) coincides with the Bethe algebra of Sing VΛ[Λ∞]. 11.7.3 Example Consider the Gaudin model corresponding to the following data: a simple Lie algebra g, a col- lection of dominant integral weights Λ = (Λ1, . . . ,ΛN ), a vector of nonnegative integers k = (k1, . . . , kr) with ki 6 1 for all i, a collection of distinct complex numbers x0 = (x0 1, . . . , x 0 N ). Let (A(z0), a) be the associated discriminantal arrangement. By our assumptions, the weights a of the discriminant arrangement are all negative and the group Sk is trivial. In this case the isomorphism γ identifies SingFk(A(z0)) and the space Sing VΛ[Λ∞]. Theorem 9.17 defines in this case a maximal commutative subalgebra of End(SingFk(A(z0))) and the iso- morphism γ sends this subalgebra to a maximal commutative subalgebra of End(Sing VΛ[Λ∞]), which contains the Gaudin Hamiltonians. One may expect that this commutative subalgebra of End(Sing VΛ[Λ∞]) coincides with the Bethe algebra of Sing VΛ[Λ∞]. Acknowledgments The idea that an analog of the Bethe ansatz construction does exist for an arbitrary arrangement of hyperplanes was formulated long time ago in [26]. That program had been realized partially in [27]. This paper is an extended exposition of my lectures at Mathematical Society of Japan Sea- sonal Institute on Arrangements of Hyperplanes in August of 2009. I thank organizers for invita- tion and Hokkaido University for hospitality. I thank for hospitality Université Paul Sabatier in Toulouse, where this paper had been finished. I thank E. Mukhin, V. Schechtman, V. Tarasov, H. Terao for discussions. The author was supported in part by NSF grant DMS-0555327. References [1] Arnold V.I., Gusein-Zade S.M., Varchenko A.M., Singularities of differentiable maps, Vol. II, Monodromy and asymptotics of integrals, Monographs in Mathematics, Vol. 83, Birkhäuser, Boston, MA, 1988. [2] Babujian H.M., Off-shell Bethe ansatz equations and N -point correlators in the SU(2) WZNW theory, J. Phys. A: Math. Gen. 26 (1993), 6981–6990. [3] Babujian H.M., Flume R., Off-shell Bethe ansatz equation for Gaudin magnets and solutions of Knizhnik– Zamolodchikov equations, Modern Phys. Lett. A 9 (1994), 2029–2039, hep-th/9310110. [4] Feigin B., Frenkel E., Reshetikhin N., Gaudin model, Bethe ansatz and critical level, Comm. Math. Phys. 166 (1994), 29–62, hep-th/9402022. [5] Feigin B., Frenkel E., Rybnikov L., Opers with irregular singularity and spectra of the shift of argument subalgebra, Duke Math. J. 155 (2010), 337–363, arXiv:0712.1183. [6] Feigin B., Schechtman V., Varchenko A., On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 (1995), 219–247, hep-th/9407010. http://dx.doi.org/10.1088/0305-4470/26/23/037 http://dx.doi.org/10.1142/S0217732394001891 http://arxiv.org/abs/hep-th/9310110 http://dx.doi.org/10.1007/BF02099300 http://arxiv.org/abs/hep-th/9402022 http://dx.doi.org/10.1215/00127094-2010-057 http://arxiv.org/abs/0712.1183 http://dx.doi.org/10.1007/BF02099447 http://arxiv.org/abs/hep-th/9407010 Quantum Integrable Model of an Arrangement of Hyperplanes 55 [7] Gaudin M., Diagonalisation d’une classe d’Hamiltoniens de spin, J. Physique 37 (1976), 1087–1098. [8] Gaudin M., La fonction d’onde de Bethe, Collection du Commissariat l’Érgie Atomique, Série Scientifique, Masson, Paris, 1983. [9] Griffiths Ph., Harris J., Principles of algebraic geometry, John Wiley & Sons, Inc., New York, 1994. [10] Mukhin E., Tarasov V., Varchenko A., Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp. 2006 (2006), no. 8, P08002, 44 pages, math.QA/0605015. [11] Mukhin E., Tarasov V., Varchenko A., The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. of Math. (2) 170 (2009), 863–881, math.AG/0512299. [12] Mukhin E., Tarasov V., Varchenko A., Schubert calculus and representations of the general linear group, J. Amer. Math. Soc. 22 (2009), 909–940, arXiv:0711.4079. [13] Mukhin E., Tarasov V., Varchenko A., Gaudin Hamiltonians generate the Bethe algebra of a tensor power of vector representation of glN , Algebra i Analiz 22 (2010), no. 3, 177–190, arXiv:0904.2131. [14] Mukhin E., Tarasov V., Varchenko A., Three sides of the geometric Langlands correspondence for glN Gaudin model and Bethe vector averaging maps, arXiv:0907.3266. [15] Mukhin E., Tarasov V., Varchenko A., Bethe algebra of the glN+1 Gaudin model and algebra of functions on the critical set of the master function, arXiv:0910.4690. [16] Mukhin E., Varchenko A., Norm of a bethe vector and the Hessian of the master function, Compos. Math. 141 (2005), 1012–1028, math.QA/0402349. [17] Orlik P., Terao H., The number of critical points of a product of powers of linear functions, Invent. Math. 120 (1995), 1–14. [18] Orlik P., Terao H., Arrangements and hypergeometric integrals, MSJ Memoirs, Vol. 9, Mathematical Society of Japan, Tokyo, 2001. [19] Reshetikhin N., Varchenko A., Quasiclassical asymptotics of solutions to the KZ equations, in Geometry, Topology, & Physics, Conf. Proc. Lecture Notes Geom. Topology, Vol. 4, Int. Press, Cambridge, MA, 1995, 293–322. [20] Rimányi R., Stevens L., Varchenko A., Combinatorics of rational functions and Poincaré–Birkhoff–Witt expansions of the canonical U(n−)-valued differential form, Ann. Comb. 9 (2005), 57–74, math.CO/0407101. [21] Silvotti R., On a conjecture of Varchenko, Invent. Math. 126 (1996), 235–248, alg-geom/9503016. [22] Schechtman V., Terao H., Varchenko A., Local systems over complements of hyperplanes and the Kac– Kazhdan conditions for singular vectors, J. Pure Appl. Algebra 100 (1995), 93–102, hep-th/9411083. [23] Schechtman V., Varchenko A., Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139–194. [24] Talalaev D., Quantization of the Gaudin system, hep-th/0404153. [25] Varchenko A., Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Advanced Series in Mathematical Physics, Vol. 21, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. [26] Varchenko A., Critical points of the product of powers of linear functions and families of bases of singular vectors, Compos. Math. 97 (1995), 385–401, hep-th/9312119. [27] Varchenko A., Bethe ansatz for arrangements of hyperplanes and the Gaudin model, Mosc. Math. J. 6 (2006), 195–210, 223–224, math.QA/0408001. [28] Yuzvinsky S., Cohomology of the Brieskorn–Orlik–Solomon algebras, Comm. Algebra 23 (1995), 5339–5354. [29] Zaslavsky T., Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), no. 154. http://dx.doi.org/10.1051/jphys:0197600370100108700 http://dx.doi.org/10.1088/1742-5468/2006/08/P08002 http://dx.doi.org/10.1088/1742-5468/2006/08/P08002 http://arxiv.org/abs/math.QA/0605015 http://dx.doi.org/10.4007/annals.2009.170.863 http://arxiv.org/abs/math.AG/0512299 http://dx.doi.org/10.1090/S0894-0347-09-00640-7 http://arxiv.org/abs/0711.4079 http://arxiv.org/abs/0904.2131 http://arxiv.org/abs/0907.3266 http://arxiv.org/abs/0910.4690 http://dx.doi.org/10.1112/S0010437X05001569 http://arxiv.org/abs/math.QA/0402349 http://dx.doi.org/10.1007/BF01241120 http://dx.doi.org/10.1007/s00026-005-0241-3 http://arxiv.org/abs/math.CO/0407101 http://dx.doi.org/10.1007/s002220050096 http://arxiv.org/abs/alg-geom/9503016 http://dx.doi.org/10.1016/0022-4049(95)00014-N http://arxiv.org/abs/hep-th/9411083 http://dx.doi.org/10.1007/BF01243909 http://arxiv.org/abs/hep-th/0404153 http://www.numdam.org/item?id=CM_1995__97_3_385_0 http://arxiv.org/abs/hep-th/9312119 http://arxiv.org/abs/math.QA/0408001 http://dx.doi.org/10.1080/00927879508825535 1 Introduction 1.1 Quantum integrable models and Bethe ansatz 1.2 Gaudin model 1.3 Gaudin model as a semiclassical limit of KZ equations 1.4 Bethe ansatz in the Gaudin model 1.5 Gaudin model and arrangements 1.6 Bethe algebra 1.7 Algebra of geometric Hamiltonians 1.8 Quantum integrable model of a weighted arrangement (or dynamical theory of arrangements) 1.9 Bethe ansatz for the quantum integrable model of an arrangement 1.10 Geometric interpretation of the algebra of Hamiltonians 1.11 Byproducts of constructions 1.12 Exposition of the material 2 Arrangements 2.1 An affine arrangement 2.2 Orlik–Solomon algebra 2.3 Weights 2.4 Space of flags, see SV 2.5 Duality, see SV 2.6 Contravariant map and form, see SV 2.7 Remarks on generic weights 2.8 Orlik–Solomon algebra as an algebra of differential forms 2.9 Critical points of the master function 2.10 Special vectors in Fk(A) and canonical element 2.11 Arrangements with normal crossings only 2.12 Real structure on Ap(A) and Fp(A) 2.13 A real arrangement with positive weights 2.14 Resolution of a hyperplane-like divisor 3 A family of parallelly translated hyperplanes 3.1 An arrangement in CnCk 3.2 Discriminant 3.3 Good fibers 3.4 Bad fibers 4 Conservation of the number of critical points 5 Hamiltonians of good fibers 5.1 Construction 5.2 Key identity (5.2) 5.3 An application of the key identity (5.2) – proof of Theorem 5.1 5.4 Another application of the key identity (5.2) 5.5 Hamiltonians, critical points and the canonical element 6 Asymptotic solutions and eigenvectors of Hamiltonians 6.1 Asymptotic solutions, one variable 6.2 Critical points of the master function and asymptotic solutions 7 Hamiltonians of bad fibers 7.1 Naive geometric Hamiltonians 7.2 Space Fk(A(z0)) and operators LC 7.3 Conjecture 7.4 Positive (aj)jJ, real (gj)jJ 7.5 Proof of Theorem 7.5 for z0Rn 7.6 Proof of Theorem 7.5 for any z0 7.7 Critical points and eigenvectors 7.8 Hamiltonians, critical points and the canonical element 8 Geometric interpretation of the algebra of Hamiltonians 8.1 An abstract setting 8.2 Proof of Theorem 8.1 8.3 Remark on maximal commutative subalgebras 8.4 Interpretation of the algebra of Hamiltonians of good fibers 8.5 Interpretation of the algebra of Hamiltonians of bad fibers if Assumption 7.4 is satisfied 9 More on Hamiltonians of bad fibers 9.1 An abstract setting 9.2 Hamiltonians of bad fibers 9.3 Remark on critical points of real arrangements 10 Arrangements with symmetries 10.1 A family of prediscriminantal arrangements 10.2 Discriminantal arrangements 10.3 Symmetries of the family of prediscriminantal arrangements 10.4 The Sk-action on geometric Hamiltonians 10.5 Functions Kxb(z) 10.6 Naive geometric Hamiltonians on SingW-(z0) 10.7 Sk-symmetries of the canonical element 11 Applications to the Bethe ansatz of the Gaudin model 11.1 Gaudin model 11.2 Master function and weight function, SV 11.3 Bethe vectors 11.4 Identification of Gaudin and naive geometric Hamiltonians 11.5 Bethe algebra 11.6 glr+1 Bethe algebra and critical points of the master function 11.7 Expectations 11.7.1 Example 11.7.2 Example 11.7.3 Example References

Quantum Integrable Model of an Arrangement of Hyperplanes

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