Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)

Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)

Takiff superalgebras are a family of non semi-simple Lie superalgebras that are believed to give rise to a rich structure of indecomposable representations of associated conformal field theories. We consider the Takiff superalgebra of gl(1|1), especially we perform harmonic analysis for the correspo...

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Дата:2015
Автори: Babichenko, A., Creutzig, T.
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Мова:English
Опубліковано: Інститут математики НАН України 2015
Назва видання:Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/147138
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Цитувати:Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) / A. Babichenko, T. Creutzig // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 50 назв. — англ.

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spelling irk-123456789-1471382019-02-14T01:24:43Z Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) Babichenko, A. Creutzig, T. Takiff superalgebras are a family of non semi-simple Lie superalgebras that are believed to give rise to a rich structure of indecomposable representations of associated conformal field theories. We consider the Takiff superalgebra of gl(1|1), especially we perform harmonic analysis for the corresponding supergroup. We find that every simple module appears as submodule of an infinite-dimensional indecomposable but reducible module. We lift our results to two free field realizations for the corresponding conformal field theory and construct some modules. 2015 Article Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) / A. Babichenko, T. Creutzig // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 50 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B67; 17B81; 22E46; 81R10; 81T40 DOI:10.3842/SIGMA.2015.067 http://dspace.nbuv.gov.ua/handle/123456789/147138 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 Takiff superalgebras are a family of non semi-simple Lie superalgebras that are believed to give rise to a rich structure of indecomposable representations of associated conformal field theories. We consider the Takiff superalgebra of gl(1|1), especially we perform harmonic analysis for the corresponding supergroup. We find that every simple module appears as submodule of an infinite-dimensional indecomposable but reducible module. We lift our results to two free field realizations for the corresponding conformal field theory and construct some modules.
format Article
author Babichenko, A.
Creutzig, T.
spellingShingle Babichenko, A.
Creutzig, T.
Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Babichenko, A.
Creutzig, T.
author_sort Babichenko, A.
title Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
title_short Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
title_full Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
title_fullStr Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
title_full_unstemmed Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
title_sort harmonic analysis and free field realization of the takiff supergroup of gl(1|1)
publisher Інститут математики НАН України
publishDate 2015
url http://dspace.nbuv.gov.ua/handle/123456789/147138
citation_txt Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) / A. Babichenko, T. Creutzig // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 50 назв. — англ.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT babichenkoa harmonicanalysisandfreefieldrealizationofthetakiffsupergroupofgl11
AT creutzigt harmonicanalysisandfreefieldrealizationofthetakiffsupergroupofgl11
first_indexed 2025-07-11T01:26:16Z
last_indexed 2025-07-11T01:26:16Z
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 067, 24 pages Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) Andrei BABICHENKO † and Thomas CREUTZIG ‡ † Department of Mathematics, Weizmann Institut, Rehovot, 76100, Israel E-mail: babichenkoandrei@gmail.com ‡ Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada E-mail: creutzig@ualberta.ca Received May 28, 2015, in final form August 01, 2015; Published online August 06, 2015 http://dx.doi.org/10.3842/SIGMA.2015.067 Abstract. Takiff superalgebras are a family of non semi-simple Lie superalgebras that are believed to give rise to a rich structure of indecomposable representations of associated con- formal field theories. We consider the Takiff superalgebra of gl(1|1), especially we perform harmonic analysis for the corresponding supergroup. We find that every simple module appears as submodule of an infinite-dimensional indecomposable but reducible module. We lift our results to two free field realizations for the corresponding conformal field theory and construct some modules. Key words: logarithmic CFT; Harmonic analysis; free field realization 2010 Mathematics Subject Classification: 17B67; 17B81; 22E46; 81R10; 81T40 1 Introduction Logarithmic conformal field theories carry this name as correlation functions sometimes have log- arithmic singularities. They are playing an essential role in different physical problems ranging from string theory, especially on supergroup target spaces [2, 5, 6, 7, 28, 34, 35, 39, 40, 43, 45] to different condensed matter and statistical mechanics problems [24, 25, 26, 33, 37, 38, 41, 42, 47]. Many of these statistical mechanics problems are described by logrithmic CFTs based on Lie superalgebras, as, e.g., supersymmetric disordered systems [29, 50]. For reviews on logarithmic CFT, see [13, 21, 23]. The presence of logarithmic singularities is tightly connected to the non semi-simple action of the chiral algebra on some of its modules. In some models [3] of non chiral conformal theory related to string theory on AdS3 × S3 non semi-simplicity appears on the level of the assumed symmetry, and not only the representation level. In [4], motivated by this fact, different chiral conformal field theories were considered, where non semi-simplicity appears on the level of symmetry algebra itself. These are conformal field theories based on Takiff (super)algebras, e.g., algebras that are non semi-simple extensions of simple Lie (super)algebras by its adjoint representation. Detailed algebraic structure of its representation theory was then investigated. In the mathematics literature, this type of non-semisimple Lie superalgebras were introduced by Takiff [46], though not in the super setting, but as part of an investigation of invariant polynomial rings. These algebras have since been considered in a slightly generalised form under the names generalised Takiff algebras [27, 49] in which a semisimple Lie algebra is tensored with a polynomial ring in a nilpotent formal variable t, and truncated current algebras [48, 49] in which one does the same to an affine Kac–Moody algebra. The algebras considered in [4] correspond to taking t2 = 0, as in Takiff’s original paper, and they were named Takiff superalgebras. These mailto:babichenkoandrei@gmail.com mailto:creutzig@ualberta.ca http://dx.doi.org/10.3842/SIGMA.2015.067 2 A. Babichenko and T. Creutzig algebras were mainly considered algebraically from the representation theory point of view, in their chiral sector. On the other hand, it is well known that potential additional algebraic structures appear for Wess–Zumino–Novikov–Witten (WZNW) theories when representations are subjected to both chiral and anti chiral action of the group. In the present work, we initiate the study of the WZNW theory for the Takiff Lie supergroup G̃L(1|1). In [19, 20, 36], WZNW theories of non-reductive Lie groups have been studied and our notion of Takiff superalgebra is what the authors of [20] there call a double extensions. Wess–Zumino–Novikov–Witten theories of non-compact Lie groups and Lie supergroups pro- vide a rich source of logarithmic conformal field theories. Usually, one considers such theories based on a semi-simple Lie (super)group or closely related supergroups as GL(n|n).1 Experience shows that in such theories most modules are actually completely reducible and only a few non- generic modules are indecomposable but reducible. It is an interesting situation if already the Lie (super)algebra/Lie (super)group on which the conformal field theory is based is itself non semi-simple, as, e.g., for Takiff (super)groups. In [4], the first author and David Ridout started to study such algebras from a conformal field theory perspective. Here, we’d like to continue their work with the aim of studying an example of a conformal field theory based on a Takiff superalgebra in detail. We believe that the case of g̃l (1|1) (the Takiff superalgebra of the Lie superalgebra gl (1|1)) is both treatable and instructive, so that we decided to first focus on this case. The chiral algebra of a WZNW model of a Lie group at level k is the simple affine vertex operator algebra of its Lie algebra g at level k. Frenkel and Zhu [22] have proven that there is a one-to-one correspondence of simple representations of this vertex algebra and simple representations of a quotient of the universal envelopping algebra of the Lie algebra g. If the vacuum Verma module of the affine vertex operator algebra is simple then there is actually a one-to-one correspondence to the simple objects of the universal envelopping algebra. As in the case of gl (1|1), the vacuum Verma module of the vertex algebra of g̃l (1|1) is simple for non-vanishing levels [4]. So, we expect that representations of g̃l (1|1) can teach us quite a bit about the conformal field theory. Very natural modules of the Lie superalgebra are functions on the Lie supergroup which are studied using harmonic analysis (often called mini superspace analysis in the CFT literature) on the Lie supergroup. This is actually a very general experience in WZNW theories based on Lie supergroups. The key strategy is to understand a problem of the finite-dimensional Lie superalgebra, to lift this understanding to the full conformal field theory, usually to a free field realization, and then to use the new understanding to derive interesting statements concerning the CFT. This strategy has been initiated by Hubert Saleur, Volker Schomerus and Thomas Quella a decade ago. It has been used extensively by various people, including the second author. For example, the key starting ingredient of the bulk the- ories of GL(1|1) [45], SU(2|1) [44], PSU(1, 1|2) [28] as well as the more general case [39] has been the harmonic analysis, and then lifting the findings to an appropriate free field realization. These free field realizations in turn were very useful in computing correlation functions [45], proving dualities to super Liouville field theories [11, 30] and finding relations to CFTs with N = (2, 2) world-sheet superconformal symmetry [17]. Similarly also boundary theories based on GL(1|1) [12, 18] and OSP(1|2) [10] have been studied by first performing an appropriate harmonic analysis. The difference to the bulk theory is, that instead of studying the action of both left- and right-invariant vector fields one investigates a twisted adjoint action. Boundary states are then lifts of distributions localized on twisted superconjugacy classes [9, 12]. The by far best understood WZNW theory of a Lie supergroup is the one of GL(1|1). It has been first studied by Rozansky and Saleur a while ago [43]. Then as mentioned, Schomerus and 1Even though gl (n|n) is not semi-simple its representation theory is similar to the one of the simple Lie superalgebra psl (n|n). However, already in the case of gl (1|1) representations of the Takiff superalgebra have much more involved indecomposability structure than the non-Takiff superalgebra [4]. Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) 3 Saleur, were able to understand spectrum and correlation functions using harmonic analysis and free field realization [45]. As the next step boundary states were constructed [12]. Studying the twisted adjoint action of invariant vector fields on functions on the supergroup allowed one to find the boundary states explicitely. Boundary and bulk-boundary correlation functions could also be computed using a similar free field realization as in the bulk case [18]. The indecomposability structure of modules is not directly visible in the free field realization used in the just mentioned works. It turns out that there is another free field realization using symplectic fermions that makes this structure much clearer [16, 32]. Using then the work of Kausch on symplectic fermions [31] most of the previous results could be obtained fairly directly [16]. This conformal field theory can also be studied from a more algebraic perspective, especially many extended algebras like sl(2|1) at levels −1/2 and 1, can be constructed and studied [1, 14, 15]. In summary, the GL(1|1) WZNW theory has been thoroughly studied and has revealed a rich structure. We believe that a similar (but much more complex) story holds for its Takiff superalgebra. In [4], representations of both finite-dimensional and affine Takiff superalgebra of gl (1|1) have been studied and a Verlinde fusion ring has been computed. We now would like to proceed to the next step, and ask which representations appear in the full bulk CFT, and wether we can realize them using free fields. Our main result is a thorough harmonic analysis of the Takiff supergroup. We find that functions split into three classes, called typical, semitypical and atypical. All of them appear in the harmonic analysis as submodule of infinite- dimensional indecomposable but reducible modules. The reason for this is a special element Ñ of the Lie superalgebra, that already acts non-semi-simple in the adjoint representation. It forces us to allow for a larger space of functions. Nonetheless, in the typical case there is still a decomposition into tensor products of modules for the action of left- and right-invariant vector fields somehow similar as one is used from the Peter–Weyl theorem for compact Lie groups. Both semitypical and especially atypical modules have a rather complicated behaviour under the left-right action. We visualize the structure of these modules in various pictures and exact sequences. Further, already the simple submodules of the typical modules posses Jordan cells of size three under the action of the Laplacian (as expected from [4]). The harmonic analysis tells us nicely how to construct a free field realization including screening charges. As in the case of the GL(1|1) WZNW theory, typical modules can be constructed easily using this free field realization. Again, in analogy to the GL(1|1) WZNW theory we find a second free field realization using symplectic fermions which allows us to also construct semitypical and atypical modules. This puts us now in a position to study (in the near future) these modules further, to ask questions about correlation functions, operator product algebra and extended algebras. This work is organized as follows: Section 2 is the main part of this paper. We first compute basic objects on the Takiff supergroup, namely invariant vector fields, invariant measure and Laplacians. Then we compute the action of invariant vector fields. The typical case can be performed rather directly, while the structure in the semitypical and atypical cases is very rich and first studying the action of various subalgebras turns out to structure the problem nicely. Section 3 then applies our findings to construct a free field realization including screening charges. But we also add another free field realization using symplectic fermions. Finally, we conclude with describing the expected future use of our findings. 2 Harmonic analysis We start from definition of Takiff superalgebra G̃L(1|1), and briefly describe its highest weight representations. The main part of this section is devoted to harmonic analysis for G̃L(1|1) supergroup. The procedure for this follows ideas established in various works on GL(1|1) (see mainly [45]) and goes as follows: 4 A. Babichenko and T. Creutzig 1) consider a supergroup element g according to a triangular decomposition; 2) compute left- and right-invariant vector fields; 3) compute the invariant measure; 4) compute the Laplacians; 5) decompose functions with respect to the combined left-right action of invariant vector fields. 2.1 Takiff superalgebra ˜gl (1|1) and its highest weight representations Takiff Lie superalgebra g̃l (1|1) was introduced in [4] as an extention of the superalgebra gl (1|1) with Grassmann even generator N , Grassmann odd generators ψ± and central element E [N,ψ±] = ±ψ±, {ψ+, ψ−} = E by a set of their Takiff partners Ñ , ψ̃±, Ẽ with commutation relations [N, ψ̃±] = [Ñ , ψ±] = ±ψ̃±, {ψ̃+, ψ−} = {ψ+, ψ̃−} = Ẽ and the remaining (anti)commutators vanish. As one can see the adjoint action of N , E, Ẽ is diagonalised in the chosen basis of generators, but the adjoint action of Ñ acts non semisimply on adjoint module. The obvious triangular decomposition g̃l(1|1) = span { ψ−, ψ̃− } ⊕ span { N,E, Ñ , Ẽ } ⊕ span { ψ+, ψ̃+ } gives rise to highest weight modules: the highest weight is defined as an eigenvector of N , E, Ñ , Ẽ with eigenvalues n, e, ñ, ẽ correspondingly, which is annihilated by the action of raising generators ψ+, ψ̃+. The highest weight Verma module is defined as a free action of lowering generators ψ−, ψ̃− on the highest weight. Since ψ−, ψ̃− are Grassmann odd and anticommute, the Verma module is four-dimensional. All the states of the Verma module with highest weight |v〉 are eigenstates of Ñ except for ψ−|v〉 on which it acts non semisimply: Ñψ−|v〉 = ñψ−|v〉 − ψ̃−|v〉. As it was shown in [4], there are three possibilities for irreducible quotients of Verma module. If e = ẽ = 0, the irreducible quotient is one-dimensional and was called atypical, if ẽ = 0, but e 6= 0, it is two-dimensional and was called semitypical, and if ẽ 6= 0, the Verma module is irreducible. In a similar way one can define lowest weight Verma modules and their irreducible quotients. There are two linearly independent quadratic Casimir operators in the universal enveloping algebra of g̃l (1|1) (modulo polynomials in central elements) which can be chosen as Q1 = NẼ + ÑE + ψ−ψ̃+ + ψ̃−ψ+, Q2 = ÑẼ + ψ̃−ψ̃+. They act on the highest weight Verma module as multiplication by nẽ+ ñe, ñẽ respectively. An important object for affinization of Lie (super)algebras is a non-degenerate symmetric bilinear form κ on the algebra. Among different possibilities, we choose the lifting of the standard bilinear form κ0(X,Y ) of gl(1|1) (defined as supertrace str(XY ) in the defining representation of it) as follows κ̃(X̃, Y ) = κ0(X,Y ), κ̃(X̃, Ỹ ) = κ̃(X,Y ) = 0. Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) 5 2.2 Invariant vector fields We choose a Lie supergroup parameterization according to above triangular decomposition g = eiθ+ψ ++iθ̃+ψ̃+ eixE+iyN+ix̃Ẽ+iỹÑeiθ−ψ −+iθ̃−ψ̃− . (2.1) Invariant vector fields on Lie supergroups have to be treated with attention to signs to (see [8]). We define them by R+g = ψ+g, R̃+g = ψ̃+g, RXg = −Xg for all X in the complement of the ψ+, ψ̃+. This definition guarantees that the invariant vector fields obey the relations of g̃l (1|1) RXRY − (−1)|X||Y |RYRX = R[X,Y ]. Then a computation reveals that ψ+g = −i d dθ+ g, ψ̃+g = −i d dθ̃+ g, Eg = −i d dx g, Ẽg = −i d dx̃ g, Ng = −i d dy g + iθ+ψ +g + iθ̃+ψ̃ +g, Ñg = −i d dỹ g + iθ+ψ̃ +g, (2.2) ψ−g = −ieiy d dθ− g + iỹψ̃−g − iθ+Eg − iθ̃+Ẽg − ỹθ+Ẽg, ψ̃−g = −ieiy d dθ̃− g − iθ+Ẽg. So that R+ = −i d dθ+ , R̃+ = −i d dθ̃+ , RE = i d dx , R̃E = i d dx̃ , RN = i d dy − θ+ d dθ+ − θ̃+ d dθ̃+ , R̃N = i d dỹ − θ+ d dθ̃+ , R− = ieiy ( d dθ− + iỹ d dθ̃− ) − θ+ d dx − θ̃+ d dx̃ , R̃− = ieiy d dθ̃− − θ+ d dx̃ . In the same way for the left-action we require L−g = −gψ−, L̃−g = −gψ̃− and LXg = Xg for all X in the complement of the ψ−, ψ̃−. Then one gets gψ− = −i d dθ− g, gψ̃− = −i d dθ̃− g, gE = −i d dx g, gẼ = −i d dx̃ g, gN = −i d dy g + θ− d dθ− g + θ̃− d dθ̃− g, gÑ = −i d dỹ g + θ− d dθ̃− g, gψ+ = −ieiy ( d dθ+ + iỹ d dθ̃+ ) g − θ− d dx g − θ̃− d dx̃ g, gψ̃+ = −ieiy d dθ̃+ g − θ− d dx̃ g. So that for the left-action we have L− = i d dθ− , L̃− = i d dθ̃− , LE = −i d dx , L̃E = −i d dx̃ , LN = −i d dy + θ− d dθ− + θ̃− d dθ̃− , L̃N = −i d dỹ + θ− d dθ̃− , L+ = −ieiy ( d dθ+ + iỹ d dθ̃+ ) − θ− d dx − θ̃− d dx̃ , L̃+ = −ieiy d dθ̃+ − θ− d dx̃ . 6 A. Babichenko and T. Creutzig 2.3 The Haar measure The left-invariant Maurer–Cartan form is ω(g) = g−1dg, and the right-invariant Maurer–Cartan form is ω(g−1). Either one can be taken to compute the Haar measure and we will use the right-invariant one ω ( g−1 ) = −dgg−1 = − ( d dθ+ g ) g−1dθ+ − ( d dθ̃+ g ) g−1dθ̃+ − · · · . This is a Lie superalgebra valued one form, that is it can be written as ω ( g−1 ) = ω ( ψ+ ) ψ+ + ω ( ψ̃+ ) ψ̃+ + ω(E)E + ω ( Ẽ ) Ẽ + ω(N)N + ω ( Ñ ) Ñ + ω ( ψ− ) ψ− + ω ( ψ̃− ) ψ̃−. The components are the dual one-forms in our basis. Using (2.2) they can be easily extracted ω ( ψ+ ) = −idθ+ − θ+dy, ω ( ψ̃+ ) = −idθ̃+ − θ̃+dy − θ+dỹ, ω(E) = −idx+ e−iyθ+dθ−, ω ( Ẽ ) = −idx̃+ e−iy ( θ̃+ − iỹθ+ ) dθ− + e−iyθ+dθ̃−, ω(N) = −idy, ω(Ñ) = −idỹ, ω ( ψ− ) = −ie−iydθ−, ω ( ψ̃− ) = −ie−iydθ̃− − e−iyỹdθ−. The right-invariant measure is the wedge product of these dual one forms µ ( g−1 ) = ω ( ψ+ ) ∧ ω ( ψ̃+ ) ∧ ω(E) ∧ ω ( Ẽ ) ∧ ω(N) ∧ ω ( Ñ ) ∧ ω ( ψ− ) ∧ ω ( ψ̃− ) = e−2iydθ+ ∧ dθ̃+ ∧ dx ∧ dx̃ ∧ dy ∧ dỹ ∧ dθ− ∧ dθ̃−. Here, we used graded anti-symmetry of the wedge product as well as integration with respect to an odd variable is the same as taking the derivative, but the double derivative of an odd variable vanishes, in other words we have used dθ ∧ dθ = 0. 2.4 The Laplace operators In [4] two Casimir operators of g̃l (1|1) were given. We change one of them by the central element Ẽ. They are then Q1 = NẼ + ÑE + ψ−ψ̃+ + ψ̃−ψ+ − Ẽ, Q2 = ÑẼ + ψ̃−ψ̃+. We compute that ∆1 := QL 1 = QR 1 = − d dy d dx̃ − i d dx̃ − d dỹ d dx + eiy ( d dθ− d dθ̃+ + d dθ̃− d dθ+ + iỹ d dθ̃− d dθ̃+ ) , ∆2 := QL 2 = QR 2 = − d dỹ d dx̃ + eiy d dθ̃− d dθ̃+ . We call these operators ∆ as they are the Laplace operators on this supergroup. This finishes our preparations, and we can turn to decomposing functions with respect to the left-right action of invariant vector fields. Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) 7 2.5 Decomposing functions Before we start decomposing functions, let us observe how left and right action are related, Proposition 2.1. The change of coordinates β : θ± 7→ −θ∓, θ̃± 7→ −θ̃∓, x 7→ −x, x̃ 7→ −x̃, y 7→ y, ỹ 7→ ỹ relates the action of left- and right-invariant vector fields as R± 7→ L∓, R̃± 7→ L̃∓, RE 7→ LE , R̃E 7→ L̃E , RN 7→ −LN , R̃N 7→ −L̃N . It will turn out that there are three-types of modules appearing in the decomposition. We will call these modules typical, semi-typical and atypical as in [4]. 2.5.1 Typical modules Let fe,n,ẽ,ñ(x, x̃, y, ỹ) = exp(ixe+ ix̃ẽ− iyn− iñỹ), then the crucial lemma is: Lemma 2.2. Define the matrix M =  1 θ− θ̃− θ−θ̃− θ+ θ+θ− θ+θ̃− θ+θ−θ̃− θ̃+ θ̃+θ− θ̃+θ̃− θ̃+θ−θ̃− θ+θ̃+ θ+θ̃+θ− θ+θ̃+θ̃− θ+θ̃+θ−θ̃−  and the second-order differential operator De,ẽ = d dθ̃− d dθ+ + d dθ− d dθ̃+ + ( iỹ − e ẽ ) d dθ̃− d dθ̃+ . Let ẽ 6= 0, then each row of the matrix Ve,n,ẽ,ñ = fe,n,ẽ,ñexp ( eiy ẽ De,ẽ ) M carries the irreducible highest-weight representation of the left-regular action of highest-weight (e, ẽ,−n + 2,−ñ). Each column transforms in the irreducible highest-weight representation of the right-regular action of highest-weight (−e,−ẽ, n, ñ). Proof. We define the function f4 = θ−θ̃−fe,n,ẽ,ñ, thenf4 is a highest-weight vector for both left- and right-action. The weight under the right-action is (−e,−ẽ, n, ñ) and the weight under the left-action is (e, ẽ,−n+ 2,−ñ). Then under the left-action, the functions f1 := fe,n,ẽ,ñ(x, x̃, y, ỹ), f2 := θ−fe,n,ẽ,ñ, f3 := θ̃−fe,n,ẽ,ñ, f4 := θ−θ̃−fe,n,ẽ,ñ, carry the four-dimensional irreducible representation of that highest-weight. Since left- and right-action commute, each of these states must be a highest-weight vector of same weight as before for the right-action. For the highest-weight f1 the remaining three states of the typical module I are ψ̃I = R̃−f1 = iẽθ+fe,n,ẽ,ñ, ψI = R−f1 = ieθ+fe,n,ẽ,ñ + iẽθ̃+fe,n,ẽ,ñ, bI = R−R̃−f1 = −ẽ2θ+θ̃+fe,n,ẽ,ñ. 8 A. Babichenko and T. Creutzig In the same way three states completing the module II with the highest-weight f2 are ψ̃II = R̃−f2 = −iẽθ−θ+fe,n,ẽ,ñ, ψII = R−f2 = ife,n−1,ẽ,ñ − ieθ−θ+fe,n,ẽ,ñ − iẽθ−θ̃+fe,n,ẽ,ñ, bII = R−R̃−f2 = ẽθ+fe,n−1,ẽ,ñ + ẽ2θ−θ+θ̃+fe,n,ẽ,ñ. For the module III with the highest-weight f3 we get ψ̃III = R̃−f3 = ife,n−1,ẽ,ñ + iẽfe,n,ẽ,ñθ+θ̃−, ψIII = R−f3 = −ỹfe,n−1,ẽ,ñ − iefe,n,ẽ,ñθ̃−θ+ − iẽfe,n,ẽ,ñθ̃−θ̃+, bIII = R−R̃−f3 = −efe,n−1,ẽ,ñθ+ − ẽfe,n−1,ẽ,ñθ̃+ + iẽỹfe,n−1,ẽ,ñθ+ + ẽ2fe,n,ẽ,ñθ̃−θ+θ̃+, and for the last module IV with highest-weight f4 ψ̃IV = R̃−f4 = −ife,n−1,ẽ,ñθ− + iẽfe,n,ẽ,ñθ−θ̃−θ+, ψIV = R−f4 = ỹfe,n−1,ẽ,ñθ− + ife,n−1,ẽ,ñθ̃− + iefe,n,ẽ,ñθ−θ̃−θ+ + iẽfe,n,ẽ,ñθ−θ̃−θ̃+, bIV = R−R̃−f4 = fe,n−2,ẽ,ñ − ẽfe,n−1,ẽ,ñθ̃−θ+ + (iỹẽ− e)fe,n−1,ẽ,ñθ−θ+ + ẽfe,n−1,ẽ,ñθ̃+θ− + ẽ2fe,n,ẽ,ñθ−θ̃−θ+θ̃+. The action of RN is the same on all four four modules: RNfi = nfi, RNψ = (n− 1)ψ, RN ψ̃ = (n− 1)ψ̃, RNb = (n− 2)b. The action of R̃N is diagonal on the states X = fi, ψ̃, b: R̃NX = ñX, but non diagonal on the states ψ: R̃Nψ = ñψ − ψ̃. In summary, for ẽ 6= 0, each row of the matrix fe,n,ẽ,ñ  1 θ− θ̃− θ−θ̃− ẽθ+ ẽθ−θ+ ẽθ+θ̃− −ẽθ−θ̃−θ+ eθ+ + ẽθ̃+ −eθ−θ+ − ẽθ−θ̃+ eθ̃−θ+ + ẽθ̃−θ̃+ eθ−θ̃−θ+ + ẽθ−θ̃−θ̃+ ẽ2θ+θ̃+ ẽ2θ−θ+θ̃+ ẽ2θ̃−θ+θ̃+ ẽ2θ−θ̃−θ+θ̃+  + fe,n−1,ẽ,ñ  0 0 0 0 0 0 1 θ− 0 1 −iỹ −iỹθ− + θ̃− 0 ẽθ+ −eθ+ − ẽθ̃+ + iẽỹθ+ −ẽθ̃−θ+ + (iỹẽ− e)θ−θ+ + ẽθ̃+θ−  + fe,n−2,ẽ,ñ  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  carries the irreducible highest-weight representation of the left regular action of highest-weight (e, ẽ,−n + 2,−ñ). Each column transforms in the irreducible highest-weight representation of the right regular action of highest-weight (−e,−ẽ, n, ñ). Changing basis ψ → ψ − e ẽ ψ̃, this can be written more symmetrically, fe,n,ẽ,ñ  1 θ− θ̃− θ−θ̃− θ+ θ+θ− θ+θ̃− θ+θ−θ̃− θ̃+ θ̃+θ− θ̃+θ̃− θ̃+θ−θ̃− θ+θ̃+ θ+θ̃+θ− θ+θ̃+θ̃− θ+θ̃+θ−θ̃−  Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) 9 + fe,n−1,ẽ,ñ ẽ  0 0 0 0 0 0 1 −θ− 0 1 ( iỹ − e ẽ ) −θ− ( iỹ − e ẽ ) + θ̃− 0 θ+ θ+ ( iỹ − e ẽ ) − θ̃+ ( iỹ − e ẽ ) θ−θ+ − θ̃−θ+ + θ̃+θ−  + fe,n−2,ẽ,ñ ẽ2  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  , so that the lemma follows. � Remark 2.3. The operator eiyDe,ẽ is the semi-classical analouge of a screening charge. It is related to the Casimir operators acting on Ve,n,ẽ,ñ via 1 ẽ ( ∆1 − e ẽ ∆2 − eiyDe,ẽ ) = −i d dy + 1. In other words the difference of the Casimir ∆1 − e ẽ∆2 and eiyDe,ẽ is a semi-simple operator. Finally, we define the matrices V (m) e,n,ẽ,ñ := ỹmVe,n,ẽ,ñ, and denote the vector space spanned by its components by T (m) e,n,ẽ,ñ, so that Te,n,ẽ,ñ := ∞⊕ m=0 T (m) e,n,ẽ,ñ is an infinite-dimensional module for the left-right action of g̃l (1|1). The irredicible submodule T (0) e,n,ẽ,ñ has been described in Lemma 2.2. The modules N⊕ m=0 T (m) e,n,ẽ,ñ are all submodules with quotient by N−1⊕ m=0 T (m) e,n,ẽ,ñ isomorphic to T (0) e,n,ẽ,ñ. 2.5.2 Semitypical modules The case ẽ = 0 and e 6= 0 is called semi-typical. Our strategy is to first decompose into irreducible gl (1|1) modules. First, fix e 6= 0, ñ. In order to avoid too many indices we take the short-hand notation fn := fe,n,0,ñ. We find the following list for the left-action L± on weight modules: A1,n = span(fn, θ−fn), A2,n = span(θ+fn, ieθ−θ+fn − ifn−1), A3,n = span ( θ̃+fn, ieθ−θ̃+fn + ỹfn−1 ) , A4,n = span ( θ̃−fn, θ−θ̃−fn ) , A5,n = span ( θ+θ̃+fn, ieθ−θ+θ̃+fn − iθ̃+fn−1 − ỹθ+fn−1 ) , A6,n = span ( θ̃−θ̃+fn, ieθ̃−θ−θ̃+fn + ỹθ̃−fn−1 ) , A7,n = span ( θ̃−θ+fn, ieθ̃−θ−θ+fn − iθ̃−fn−1 ) , A8,n = span ( θ̃−θ+θ̃+fn, ieθ̃−θ−θ+θ̃+fn − iθ̃−θ̃+fn−1 − ỹθ̃−θ+fn−1 ) . 10 A. Babichenko and T. Creutzig We then also define for non-negative integer m A (m) i,n := ỹmAi,n. The L̃± action gives gl (1|1)-module one-to-one maps, that only change the LN -action, L̃+ : A (m) 3,n → A (m) 1,n−1, L̃− : A (m) 4,n → A (m) 1,n , L̃+ : A (m) 5,n → A (m) 2,n−1, L̃− : A (m) 6,n → A (m) 3,n , L̃+ : A (m) 6,n → A (m) 4,n−1, L̃− : A (m) 7,n → A (m) 2,n , L̃+ : A (m) 8,n → A (m) 7,n−1, L̃− : A (m) 8,n → A (m) 5,n . The L̃N -action on gl (1|1)-modules is described as follows. L̃N splits into a semi-simple and a nilpotent part. The semi-simple part acts by multiplication by −ñ, while the nilpotent part acts as follows L̃nil N : A (m) i,n → A (m−1) i,n ⊕  0 if i = 1, 2, A (m) 1,n−1 if i = 3, A (m) 1,n if i = 4, A (m) 2,n−1 if i = 5, A (m) 3,n ⊕A (m) 4,n−1 ⊕A (m+1) 1,n−1 if i = 6, A (m) 1,n−1 ⊕A (m) 2,n if i = 7, A (m) 5,n ⊕A (m) 7,n−1 ⊕A (m) 3,n−1 ⊕A (m+1) 2,n−1 ⊕A (m+1) 1,n−2 if i = 8. By this we mean that the gl (1|1) action on the image of the nilpotent part is the right-hand side. We also understand A (−1) i,n = 0. An efficient way to picture these computations is the following Proposition 2.4. Let ẽ = 0 and e 6= 0, then under the left-regular action as s̃l (1|1) modules, we have the following two types of 8-dimensional s̃l (1|1)-modules, S (m) n,a A (m) 4,n−1 A (m) 6,n A (m) 1,n−1 A (m) 3,n L̃− L̃−L̃+ L̃+ and S (m) n,b A (m) 7,n−1 A (m) 8,n A (m) 2,n−1 A (m) 5,n L̃− L̃−L̃+ L̃+ Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) 11 While LN -acts semi-simple on these modules, the nilpotent part of L̃N acts as L̃nil N : S(m) n,a → S(m−1) n,a ⊕ S(m) n,a ⊕ S(m+1) n,a , L̃nil N : S (m) n,b → S (m−1) n,b ⊕ S(m) n,b ⊕ S (m+1) n,b ⊕ S(m) n,a ⊕ S(m+1) n,a . However, with a suitable change of basis the situation can be improved. Define Ǎ (m) 6,n := span ( ỹm ( 1 2 ỹ2fn−1 − efnθ̃−θ̃+, eỹfn−1θ̃− + i 2 ỹ2fn−1θ− − iefnθ−θ̃−θ̃+ )) , Ǎ (m) 8,n := span ( ỹm ( iefnθ̃−θ+θ̃+ + ỹfn−1θ̃+ − i 2 ỹ2fn−1θ+, 1 2 ỹ2fn−2 + e 2 ỹ2fn−1θ−θ+ + ieỹfn−1(θ̃−θ+ + θ−θ̃+)− efn−1θ̃−θ̃+ − e2fnθ−θ̃−θ+θ̃+ )) , Ǎ (m) 7,n := span ( ỹm ( efnθ̃−θ+ − iỹfn−1, ỹfn−1θ− + ifn−1θ̃− + ieθ−θ̃−θ+ )) , Ǎ (m) i,n := A (m) i,n for i = 1, 2, 3, 4, 5. This redefinition preserves the s̃l (1|1) action. However the nilpotent part of L̃N acts as L̃nil N : Ǎ (m) i,n → Ǎ (m−1) i,n ⊕  0 if i = 1, 2, Ǎ (m) 1,n−1 if i = 3, Ǎ (m) 1,n if i = 4, Ǎ (m) 2,n−1 if i = 5, Ǎ (m) 3,n ⊕ Ǎ (m) 1,n−1 if i = 6, Ǎ (m) 2,n if i = 7, Ǎ (m) 5,n ⊕ Ǎ (m) 2,n−1 if i = 8. (2.3) We thus can considerably improve the previous proposition. Theorem 2.5. Let ẽ = 0 and e 6= 0, then under the left-regular action of g̃l (1|1), we have the following modules, Š (m) n,a Ǎ (m) 4,n−1 Ǎ (m) 6,n Ǎ (m) 1,n−1 Ǎ (m) 3,n L̃− L̃−L̃+ L̃+ and Š (m) n,b Ǎ (m) 7,n−1 Ǎ (m) 8,n Ǎ (m) 2,n−1 Ǎ (m) 5,n L̃− L̃−L̃+ L̃+ 12 A. Babichenko and T. Creutzig While LN -acts semi-simple on these modules, the nilpotent part of L̃N acts as described in equation (2.3). We can split it into two pieces L̃nil N = Ña + Ñb, such that Ña : Ǎ (m) i,n → Ǎ (m−1) i,n and Ñb : Ǎ (m) i,n →  0 if i = 1, 2, Ǎ (m) 1,n−1 if i = 3, Ǎ (m) 1,n if i = 4, Ǎ (m) 2,n−1 if i = 5, Ǎ (m) 3,n ⊕ Ǎ (m) 1,n−1 if i = 6, Ǎ (m) 2,n if i = 7, Ǎ (m) 5,n ⊕ Ǎ (m) 2,n−1 if i = 8. The long-exact sequence describes the action of Ña, · · · −→ Š(m) n,c −→ Š(m−1) n,c −→ . . . −→ Š(1) n,c −→ Š(0) n,c −→ 0 for c ∈ {a, b}. In order to understand the left-right action, we first have to understand the left-right gl (1|1) action. Definition 2.6. The highest-weight representation of gl (1|1), with highest-weight (e, n) and lowest-weight (e, n− 1) will be denoted by ρe,n− 1 2 . We compute: Under the left-right gl (1|1) action the modules decompose as X (m) 2,n := Ǎ (m) 6,n ⊕ Ǎ (m) 8,n = ρL e,−n+ 3 2 ⊗ ρR−e,n− 3 2 , X (m) −,n := Ǎ (m) 3,n ⊕ Ǎ (m) 5,n = ρL e,−n+ 1 2 ⊗ ρR−e,n− 3 2 , X (m) +,n−1 := Ǎ (m) 4,n−1 ⊕ Ǎ (m) 7,n−1 = ρL e,−n+ 5 2 ⊗ ρR−e,n− 3 2 , X (m) 0,n−1 := Ǎ (m) 1,n−1 ⊕ Ǎ (m) 2,n−1 = ρL e,−n+ 3 2 ⊗ ρR−e,n− 3 2 , where the last three decompositions follow from the first one and the first one is a straight- forward computation. Further, the R̃± action gives gl (1|1)-module one-to-one maps, that only change the RN -action, R̃− : A (m) 4,n → A (m) 1,n−1, R̃+ : A (m) 3,n → A (m) 1,n , R̃− : A (m) 6,n → A (m) 3,n−1, R̃+ : A (m) 5,n → A (m) 2,n , R̃− : A (m) 7,n → A (m) 2,n−1, R̃+ : A (m) 6,n → A (m) 4,n , R̃− : A (m) 8,n → A (m) 5,n−1, R̃+ : A (m) 8,n → A (m) 7,n . We thus get the picture for left-right modules given in Fig. 1. Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) 13 X (m) 2,n+1X (m) 2,nX (m) 2,n−1 X (m) 0,nX (m) 0,n−1X (m) 0,n−2 · · · · · · · · · X (m) +,nX (m) −,nX (m) +,n−1X (m) −,n−1 · · · · · · · · · Figure 1. The solid arrows to the right denote the action of L̃−, those to the left of L̃+. While the dotted arrows to the right indicate the action of R̃+ and those to the left the one of R̃−. 2.5.3 Atypical modules Let now ẽ = e = 0. We will start by studying the left-right gl (1|1) action. We define w(2,2),n = θ−θ̃−θ+θ̃+fn, w(2,−),n = θ̃−θ+θ̃+fn, w(2,+),n = ( θ−θ̃−θ̃+ − iỹθ−θ̃−θ+ ) fn, w(2,0),n = ( θ̃−θ̃+ − iỹθ̃−θ+ ) fn, w(−,2),n = ( θ̃−θ+θ̃+ − iỹθ−θ+θ̃+ ) fn, w(−,−),n = ỹθ+θ̃+fn, w(−,+),n = ( θ̃−θ̃+ − iỹ ( θ−θ̃+ + θ̃−θ+ ) − ỹ2θ−θ+ ) fn, w(−,0),n = ỹ ( θ̃+ − iỹθ+ ) fn, w(+,2),n = θ−θ̃−θ̃+fn, w(+,−),n = θ̃−θ̃+fn, w(+,+),n = ỹθ−θ̃−fn, w(+,0),n = ỹθ̃−fn, w(0,2),n = ( θ̃−θ̃+ − iỹθ−θ̃+ ) fn, w(0,−),n = ỹθ̃+fn, w(0,+),n = ỹ ( θ̃− − iỹθ− ) fn, w(0,0),n = ỹ2fn. Further, define the one-dimensional vector spaces w (m) (a,b),n = span ( ỹw(a,b),n ) . We compute the action of the left- and right-invariant vector fields of the gl (1|1) subalgebra. The result can be best visualized in the following diagram w (m) (a,2),n w (m) (a,+),n−1 w (m) (a,−),n w (m) (a,0),n−1 L− L−L+ L+ w (m) (2,b),n w (m) (+,b),n w (m) (−,b),n−1 w (m) (0,b),n−1 R− R−R+ R+ so that we observe Proposition 2.7. Under the left- and right-invariant vector fields of the gl (1|1) subalgebra the vector space B(m) n := ⊕ a∈{2,+} b∈{2,−} w (m) (a,b),n ⊕ ⊕ a∈{2,+} b∈{+,0} w (m) (a,b),n−1 ⊕ ⊕ a∈{−,0} b∈{2,−} w (m) (a,b),n−1 ⊕ ⊕ a∈{−,0} b∈{+,0} w (m) (a,b),n−2 is the tensor product of gl (1|1) projective covers, B(m) n ∼= PL2−n ⊗ PRn−2. 14 A. Babichenko and T. Creutzig U (m) n U (m) n+1 U (m) n+2 V (m) n−1 V (m) n V (m) n+1 U (m) n−1 V (m) n−2 · · · · · · · · · · · · Figure 2. The solid arrows denote the action of R−, the dotted ones the one of L−. We define Pn(m) := ⊕ a∈Z B (m) n+a. The second type of gl (1|1) modules appearing are given by defining the following u (m) 2,n := span ( ỹmθ−θ+θ̃+fn ) , u (m) R,n := span ( ỹmθ−θ̃+fn ) , u (m) L,n := span ( ỹm ( θ−θ̃+ − iỹθ−θ+ ) fn ) , u (m) 0,n := span ( ỹmỹθ−fn ) , v (m) 2,n := span ( ỹmθ+θ̃+fn ) , v (m) R,n := span ( ỹmθ̃+fn ) , v (m) L,n := span ( ỹm ( θ̃+ − iỹθ+ ) fn ) , v (m) 0,n := span ( ỹmỹfn ) . Again, we compute the action of the left- and right-invariant vector fields of the gl (1|1) sub- algebra. The result can be best visualized by diagrams. First, we restrict to the action of R+ and L+, we then get the modules for this subalgebra of gl (1|1)⊕ gl (1|1) u (m) 2,n u (m) R,n u (m) L,n−1 u (m) 0,n−1 L+ L+R+ R+ v (m) 2,n v (m) R,n v (m) L,n−1 v (m) 0,n−1 L+ L+R+ R+ We call the first one U (m) n and the second one V (m) n . The vector fields L− and R− act trivially on the second one while the first one gives one-to-one maps illustrated in Fig. 2. Thus the set P−n (m) := ⊕ a∈Z U (m) n+a ⊕ ⊕ b∈Z V (m) n+b forms an infinite-dimensional gl (1|1) left-right module called P−n (m) . The third type of gl (1|1) modules appearing is very similar to the previous one and are given by defining the following s (m) 2,n := span ( ỹmθ−θ̃−θ+fn ) , s (m) L,n := span ( ỹmθ̃−θ+fn ) , s (m) R,n := span ( ỹm ( θ̃−θ+ − iỹθ−θ+ ) fn ) , s (m) 0,n := span ( ỹmỹθ+fn ) , t (m) 2,n := span ( ỹmθ−θ̃−fn ) , t (m) L,n := span ( ỹmθ̃−fn ) , t (m) R,n := span ( ỹm ( θ̃− − iỹθ− ) fn ) , t (m) 0,n := span ( ỹmỹfn ) . Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) 15 S (m) n S (m) n+1 S (m) n+2 T (m) n−1 T (m) n T (m) n+1 S (m) n−1 T (m) n−2 · · · · · · · · · · · · Figure 3. The solid arrows denote the action of L+, the dotted ones the one of R+. r (m) 2,n+1r (m) 2,nr (m) 2,n−1 r (m) 0,nr (m) 0,n−1r (m) 0,n−2 · · · · · · · · · r (m) +,nr (m) −,nr (m) +,n−1r (m) −,n−1 · · · · · · · · · Figure 4. The solid arrows to the right denote the action of L−, those to the left of L+. While the dotted arrows to the right indicate the action of R+ and those to the left the one of R−. Again, we compute the action of the left- and right-invariant vector fields of the gl (1|1) subal- gebra. The result can also again be best visualized by diagrams. First, we restric to the action of R− and L−, we then get the modules for this subalgebra of gl (1|1)⊕ gl (1|1) s (m) 2,n s (m) R,n−1 s (m) L,n s (m) 0,n−1 L− L−R− R− t (m) 2,n t (m) R,n−1 t (m) L,n t (m) 0,n−1 L− L−R− R− We call the first one S (m) n and the second one T (m) n . The vector fields L+ and R+ act trivially on the second one while the first one gives one-to-one maps illustrated in Fig. 3. Thus the set P+ n (m) := ⊕ a∈Z S (m) n+a ⊕ ⊕ b∈Z T (m) n+b forms an infinite-dimensional gl (1|1) left-right module called P+ n (m) . The forth type of gl (1|1) module is given by r (m) 2,n := span ( ỹmθ−θ+fn ) , r (m) −,n := span ( ỹmθ+fn ) , r (m) +,n := span ( ỹmθ−fn ) , r (m) 0,n := span ( ỹmfn ) . Then as a left-right module we get the picture as in Fig. 4. We denote this module by P0 n (m) . Observe that we have the inclusion of gl (1|1) left-right modules P0 n (m) ⊂ P+ n (m−1) ⊕ P−n (m−1) , P±n (m) ⊂ Pn(m−1) 16 A. Babichenko and T. Creutzig for m > 0. Especially as a gl (1|1) left-right module the atypical part, that is the e = ẽ = 0 part of the space of functions on g̃l (1|1) decomposes as ∫ 1 0 ( P0 n (0) ⊕ P1 n (0) ⊕ ∞⊕ m=1 Pn(m) ) dn. Here P1 n (0) denotes the module generated by P+ n (0) and P−n (0) whose intersection is the submod- ule P0(1) . These modules are constructed such that the tilded action is easy to read of, the L̃± and R̃± are sl (1|1) module homomorphism (they only change the action of LN or RN ) and for m > 0 they act as Image ( L̃± : Pn(m) −→ Pn(m−1) ) = Image ( R̃± : Pn(m) −→ Pn(m−1) ) = P±n (m) , Image ( L̃± : P∓n (m) −→ P∓n (m−1)) = Image ( R̃± : P∓n (m) −→ P∓n (m−1)) = P0 n (m) , while in the m = 0 case we have Image ( L̃± : Pn(0) −→ P±n (0)) = Image ( R̃± : Pn(0) −→ P±n (0)) = P±n (0) , Image ( L̃± : P∓n (0) −→ P0 n (0)) = Image ( R̃± : P∓n (0) −→ Pn(0) ) = P0 n (0) . The image of these maps is in the kernel and we inspect that it is precisely the kernel, hence both L̃+ and R̃+ action is described by the long-exact sequences · · · −→ Pn(m) −→ Pn(m−1) −→ . . . −→ Pn(1) −→ Pn(0) −→ P+ n (0) −→ 0, · · · −→ P−n (m) −→ P−n (m−1) −→ . . . −→ P−n (1) −→ P−n (0) −→ P0 n (0) −→ 0, while the one of L̃− and R̃− action is described by the long-exact sequences · · · −→ Pn(m) −→ Pn(m−1) −→ . . . −→ Pn(1) −→ Pn(0) −→ P−n (0) −→ 0, · · · −→ P+ n (m) −→ P+ n (m−1) −→ . . . −→ P+ n (1) −→ P+ n (0) −→ P0 n (0) −→ 0. The action of L̃N and R̃N is uniquely specified by its action on the top components, the map can be split into two components L̃N = L̃aN +L̃bN and R̃N = R̃aN +R̃bN such that L̃aN and R̃aN are both one-to-one maps from x (m) i to x (m−1) i for all x in {s, t, u, v, w} and i labelling the corresponding indices. More precisely they act as multiplication by ±im (+ for R̃ and − for L̃) times ỹ−1. Especially they act trivially when m = 0. The images of the second part are (for m 6= 0) Image ( L̃bN : Pn(m) −→ Pn(m−1) ) = Image ( R̃bN : Pn(m) −→ Pn(m−1) ) = P1 n (m) , Image ( L̃bN : P±n (m) −→ P±n (m−1)) = Image ( R̃bN : P±n (m) −→ P±n (m−1)) = P0 n (m) , Image ( L̃bN : P0 n (m) −→ P0 n (m−1)) = Image ( R̃bN : P0 n (m) −→ P0 n (m−1)) = 0, and for m = 0 Image ( L̃bN : Pn(0) −→ P1 n (0)) = Image ( R̃bN : Pn(0) −→ P1 n (0)) = P1 n (0) , Image ( L̃bN : P±n (0) −→ P0 n (0)) = Image ( R̃bN : P±n (0) −→ P0 n (0)) = P0 n (0) , Image ( L̃bN : P0 n (0) −→ P0 n (0)) = Image ( R̃bN : P0 n (0) −→ P0 n (0)) = 0. Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) 17 3 Free field realization In this section we construct free field realization of the holomorphic and anti-holomorphic cur- rents of affine Takiff super Lie algebra ˆ̃ gl (1|1), and its modules. Such free field realizations are constructed by starting with a triangular decomposition of a Lie supergroup element as in (2.1). This procedure is outlined in [39], and we can immediately infer the bulk screening charge from (3.2.32) of [8], as well as the currents in the free field realization from (3.2.39) of that work. We however still need to prove that the screening charge has the desired properties. The harmonic analysis is expected to be a semi-classical (large level) limit of the screened free field theory. Examples on this for type one Lie supergroups are [12, 39, 44, 45]. The Takiff algebra admits an automorphism ωα for α in C. It is given by E 7→ E + αẼ, ψ+ 7→ ψ+ + αψ̃+ and leaving all others invariant. It allows us to restrict to the bilinear form κ̃. In order to see this, consider the bilinear form κβ := κ̃+ βκ0, then κ(ω−β(X), ω−β(Y )) = κ̃(X,Y ) for all X, Y in g̃l (1|1). In [4], the affinization of g̃l (1|1) was discussed and levels k̃ and k associated to the bilinear forms κ̃ and κ were introduced. The above automorphism relates the affine vertex algebra of g̃l (1|1) at levels (k̃, k) to the one at levels (k̃, 0). 3.1 Currents and screening operator Consider a holomorphic bc-ghost vertex algebra taking values in the odd part of the Takiff superalgebra. Its components have OPE b(z)c̃(w) ∼ b̃(z)c(w) ∼ 1 (z − w) and all other OPEs are regular. We also need free bosons (generators of the Heisenberg VOA) taking values in the even part of the Takiff superalgebra. Its components have OPE in their holomorphic part ∂X(z)∂Ỹ (w) ∼ ∂X̃(z)∂Y (w) ∼ −1/k (z − w)2 . In the same way we define anti holomorphic partners of the ghosts. The fields X(z, z̄), Y (z, z̄), X̃(z, z̄), Ỹ (z, z̄) will be considered as both holomorphic and anti holomorphic, i.e., depending on z and z̄, with the same OPEs in anti-holomorphic sector. Currents should be thought of as quantizations of invariant vector fields. We define holo- morphic currents by the formulas JN = −i∂X̃ + bc̃+ b̃c+ i∂Y, JE = −ik∂Ỹ , J+ = b̃, J− = k∂c̃− ikc̃∂Y − ikc∂Ỹ , J̃N = −i∂X + bc, J̃E = −ik∂Y, J̃+ = b, J̃− = k∂c− ikc∂Y. (3.1) and the anti holomorphic ones by J̄N = ik∂̄X̃ − i∂̄Y − b̄¯̃c− ¯̃ bc̄, J̄E = ik∂̄Ỹ , 18 A. Babichenko and T. Creutzig J̄+ = k∂̄ ˜̄c− ik˜̄c∂̄Y − ikc̄∂̄Ỹ , J̄− = −¯̃ b, ˜̄JN = ik∂̄X − b̄c̄, ˜̄JE = ik∂̄Y, ˜̄J+ = k∂̄c̄− ikc̄∂̄Y, ˜̄J− = −b̄. (3.2) By direct calculation one can see that Proposition 3.1. The OPEs of currents (3.1) are those of the level k with respect to κ̃ affine Takiff superalgebra, that is the only non-regular OPEs are JN (z)J±(w) ∼ ± J±(w) (z − w) , J+(z)J−(w) ∼ JE(w) (z − w) , JN (z)J̃E(w) ∼ k (z − w)2 , JN (z)J̃±(w) ∼ ± J̃±(w) (z − w) , J̃N (z)JE(w) ∼ k (z − w)2 , J̃N (z)J±(w) ∼ ± J̃±(w) (z − w) , J+(z)J̃−(w) ∼ k (z − w)2 + J̃E(w) (z − w) , J−(z)J̃+(w) ∼ − k (z − w)2 + J̃E(w) (z − w) . and the same set of OPEs of anti-holomorphic currents (3.2). One can substitute these currents into the energy-momentum tensor obtained in [4] (see also [36]) T (z) = 1 k : ( JN J̃E + JE J̃N − J+J̃− + J−J̃+ + 1 k J̃E J̃E ) : (z) getting T (z) = : ( ∂X̃∂Y + ∂Ỹ ∂X − k∂2Y − b̃∂c− b∂c̃) ) : (z), (3.3) which clearly has central charge zero. The key role in free field realization is played by screening operators. Theorem 3.2. The currents (3.1) lie in the joint kernel of the intertwiners S = ∮ b(z)eiY (z)dz, S̃ = ∮ ( iỸ (z)b(z) + b̃(z) ) eiY (z)dz. Proof. The computations are straightforward. We demonstrate here only action of S and S̃ on J− and J̃−. For S we get SJ−(w) = ∮ keiY (z) ( 1 (z − w)2 + −i∂Y (w) (z − w) + regular ) dz = ∮ ( eiY (w) k (z − w)2 + regular ) dz = 0, SJ̃−(w) = ∮ regular dz = 0, and for the action of S̃ it becomes S̃J−(w) = ∮ eiY (z) ( ikỸ (z) (z − w)2 − ik∂Ỹ (w)− kỸ (w)∂Y (w) (z − w) + regular ) dz Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) 19 = ∮ ( eiY (w) ikỸ (w) (z − w)2 + regular ) dz = 0, S̃J̃−(w) = ∮ eiY (z) ( k (z − w)2 − ik∂Y (w) (z − w) + regular ) dz = ∮ ( eiY (w) k (z − w)2 + regular ) dz = 0. � The same theorem can be proved for the anti-holomorphic part. We can now define bulk screening charge which involves both holomorphic and anti-holomorphic parts: Q = ∫ ( b̄(z̄)b̃(z) + ˜̄b(z̄)b(z) + ib̄(z̄)b(z)Ỹ (z, z̄) ) eiY (z,z̄)d2z. This Q should define correlation functions, which will not be considered in this paper. 3.2 Vertex operators We define the vertex operator Ve,n,ẽ,ñ = : eieX−inY+iẽX̃−iñỸ : , whose conformal dimension can be calculated with (3.3): ∆ = eñ+ nẽ+ ikẽ. (3.4) It transforms as a primary field corresponding to a highest-weight state, that is JE(z)Ve,n,ẽ,ñ(w) ∼ keVe,n,ẽ,ñ(w) (z − w) , JN (z)Ve,n,ẽ,ñ(w) ∼ (n− ẽ)Ve,n,ẽ,ñ(w) (z − w) , J̃E(z)Ve,n,ẽ,ñ(w) ∼ kẽVe,n,ẽ,ñ(w) (z − w) , J̃N (z)Ve,n,ẽ,ñ(w) ∼ ñVe,n,ẽ,ñ(w) (z − w) , J−(z)Ve,n,ẽ,ñ(w) ∼ ke : c(w)Ve,n,ẽ,ñ(w) : + ẽk : c̃(w)Ve,n,ẽ,ñ(w) : (z − w) , J̃−(z)Ve,n,ẽ,ñ(w) ∼ kẽ : c(w)Ve,n,ẽ,ñ(w) : (z − w) , J−(z) : c(w)Ve,n,ẽ,ñ(w) : ∼ kẽ : c̃(w)c(w)Ve,n,ẽ,ñ(w) : (z − w) . 3.3 Free field realization of typical modules We can now combine holomorphic and anti-holomorphic ghost fields with the vertex operator introduced above to form bulk free field realization for typical modules. It is convenient to represent it in the matrix form φe,n,ẽ,ñ(z) = : Ỹ mVe,n,ẽ,ñ : (z, z̄)  1 c̄ ˜̄c ˜̄cc̄ c cc̄ c˜̄c c˜̄cc̄ c̃ c̃c̄ c̃˜̄c c̃˜̄cc̄ cc̃ cc̃c̄ cc̃˜̄c cc̃˜̄cc̄  : . 20 A. Babichenko and T. Creutzig Here each row and column of the matrix C is understood as a span of its elements. Using the OPEs one can easily check that columns carry typical representations for the holomorphic currents and the rows – for the anti-holomorphic ones. The elements C1,i are the highest weights for the columns, and the elements Ci,1 – for the rows. For m = 0 they are Virasoro primary fields with conformal dimension (3.4). 3.4 A free field realization for atypical and semitypical modules Consider two pairs of symplectic fermions χ±, χ̃± with OPE χ+(z)χ̃−(w) ∼ χ−(z)χ̃+(w) ∼ k (z − w)2 and a pair of pairs of free bosons ∂X, ∂Y , ∂X̃, ∂Ỹ with OPE ∂X(z)∂Ỹ (w) ∼ ∂Y (z)∂X̃(w) ∼ 1 (z − w)2 . Recall, that symplectic fermions are a logarithmic conformal field theory. Especially, the vacuum is part of a larger indecomposable module whose four composition factors are all isomorphic to the vacuum itself, for details see [31]. The generating fields of these modules are Ω±(z, z̄), θ̃±(z, z̄), θ±(z, z̄) with the following behaviour under the action of the symplectic fermions: χ±(z)Ω±(w, w̄) ∼ kθ±(w, w̄) (z − w) , χ̃∓(z)Ω±(w, w̄) ∼ kθ̃∓(w, w̄) (z − w) , χ±(z)θ̃∓(w, w̄) ∼ k (z − w) , χ̃∓(z)θ±(w, w̄) ∼ − k (z − w) . We split the field Ỹ (z, z̄) into chiral and anti-chiral part, that is Ỹ (z, z̄) = Ỹ (z) + ˜̄Y (z̄). Then we define KE(z) = k∂Ỹ (z), KN (z) = ∂X̃(z), K̃E(z) = k∂Y (z), K̃N (z) = ∂X(z), K±(z) = e±Y (z) ( χ±(z)− Ỹ (z)χ̃±(z) ) , K̃±(z) = ∓e±Y (z)χ̃±(z). (3.5) and in analogy also anti-holomorphic currents. We compute Proposition 3.3. The OPEs of currents (3.5) are those of the level k with respect to κ̃ affine Takiff superalgebra. These OPEs are listed in Proposition 3.1. The energy-momentum tensor for these currents is T (z) = : ∂X̃(z)∂Y (z) : + : ∂Ỹ (z)∂X(z) : − 1 k ( :χ+(z)χ̃−(z) : + :χ−(z)χ̃+(z) : ) . These currents are in the kernel of Q := ∮ ( : χ̃+(z)θ̃+(z, z̄) : + : χ̃−(z)θ̃−(z, z̄) : + 2k∂X(z) ) dz acting on the chiral algebra generated under operator product by e±Y (z), Ỹ (z), χ±(z), χ̃±(z), ∂X(z), ∂X̃(z) and their derivatives. The currents are also invariant under the U(1)-action induced by ϕ which is defined by ϕ ( e±Y (z) ) = ∓e±Y (z), ϕ(χ±(z)) = ±χ±(z), ϕ(χ̃±(z)) = ±χ̃±(z). Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1) 21 As discussed in [4], semi-typical and atypical modules appear when ẽ/k = m ∈ Z. The vertex operator Ve,n,ẽ,ñ(z) := : e e k X(z)+nY (z)+ ẽ k X̃(z)+ñỸ (z) : has weight (e, n, ẽ, ñ) under the currents KE , KN , K̃E , K̃N . Define φ±m(z) := :χ±(z)∂χ±(z) · · · ∂mχ±(z) : , φ̃±m(z) := : χ̃±(z)∂χ̃±(z) · · · ∂mχ̃±(z) : . Then it is a computation to verify that for m ≤ 0 ϕe,n,mk,ñ(z) := Ve,n,mk,ñ(z)φ+ −m(z)φ̃+ −m is a primary field for the highest-weight representation of weight (e, n,mk, ñ) of the correct conformal dimension eñ/k +mn+m(m− 1). It is a submodule of the modules generated by ϕe,n,mk,ñ(z)θ+(z, z̄), ϕe,n,mk,ñ(z)θ̃+(z, z̄), ϕe,n,mk,ñ(z)θ+(z, z̄)θ̃+(z, z̄). If m ≥ 0, then we can construct analogous primary fields for lowest-weight representations by replacing ±, i.e., the primary field is ϕe,n,mk,ñ(z) := Ve,n,mk,ñ(z)φ−m(z)φ̃−m and it is a submodule of the modules generated by ϕe,n,mk,ñ(z)θ−(z, z̄), ϕe,n,mk,ñ(z)θ̃−(z, z̄), ϕe,n,mk,ñ(z)θ−(z, z̄)θ̃−(z, z̄). So in all these cases, we can use our free field realization to construct indecomposable but reducible modules. In the case of m = 0, we have both highest and lowest weight modules. These modules now deserve further investigation. We plan to study the combined left-right action of currents on these modules in the near future. The results then have to be compared to the semi-typicals of our harmonic analysis. Finally, let us note that there is a third free field realization simply by replacing Ỹ (z) and ∂X(z) by a βγ bosonic ghost algebra. This one will also be investigated later. 4 Conclusion This work together with [4] puts one in the situation to study conformal field theories with chiral algebra the affinization of g̃l (1|1). We have studied the harmonic analysis on the su- pergroup in detail and especially we have found that all irreducible modules are part of larger indecomposable but reducible modules. We then have found two free field realizations for the associated CFT. The next question is to understand the spectrum of the full CFT better. For this, one needs to study the free field modules in detail. Especially the structure of semi-typicals and atypicals from the symplectic fermion free field realization needs to be worked out. In log- arithmic CFT, indecomposable structure of modules is related to logarithmic singularities in correlation functions. Screening charges provide integral formula for correlation functions, so that one can use our findings to compute those. Since all our modules in the harmonic analysis are submodules of indecomposable but reducible ones we expect that logarithmic singularities will be a generic feature. This will be in strong contrast to logarithmic CFTs based on non Takiff supergroups. Our findings have the drawback, that the Takiff superalgebra forced us to allow for polyno- mials in the variable ỹ. In the conformal field theory this suggests to also allow for polynomials 22 A. Babichenko and T. Creutzig in the field ỹ. This is unusual, as, e.g., the free boson CFT does not allow for fields of analo- gous type. Of course the OPE involving such polynomials will automatically yield logarithmic singularities. As mentioned in the end of last section, a possible way out is to replace Ỹ and ∂Y by a βγ bosonic ghost algebra. In that case we would have no semi-typical modules but still typical and atypical ones. There are many future directions as boundary states and boundary correlation functions. These can surely be studied along the lines of their non-Takiff analouges [12, 18]. Recall, that in that case also harmonic analysis was the important starting point. We are more interested in Takiff superalgebra CFTs based on more complicated algebras. It was shown that ĝl (1|1) has many interesting simple current extensions [14, 15], including ŝl (2|1) at levels 1 and −1/2. Characters of all these extensions turned out to be mock Jacobi forms for atypical modules [1]. It is probable that the Takiff superalgebra of ŝl (2|1) appears in extensions of the g̃l (1|1) CFT, and interesting mock modular-like objects are expected to appear as characters of extended algebra modules. It is also obvious that more realistic applications of Takiff affine algebras and superalgebras will require detailed understanding of Takiff affine ˜̂ sl(2). It seems natural to start its inves- tigation from harmonic analysis, as a mini superspace toy model. One can expect here more involved structures of modules under the action of left- and right-invariant vector fields on the group manifold, since the representation theory of s̃l(2) consists of infinite-dimensional mod- ules. It is also natural to expect that its free field realization will contain usual Wakimoto ŝl(2) realization embedded into it. Acknowledgements A. Babichenko is thankful for hospitality to DESY Hamburg Theory Group and to ETH Zurich Institute for Theoretical Physics, where the final part of this work was done. His work was sup- ported by SFB676, and his visit to ETH – by Pauli Center for Theoretical Studies. T. Creutzig is supported by NSERC Research Grant (Project #: RES0020460). 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