Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups
We considered two types of string models: on the Riemmann space of string coordinates with null torsion and on the Riemman-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable systems and Dubrovin solutions of WDVV associativi...
Saved in:
| Date: | 2008 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2008
|
| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/149042 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups / V.D. Gershun // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 30 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-149042 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1490422019-02-20T01:28:36Z Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups Gershun, V.D. We considered two types of string models: on the Riemmann space of string coordinates with null torsion and on the Riemman-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable systems and Dubrovin solutions of WDVV associativity equation to construct new integrable string equations of hydrodynamic type on the torsionless Riemmann space of chiral currents in first case. We used the invariant local chiral currents of principal chiral models for SU(n), SO(n), SP(n) groups to construct new integrable string equations of hydrodynamic type on the Riemmann space of the chiral primitive invariant currents and on the chiral non-primitive Casimir operators as Hamiltonians in second case. We also used Pohlmeyer tensor nonlocal currents to construct new nonlocal string equation. 2008 Article Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups / V.D. Gershun // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 30 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81T20; 81T30; 81T40; 37J35; 53Z05; 22E70 http://dspace.nbuv.gov.ua/handle/123456789/149042 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 |
We considered two types of string models: on the Riemmann space of string coordinates with null torsion and on the Riemman-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable systems and Dubrovin solutions of WDVV associativity equation to construct new integrable string equations of hydrodynamic type on the torsionless Riemmann space of chiral currents in first case. We used the invariant local chiral currents of principal chiral models for SU(n), SO(n), SP(n) groups to construct new integrable string equations of hydrodynamic type on the Riemmann space of the chiral primitive invariant currents and on the chiral non-primitive Casimir operators as Hamiltonians in second case. We also used Pohlmeyer tensor nonlocal currents to construct new nonlocal string equation. |
| format |
Article |
| author |
Gershun, V.D. |
| spellingShingle |
Gershun, V.D. Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Gershun, V.D. |
| author_sort |
Gershun, V.D. |
| title |
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups |
| title_short |
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups |
| title_full |
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups |
| title_fullStr |
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups |
| title_full_unstemmed |
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups |
| title_sort |
integrable string models in terms of chiral invariants of su(n), so(n), sp(n) groups |
| publisher |
Інститут математики НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149042 |
| citation_txt |
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups / V.D. Gershun // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 30 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT gershunvd integrablestringmodelsintermsofchiralinvariantsofsunsonspngroups |
| first_indexed |
2025-07-12T20:57:48Z |
| last_indexed |
2025-07-12T20:57:48Z |
| _version_ |
1837476372459552768 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 041, 16 pages
Integrable String Models in Terms
of Chiral Invariants of SU(n), SO(n), SP (n) Groups?
Victor D. GERSHUN
ITP, NSC Kharkiv Institute of Physics and Technology, Kharkiv, Ukraine
E-mail: gershun@kipt.kharkov.ua
Received October 30, 2007, in final form April 22, 2008; Published online May 06, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/041/
Abstract. We considered two types of string models: on the Riemmann space of string
coordinates with null torsion and on the Riemman–Cartan space of string coordinates with
constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable
systems and Dubrovin solutions of WDVV associativity equation to construct new integrable
string equations of hydrodynamic type on the torsionless Riemmann space of chiral currents
in first case. We used the invariant local chiral currents of principal chiral models for SU(n),
SO(n), SP (n) groups to construct new integrable string equations of hydrodynamic type
on the Riemmann space of the chiral primitive invariant currents and on the chiral non-
primitive Casimir operators as Hamiltonians in second case. We also used Pohlmeyer tensor
nonlocal currents to construct new nonlocal string equation.
Key words: string; integrable models; Poisson brackets; Casimir operators; chiral currents
2000 Mathematics Subject Classification: 81T20; 81T30; 81T40; 37J35; 53Z05; 22E70
1 Introduction
String theory is a very promising candidate for a unified quantum theory of gravity and all
the other forces of nature. For quantum description of string model we must have classical
solutions of the string in the background fields. String theory in suitable space-time backgrounds
can be considered as principal chiral model. The integrability of the classical principal chiral
model is manifested through an infinite set of conserved charges, which can form non-Abelian
algebra. Any charge from the commuting subset of charges and any Casimir operators of charge
algebra can be considered as Hamiltonian in bi-Hamiltonian approach to integrable models.
The bi-Hamiltonian approach to integrable systems was initiated by Magri [1]. Two Poisson
brackets (PBs)
{φa(x), φb(y)}0 = P ab
0 (x, y)(φ), {φa(x), φb(y)}1 = P ab
1 (x, y)(φ)
are called compatible if any linear combination of these PBs
{∗, ∗}0 + λ{∗, ∗}1
is PB also for arbitrary constant λ. The functions φa(t, x), a = 1, 2, . . . , n are local coordinates
on a certain given smooth n-dimensional manifold Mn. The Hamiltonian operators P ab
0 (x, y)(φ),
P ab
1 (x, y)(φ) are functions of local coordinates φa(x). It is possible to find such Hamiltonians H0
and H1 which satisfy bi-Hamiltonian condition [2]
dφa(x)
dt
= {φa(x),H0}0 = {φa(x),H1}1,
?This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in
Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The full collection is available at
http://www.emis.de/journals/SIGMA/symmetry2007.html
mailto:gershun@kipt.kharkov.ua
http://www.emis.de/journals/SIGMA/2008/041/
http://www.emis.de/journals/SIGMA/symmetry2007.html
2 V.D. Gershun
where HM =
∫ 2π
0 hM (φ(y))dy, M = 0, 1. Two branches of hierarchies arise under two equations
of motion under two different parameters of evolution t0M and tM0 [2]
dφa(x)
dt01
= {φa(x),H0}1 =
∫ 2π
0
P ab
1 (x, y)
∂h0
∂φb(y)
dy =
∫ 2π
0
Ra
c (x, z)P cb
0 (z, y)
∂h0
∂φb(y)
dy,
dφa(x)
dt10
= {φa(x),H1}0 =
∫ 2π
0
P ab
0 (x, y)
∂h1
∂φb(y)
dy =
∫ 2π
0
(R−1)a
c (x, z)P cb
0 (z, y)
∂h0
∂φb(y)
dy.
There Ra
b (x, y) is a recursion operator and (R−1)a
b (x, y) is its inverse
Ra
c (x, y) =
∫ 2π
0
P ab
1 (x, z)(P0)−1
bc (z, y)dz.
The first branch of the hierarchies of dynamical systems has the following form
dφa(x)
dt0N
=
∫ 2π
0
(R(x, y1)· · ·R(yN−1))a
cP
cb
0 (yN−1, yN )
∂h0
∂φb(yN )
dy1· · ·dyN , N = 1, 2, . . . ,∞.
The second branch of the hierarchies can be obtained by replacement R → R−1 and t0N → tN0.
We will consider only the first branch of the hierarchies.
The local PBs of hydrodynamic type were introduced by Dubrovin, Novikov [3, 4] for Hamil-
tonian description of equations of hydrodynamics. They were generalized by Ferapontov [5] and
Mokhov, Ferapontov [6] to the non-local PBs of hydrodynamic type. The hydrodynamic type
systems were considered by Tsarev [8], Maltsev [9], Ferapontov [10], Mokhov [12] (see also [7]),
Pavlov [13] (see also [14]), Maltsev, Novikov [15]. The polynomials of local chiral currents were
considered by Goldshmidt and Witten [16] (see also [17]). The local conserved chiral charges in
principal chiral models were considered by Evans, Hassan, MacKay, Mountain [24]. The tensor
nonlocal chiral charges were introduced by Pohlmeyer [26] (see also [27, 28]). The string mod-
els of hydrodynamic type were considered by author [18, 19]. In Section 3, the author applied
hydrodynamic approach to integrable systems to obtain new integrable string equations. In Sec-
tion 4, the author used the nonlocal Pohlmeyer charges to obtain a new string equation in terms
of the nonlocal currents. In Section 5, the author applied the local invariant chiral currents to
a simple Lie algebra to construct new integrable string equations.
2 String model of principal chiral model type
A string model is described by the Lagrangian
L =
1
2
∫ 2π
0
ηαβgab(φ(t, x))
∂φa(t, x)
∂xα
∂φb(t, x)
∂xβ
dx (1)
and by two first kind constraints
gab(φ(x))
[
∂φa(x)
∂t
∂φb(x)
∂t
+
∂φa(x)
∂x
∂φb(x)
∂x
]
≈ 0, gab(φ(x))
∂φa(x)
∂t
∂φb(x)
∂x
≈ 0.
The target space local coordinates φa(x), a = 1, . . . , n belong to certain given smooth n-dimen-
sional manifold Mn with nondegenerate metric tensor
gab(φ(x)) = ηµνe
µ
a(φ(x))eν
b (φ(x)),
where µ, ν = 1, . . . , n are indices of tangent space to manifold Mn on some point φa(x). The
veilbein eµ
a(φ) and its inverse ea
µ(φ) satisfy the conditions
eµ
aeb
µ = δb
a, eµ
aeaν = ηµν .
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP (n) Groups 3
The coordinates xα (x0 = t, x1 = x) belong to world sheet with metric tensor gαβ in conformal
gauge. The string equations of motion have the form
ηαβ [∂αβφa + Γa
bc(φ)∂αφb∂βφc] = 0, ∂α =
∂
∂xα
, α = 0, 1,
where
Γa
bc(φ) =
1
2
ea
µ
[
∂eµ
b
∂φc
+
∂eµ
c
∂φb
]
is the connection. In terms of canonical currents
Jµ
α(φ) = eµ
a(φ)∂αφa
the equations of motion have the form
ηαβ∂αJµ
β (φ(t, x)) = 0, ∂αJµ
β (φ)− ∂βJµ
α(φ)− Cµ
νλ(φ)Jν
α(φ)Jλ
β (φ) = 0,
where
Cµ
νλ(φ) = ea
νe
b
λ
[
∂eµ
a
∂φb
−
∂eµ
b
∂φa
]
is the torsion. The Hamiltonian has the form
H =
1
2
∫ 2π
0
[ηµνJ0µJ0ν + ηµνJ
µ
1 Jν
1 ]dx,
where J0µ(φ) = ea
µ(φ)pa, Jµ
1 (φ) = eµ
a
∂
∂xφa and pa(t, x) = ηµνe
µ
aeν
b
∂
∂tφ
b is the canonical momen-
tum. The canonical commutation relations of currents are as follows
{J0µ(φ(x)), J0ν(φ(y))} = Cλ
µν(φ(x))J0λ(φ(x))δ(x− y),
{J0µ(φ(x)), Jν
1 (φ(y))} = Cν
µλ(φ(x))Jλ
1 (φ(x))δ(x− y)− 1
2
δν
µ
∂
∂x
δ(x− y),
{Jµ
1 (φ(x)), Jν
1 (φ(y))} = 0.
Let us introduce chiral currents
Uµ = ηµνJ0ν + Jµ
1 , V µ = ηµνJ0ν − Jµ
1
The commutation relations of chiral currents are the following
{Uµ(φ(x)), Uν(φ(y))} = Cµν
λ (φ(x))
[
3
2
Uλ(φ(x))− 1
2
V λ(φ(x))
]
δ(x− y)− ηµν ∂
∂x
δ(x− y),
{Uµ(φ(x)), V ν(φ(y))} = Cµν
λ (φ(x))[Uλ(φ(x)) + V λ(φ(x))]δ(x− y),
{V µ(φ(x)), V ν(φ(y))} = Cµν
λ (φ(x))
[
3
2
V λ(φ(x))− 1
2
Uλ(φ(x))
]
δ(x− y) + ηµν ∂
∂x
δ(x− y).
Equations of motion in light-cone coordinates
x± =
1
2
(t± x),
∂
∂x±
=
∂
∂t
± ∂
∂x
have the form
∂−Uµ = Cµ
νλ(φ(x))UνV λ, ∂−V µ = Cµ
νλ(φ(x))V νUλ.
In the case of the null torsion
Cµ
νλ = 0, eµ
a(φ) =
∂eµ
∂φa
, Γa
bc(φ) = ea
µ
∂2eµ
∂φb∂φc
, Rµ
νλρ(φ) = 0
the string model is integrable.The Hamiltonian equations of motion under Hamiltonian (1) are
described by two independent left and right movers: Uµ(t + x) and V µ(t− x).
4 V.D. Gershun
3 Integrable string models of hydrodynamic type
with null torsion
We want to construct new integrable string models with Hamiltonians as polynomials of the
initial chiral currents Uµ(φ(x)). The PB of chiral currents Uµ(x) coincides with the flat PB of
Dubrovin, Novikov
{Uµ(x), Uν(y)}0 = −ηµν ∂
∂x
δ(x− y).
Let us introduce a local Dubrovin, Novikov PB [3, 4]. It has the form
{Uµ(x), Uν(y)}1 = gµν(U(x))
∂
∂x
δ(x− y)− Γµν
λ (U(x))
∂Uλ(x)
∂x
δ(x− y).
This PB is skew-symmetric if gµν(U) = gνµ(U) and it satisfies Jacobi identity if Γa
bc(U) = Γa
cb(U),
Ca
bc(U) = 0, Ra
bcd(U) = 0. In the case of non-zero curvature tensor we must include Weingarten
operators into right side of the PB with the step-function sgn (x− y)=( d
dx)−1δ(x− y)=ν(x− y)
[5, 6]. The PBs {∗, ∗}0 and {∗, ∗}1 are compatible by Magri [1] if the pencil {∗, ∗}0 + λ{∗, ∗}1 is
also PB . As a result, Mokhov [12, 11] obtained the compatible pair of PBs
P0µν(U)(x, y) = −ηµν
∂
∂x
δ(x− y),
P1µν(U)(x, y) = 2
∂2F (U)
∂Uµ∂Uν
∂
∂x
δ(x− y) +
∂3F (U)
∂Uµ∂Uν∂Uλ
∂Uλ
∂x
δ(x− y).
The function F (U) satisfies the equation
∂3F (U)
∂Uµ∂Uρ∂Uλ
ηλρ ∂3F (U)
∂Uν∂Uσ∂Uρ
=
∂3F (U)
∂Uν∂Uρ∂Uλ
ηλρ ∂3F (U)
∂Uµ∂Uσ∂Uρ
.
This equation is WDVV [20, 21] associativity equation and it was obtained in 2D topological
field theory. Dubrovin [22, 23] obtained a lot of solutions of WDVV equation. He showed
that local fields Uµ(x) must belong Frobenius manifolds to solve the WDVV equation and gave
examples of Frobenius structures. Associative Frobenius algebra may be written in the following
form
∂
∂Uµ
∗ ∂
∂Uν
:= dλ
µν(U)
∂
∂Uλ
.
Totally symmetric structure function has the form
dµνλ(U) =
∂F (U)
∂Uµ∂Uν∂Uλ
, µ, ν, λ = 1, . . . , n
and associativity condition(
∂
∂Uµ
∗ ∂
∂Uν
)
∗ ∂
∂Uλ
=
∂
∂Uµ
∗
(
∂
∂Uν
∗ ∂
∂Uλ
)
leads to the WDVV equation. Function F (U) is quasihomogeneous function of its variables(
dµUµ ∂
∂Uµ
)
F (U) = dF F (U) + AµνU
µUν + BµUµ + C,
here numbers dµ, dF , Aµν , Bµ, C depend on the type of polynomial function F (U). Here are
some Dubrovin examples of solutions of the WDVV equation
n = 1, F (U) = U3
1 ;
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP (n) Groups 5
n = 2, F (U) =
1
2
U2
1 U2 + eU2 , d1 = 1, d2 = 2, dF = 2, A11 = 1 (2)
and quasihomogeneity condition for n = 2 has the form(
d1U1
∂
∂U1
+ d2
∂
∂U2
)
F (U) = dF F (U) + A11U
2
1 .
We used local fields Uµ with low indices here for convenience. One of the Dubrovin polynomial
solutions is
F (U) =
1
2
(U2
1 U3 + U1U
2
2 ) +
1
4
U2
2 U2
3 +
1
60
U5
3 , (3)
here d1 = 1, d2 = 3
2 , d3 = 2, dF = 4 and the polynomial function
f(U2, U3) =
1
4
U2
2 U2
3 +
1
60
U5
3
is a solution of the additional PDE.
In the bi-Hamiltonian approach to an integrable string model we must construct the recursion
operator to generate a hierarchy of PBs and a hierarchy of Hamiltonians
Rµ
ν (x, y) =
∫ 2π
0
Pµλ
1 (x, z)(P−1
0 (z, y))λνdz
= 2
∂2F (U(x))
∂Uµ(x)∂Uν(x)
δ(x− y) +
∂3F (U(x))
∂Uµ(x)∂Uν(x)∂Uλ(x)
∂Uλ(y)
∂y
ν(x− y).
The Hamiltonian equation of motion with Hamiltonian H0 is the following
H0 =
∫ 2π
0
ηµνU
µ(x)Uν(x)dx,
∂Uµ
∂t
=
∂Uµ
∂x
.
First of the new equations of motion under the new time t1 has the form [12]
∂Uµ
∂t1
=
∫ 2π
0
Rµ
ν (x, y)
∂Uµ(y)
∂y
dy = ηµν d
dx
(
∂F (x)
∂Uν
)
. (4)
This equation of motion can be obtained as Hamiltonian equation with new Hamiltonian H1
H1 =
∫ 2π
0
∂F (U(x))
∂Uµ
Uµ(x)dx,
where F (U) is each of Dubrovin solutions WDVV associativity equation (2), (3). Any system
of the following hierarchy [12]
∂Uµ
∂tM
=
∫ 2π
0
(R(x, y1) · · ·R(yM−1, yM ))µ
ν
∂Uν
∂yM
dy1 · · · dyM
is an integrable system. As result we obtain chiral currents Uµ(φ(tM , x)) = fµ(φ(tM , x), where
fµ(φ) is a solution of the equation of motion. In the case of the Hamiltonian H1 and of the
equation of motion (4) we can introduce new currents
Jµ
0 (t1, x) = Uµ(t1, x), Jµ
1 (t1, x) = ηµν ∂F (U(t1, x))
∂Uν
.
Consequently, we can introduce a new metric tensor and a new veilbein depending of the new
time coordinate. The equation for the new metric tensor has the form
eµ
a(φ(t1, x))
∂φa(t1, x)
∂x
=
deµ(φ(t1, x))
dx
= ηµν ∂F (f(φ(t1, x)))
∂fν(φ(t1, x))
.
6 V.D. Gershun
4 New string equation in terms
of Pohlmeyer tensor nonlocal currents
In the case of the flat space Cµ
νλ = 0 there exist nonlocal totally symmetric tensor chiral currents
called “Pohlmeyer” currents [26, 27, 28]
R(M)(U(x)) ≡ R(µ1µ2...µM )(U(x))
= U (µ1(x)
∫ x
0
Uµ2(x1)dx1 · · ·
∫ xM−2
0
UµM )(xM−1)dxM−1,
where round brackets the mean totally symmetric product of chiral currents Uµ(U). The new
Hamiltonians may have the following forms
H(M) =
1
2
∫ 2π
0
R(M)(U(x))d2MR(M)(U(x))dx,
where dM ≡ d(µ1µ2...µM ) is totally symmetric invariant constant tensor, which can be constructed
from Kronecker deltas. For example
R(2) ≡ Rµν(U(x)) =
1
2
[Uµ(x)
x∫
0
Uν(x1)dx1 + Uν(x)
∫ x
0
Uµ(x1)dx1],
H(2) =
1
2
∫ 2π
0
[
Uµ(x)Uµ(x)
∫ x
0
Uν(x1)dx1
∫ x
0
Uν(x2)dx2
+ Uµ(x)Uν(x)
∫ x
0
Uµ(x1)dx1
∫ x
0
Uν(x2)dx2
]
dx.
The Hamiltonian H(2) commutes with the Hamiltonian H(1) = 1
2
∫ 2π
0 Uµ(x)Uµ(x)dx and it
commutes with the Casimir
∫ 2π
0 Uµ(x)dx. The equation of motion under the Hamiltonian H(2)
is as follows
∂Uµ(x)
∂t
=
∂
∂x
[
Uµ(x)
∫ x
0
Uν(x1)dx1
∫ x
0
Uν(x2)dx2 + Uν(x)
∫ x
0
Uµ(x1)dx1
∫ x
0
Uν(x2)dx2
]
− Uν(x)Uν(x)
∫ x
0
Uµ(x1)dx1 − Uµ(x)Uν(x)
∫ x
0
Uν(x1)dx1.
In the variables
Sµ(x) =
∫ x
0
Uµ(y)dy
the latter equation can be rewritten as follows
∂Sµ
∂t
=
∂
∂x
(Sµ(SνSν)) +
∫ x
0
Sµ
(
Sν ∂2Sν
∂2y
)
dy, µ, ν = 1, 2, . . . , n.
5 Integrable string models with constant torsion
Let us go back to the commutation relations of chiral currents. Let the torsion Cµ
νλ(φ(x)) 6= 0
and Cµνλ = fµνλ be structure constant of s simple Lie algebra. We will consider a string model
with the constant torsion in light-cone gauge in target space. This model coincides with the
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP (n) Groups 7
principal chiral model on compact simple Lie group. We cannot divide the motion on right and
left mover because of chiral currents ∂−Uµ = fµ
νλUνV λ, ∂−V µ = fµ
νλV νUλ are not conserved.
The correspondent charges are not Casimirs. The present paper was stimulated by paper [24].
Evans, Hassan, MacKay, Mountain (see [24] and references therein) constructed local invariant
chiral currents as polynomials of the initial chiral currents of SU(n), SO(n), SP (n) principal
chiral models and they found such combination of them that the corresponding charges are
Casimir operators of these dynamical systems. Their paper was based on the paper of de
Azcarraga, Macfarlane, MacKay, Perez Bueno (see [25] and references therein) about invariant
tensors for simple Lie algebras. Let tµ be n ⊗ n traceless hermitian matrix representations of
generators Lie algebra
[tµ, tν ] = 2ifµνλtλ, Tr(tµtν) = 2δµν .
Here is an additional relation for SU(n) algebra
{tµ, tν} =
4
n
δµν + 2dµνλtλ, µ = 1, . . . , n2 − 1.
De Azcarraga et al. gave some examples of invariant tensors of simple Lie algebras and they
gave a general method to calculate them. Invariant tensors may be constructed as invariant
symmetric polynomials on SU(n)
d
(M)
(µ1...µM ) =
1
M !
STr(tµ1 · · · tµM ),
where STr means the completely symmetrized product of matrices and d
(M)
(µ1...µM ) is the totally
symmetric tensor and M = 2, 3, . . . ,∞. Another family of invariant symmetric tensors [29, 30]
(see also [25]) called D-family based on the product of the symmetric structure constant dµνλ
of the SU(n) algebra is as follows:
D
(M)
(µ1...µM ) = dk1
(µ1µ2
dk1k2
µ3
· · · dkM−2kM−3
µM−2 d
kM−3
µM−1µM ),
where D
(2)
µν = δµν ,D
(3)
µνλ = dµνλ and M = 4, 5, . . . ,∞.
Here are n − 1 primitive invariant tensors on SU(n). The invariant tensors for M ≥ n are
functions of primitive tensors. The Casimir operators on SU(n) algebra have the form
C(M)(t) = dM
(µ1...µM )tµ1 · · · tµM .
Evans et al. introduced local chiral currents based on the invariant symmetric polynomials on
simple Lie groups
J (M)(U) = STr(U · · ·U) ≡ STr UM = d(M)
µ1...µM
Uµ1 · · ·UµM , (5)
where U = tµUµ and µ = 1, . . . , n2 − 1 . It is possible to decompose the invariant symmetric
chiral currents J (M)(U) into product of the basic invariant chiral currents D(M)(U)
D(2)(U) = d(2)
µν UµUν = ηµνU
µUν , D(3)(U) = dµνλUµUνUλ,
D(M)(U(x)) = dk1
µ1µ2
dk1k2
µ3
· · · dkM−2kM−3
µM−2 d
kM−3
µM−1µM Uµ1Uµ2 · · ·UµM ,
where M = 4, 5, . . . ,∞. The author obtained the following expressions for local invariant chiral
currents J (M)(U)
J (2) = 2D(2), J (3) = 2D(3), J (4) = 2D(4) +
4
n
D(2)2,
8 V.D. Gershun
J (5) = 2D(5) +
8
n
D(2)D(3), J (6) = 2D(6) +
4
n
D(3)2 +
8
n
D(2)D(4) +
8
n2
D(2)3,
J (7) = 2D(7) +
8
n
D(3)D(4) +
8
n
D(2)D(5) +
24
n2
D(2)2D(3),
J (8) = 2D(8) +
4
n
D(4)2+
8
n
D(3)D(5)+
8
n
D(2)D(6)+
24
n2
D(2)D(3)2+
24
n2
D(2)2D(4)+
16
n3
D(2)4,
J (9) = 2D(9) +
8
n
D(4)D(5) +
8
n
D(3)D(6) +
8
n
D(2)D(7) +
8
n2
D(3)3 +
48
n2
D(2)D(3)D(4)
+
24
n2
D(2)2D(5) +
64
n3
D(2)3D(3).
Both families of invariant chiral currents J (M)(U(x)) and D(M)(U(x)) satisfy the conservation
equations ∂−J (M)(U(x)) = 0, ∂−D(M)(U(x)) = 0.
The commutation relations of invariant chiral currents J (M)(U(x)) show that these currents
are not densities of dynamical Casimir operators for SU(n) group. Therefore, we will not
consider these currents in the following.
We considered abasic family of invariant chiral currents D(M)(U) and we proved that the
invariant chiral currents D(M)(U) form closed algebra under canonical PB and corresponding
charges are dynamical Casimir operators. The commutation relations of invariant chiral currents
D(M)(U(x)) and D(N)(U(y)) for M,N = 2, 3, 4 and for M = 2, N = 2, 3, . . . ,∞ are as follows
{D(M)(x), D(N)(y)} = −MND(M+N−2)(x)
∂
∂x
δ(x− y)
− MN(N − 1)
M + N − 2
∂D(M+N−2)(x)
∂x
δ(x− y).
The commutation relations for M ≥ 5, N ≥ 3 are as follows
{D(5)(x), D(3)(y)} = −[12D(6)(x) + 3D(6,1)(x)]
∂
∂x
δ(x− y)
− 1
3
∂
∂x
[12D(6)(x) + 3D(6,1)(x)]δ(x− y),
{D(5)(x), D(4)(y)} = −[16D(7)(x) + 4D(7,1)(x)]
∂
∂x
δ(x− y)
− 3
7
∂
∂x
[16D(7)(x) + 4D(7,1)(x)]δ(x− y),
{D(6)(x), D(3)(y)} = −[12D(7)(x) + 6D(7,1)(x)]
∂
∂x
δ(x− y)
− 2
7
∂
∂x
[12D(7)(x) + 6D(7,1)(x)]δ(x− y),
{D(5)(x), D(5)(y)} = −[16D(8)(x) + 8D(8,1)(x) + D(8,2)(x)]
∂
∂x
δ(x− y)
− 1
2
∂
∂x
[16D(8)(x) + 8D(8,1)(x) + D(8,2)(x)]δ(x− y),
{D(6)(x), D(4)(y)} = −[16D(8)(x) + 8D(8,3)(x)]
∂
∂x
δ(x− y)
− 3
8
∂
∂x
[16D(8)(x) + 8D(8,3)(x)]δ(x− y),
{D(7)(x), D(3)(y)} = −[12D(8)(x) + 6D(8,1)(x) + 3D(8,3)(x)]
∂
∂x
δ(x− y)
− 1
4
∂
∂x
[12D(8)(x) + 6D(8,1)(x) + 3D(8,3)(x)]δ(x− y),
{D(8)(x), D(3)(y)} = −[12D(9)(x) + 6D(9,1)(x) + 6D(9,2)(x)]
∂
∂x
δ(x− y)
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP (n) Groups 9
− 2
9
∂
∂x
[12D(9)(x) + 6D(9,1)(x) + 6D(9,2)(x)]δ(x− y),
{D(7)(x), D(4)(y)} = −[16D(9)(x) + 8D(9,2)(x) + 4D(9,3)(x)]
∂
∂x
δ(x− y)
− 1
3
∂
∂x
[16D(9)(x) + 8D(9,2)(x) + 4D(9,3)(x)]δ(x− y),
{D(6)(x), D(5)(y)} = −[16D(9)(x) + 4D(9,1)(x) + 8D(9,2)(x) + 2D(9,4)(x)]
∂
∂x
δ(x− y)
− 4
9
∂
∂x
[16D(9)(x) + 4D(9,1)(x) + 8D(9,2)(x) + 2D(9,4)(x)]δ(x− y),
{D(9)(x), D(3)(y)} = −[12D(10)(x) + 6D(10,1)(x) + 6D(10,2)(x) + 3D(10,3)(x)]
∂
∂x
δ(x− y)
− 1
5
∂
∂x
[12D(10)(x) + 6D(10,1)(x) + 6D(10,2)(x) + 3D(10,3)(x)]δ(x− y),
{D(8)(x), D(4)(y)} = −[16D(10)(x) + 8D(10,2)(x) + 8D(10,4)(x)]
∂
∂x
δ(x− y)
− 3
10
∂
∂x
[16D(10)(x) + 8D(10,2)(x) + 8D(10,4)(x)]δ(x− y),
{D(7)(x), D(5)(y)} = −[16D(10)(x) + 8D(10,3)(x) + 4D(10,1)(x)
+ 4D(10,4)(x) + 2D(10,5)(x) + D(10,6)(x)]
∂
∂x
δ(x− y)
− 2
5
∂
∂x
[16D(10)(x) + 8D(10,3)(x) + 4D(10,1)(x) + 4D(10,4)(x)
+ 2D(10,5)(x) + D(10,6)(x)]δ(x− y),
{D(6)(x), D(6)(y)} = −[16D(10)(x) + 16D(10,2)(x) + 4D(10,7)(x)]
∂
∂x
δ(x− y)
− 1
2
∂
∂x
[16D(10)(x) + 16D(10,2)(x) + 4D(10,7)(x)]δ(x− y).
The new dependent invariant chiral currents D(6,1), D(7,1), D(8,1) − D(8,3), D(9,1) − D(9,4),
D(10,1) −D(10,7) (see Appendix A) have the form
D(6,1) = dk
µνd
l
λρd
n
σϕdklnUµUνUλUρUσUϕ,
D(7,1) = dk
µνd
l
λρd
n
σϕdnm
τ dklmUµUνUλUρUσUϕU τ ,
D(8,1) = [dk
µνd
kl
λ dln
ρ ][dm
σϕ][dp
τθ]d
nmpUµUνUλUρUσUϕU τU θ,
D(8,2) = [dk
µν ][d
l
λρ][d
n
σϕ][dm
τθ]d
klpdnmpUµUνUλUρUσUϕU τU θ,
D(8,3) = [dk
µνd
kl
λ ][dn
ρσdnm
ϕ ][dp
τθ]d
lmpUµUνUλUρUσUϕU τU θ,
D(9,1) = [dk
µνd
kl
λ dln
ρ dnm
σ ][dp
ϕτ ][d
r
θω]dmprUµUνUλUρUσUϕU τU θUω,
D(9,2) = [dk
µνd
kl
λ dln
ρ ][dm
σϕdmp
τ ][dr
θω]dnprUµUνUλUρUσUϕU τU θUω,
D(9,3) = [dk
µνd
kl
λ ][dn
ρσdnm
ϕ ][dp
τθd
pr
ω ]dlmrUµUνUλUρUσUϕU τU θUω,
D(9,4) = [dk
µνd
kl
λ ][dn
ρσ][dm
ϕτ ][d
p
θω]dlnrdmprUµUνUλUρUσUϕU τU θUω,
D(10,1) = [dk
µνd
kl
λ dln
ρ dnm
σ dmp
ϕ ][dr
τθ][d
s
ωβ]dprsUµUνUλUρUσUϕU τU θUωUβ ,
D(10,2) = [dk
µνd
kl
λ dln
ρ dnm
σ ][dp
ϕτd
pr
θ ][ds
ωβ]dmrsUµUνUλUρUσUϕU τU θUωUβ,
D(10,3) = [dk
µνd
kl
λ dln
ρ ][dm
σϕdmp
ϕ dpr
τ ][ds
ωβ]dnrsUµUνUλUρUσUϕU τU θUωUβ,
D(10,4) = [dk
µνd
kl
λ dln
ρ ][dm
σϕdmp
τ ][dr
θωdrs
β ]dnpsUµUνUλUρUσUϕU τU θUωUβ,
D(10,5) = [dk
µνd
kl
λ dln
ρ ][dm
σϕ][dp
τθ][d
r
ωβ]dnmsdprsUµUνUλUρUσUϕU τU θUωUβ ,
10 V.D. Gershun
D(10,6) = [dk
µνd
kl
λ ][dn
ρσdnm
ϕ ][dp
τθ][d
r
ωβ]dlmsdprsUµUνUλUρUσUϕU τU θUωUβ ,
D(10,7) = [dk
µνd
kl
λ ][dn
ρσ]dm
ϕ ][dmp
τθ ][dr
ωβ]dlnsdprsUµUνUλUρUσUϕU τU θUωUβ .
Let us apply the hydrodynamic approach to integrable string models with constant tor-
sion. In this case we must consider the conserved primitive chiral currents D(M)(U(x)), (M =
2, 3, . . . , n− 1) as local fields of the Riemmann manifold. The non-primitive local charges of in-
variant chiral currents with M ≥ n form the hierarchy of new Hamiltonians in the bi-Hamiltonian
approach to integrable systems. The commutation relations of invariant chiral currents are local
PBs of hydrodynamic type.
The invariant chiral currents D(M) with M ≥ 3 for the SU(3) group can be obtained from
the following relation
dklndkmp + dklmdknp + dklpdknm =
1
3
(δlnδmp + δlmδnp + δlpδnm).
The corresponding invariant chiral currents for SU(3) group have the form
D(2N) =
1
3N−1
(ηµνU
µUν)N =
1
3N−1
D(2)N ,
D(2N+1) =
1
3N−1
(ηµνU
µUν)N−1dklnUkU lUn =
1
3N−1
D(2)N−1D(3).
The invariant chiral currents D(2), D(3) are local coordinates of the Riemmann manifold M2.
The local charges D(2N), N ≥ 2 form a hierarchy of Hamiltonians. The new nonlinear equations
of motion for chiral currents are as follows
∂D(k)(U(x))
∂tN
=
{
D(k)(U(x)),
∫ 2π
0
D(2)N (U(y))dy
}
, k = 2, 3, N = 2, . . . ,∞.
∂D(2)(U(x))
∂tN
= −2(2N − 1)
∂D(2)N (U(x))
∂x
,
∂D(3)(U(x))
∂tN
= −6ND(3)(U(x))
∂D(2)N−1(U(x))
∂x
− 2ND(2)N−1(U(x))
∂D(3)(U(x))
∂x
.
The construction of integrable equations with SU(n) symmetries for n ≥ 4 has difficulties in
reduction of non-primitive invariant currents to primitive currents.
The similar method of construction of chiral currents for SO(2l + 1) = Bl, SP (2l) = Cl
groups was used by Evans et al. [24] on the base of symmetric invariant tensors of de Azcarraga
et al. [25]. In the defining representation these group generators corresponding to algebras tµ
satisfy the rules
[tµ, tν ] = 2ifλ
µνtλ, Tr(tµtν) = 2δµν , tµη = −ηttµ,
where η is a Euclidean or symplectic structure.
The symmetric tensor structure constants for these groups were introduced through com-
pletely symmetrized product of three generators of corresponding algebras
t(µtνtλ) = vρ
µνλtρ,
where vµνλρ is a totally symmetric tensor. The basic invariant symmetric tensors have the
form [25]
V (2)
µν = δµν , V
(2N)
(µ1µ2...µ2N−1µ2N ) = vν1
(µ1µ2µ3
vν1ν2
µ4µ5
· · · vν2N−3
µ2N−2µ2N−1µ2N ), N = 2, . . . ,∞.
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP (n) Groups 11
The invariant chiral currents J (2N) (5) coincide with the basis invariant chiral currents V (2N)
J (2N) = 2V
(2N)
µ1...µ2N
Uµ1 · · ·Uµ2N .
The commutation relations of invariant chiral currents are PBs of hydrodynamic type
{J (M)(x), J (N)(y)} = −MNJ (M+N−2)(x)
∂
∂x
δ(x− y)
− MN(N − 1)
M + N − 2
∂J (M+N−2)(x)
∂x
δ(x− y). (6)
The commuting charges of these invariant chiral currents are dynamical Casimir operators on
SO(2l + 1), SP (2l). The metric tensor of Riemmann space of invariant chiral currents is as
follows
gMN (J(x)) = −MN(M + N − 2)J (M+N−2)(x).
The commutation relations (6) coincide with commutation relations, which was obtained by
Evans at al. [24].
We used relations for new symmetric invariant tensors V
(2N,1)
(µ1...µ2N ) (see Appendix B), which
we obtained during calculation PB (6)
vk
(µ1µ2µ3
vl
µ4µ5µ6
vn
µ7µ8µ9
vkln
µ10) = V
(10)
(µ1...µ10),
vk
(µ1µ2µ3
vl
µ4µ5µ6
vn
µ7µ8µ9
vm
µ10µ11µ12)v
klnm = V
(12)
(µ1...µ12),
vk
(µ1µ2µ3
vl
µ4µ5µ6
vn
µ7µ8µ9
vm
µ10µ11µ12
vklp
µ13
vnmp
µ14) = V
(14)
(µ1...µ14).
Appendix A
The new dependent invariant chiral currents and the new dependent totally symmetric invariant
tensors for SU(N) group can be obtained under different order of calculation of trace of the pro-
duct of the generators of SU(n) algebra. Let us mark the matrix product of two generators tµ, tν
in round brackets
(tµtν) =
2
n
δµν + (dk
µν + ifk
µν)tk. (7)
The expression of invariant chiral currents JM (U) depends on the position of the matrix product
of two generators in the general list of generators. For example
J (6) = Tr[t(tt)(tt)t] = 2D(6) +
4
n
D(3)2 +
8
n
D(2)D(4) +
8
n2
D(2)3,
J (6) = Tr[(tt)(tt)(tt)] = 2D(6,1) +
12
n
D(2)D(4) +
8
n2
D(2)3,
J (7) = Tr[t(tt)t(tt)t] = 2D(7) +
8
n
D(3)D(4) +
8
n2
D(2)D(5) +
24
n2
D(2)2D(3),
J (7) = Tr[(tt)(tt)(tt)t] = 2D(7,1) +
4
n
D(3)D(4) +
12
n2
D(2)D(5) +
24
n2
D(2)2D(3),
J (8) = Tr[t(tt)tt(tt)t] = 2D(8) +
4
n
D(4)2 +
8
n
D(3)D(5) +
8
n
D(2)D(6)
+
24
n2
D(2)D(3)2 +
24
n2
D(2)2D(4) +
16
n3
D(2)4,
12 V.D. Gershun
J (8) = Tr[(tt)(tt)t(tt)t] = 2D(8,1) +
4
n
D(4)2 +
4
n
D(3)D(5) +
24
n2
D(2)D(3)2
+
12
n
D(2)D(6) +
24
n2
D(2)2D(4) +
16
n3
D(2)4,
J (8) = Tr[(tt)(tt)(tt)(tt)] = 2D(8,2) +
4
n
D(4)2 +
16
n
D(2)D(6,1) +
32
n2
D(2)2D(4) +
16
n3
D(2)4,
J (8) = Tr[t(tt)(tt)(tt)t] = 2D(8,3) +
12
n
D(2)D(6) +
8
n
D(3)D(5)
+
24
n2
D(2)2D(4) +
24
n2
D(2)D(3)2 +
16
n
D(2)4,
J (9) = Tr[t(tt)ttt(tt)t] = 2D(9) +
8
n
D(4)D(5) +
8
n
D(3)D(6) +
8
n
D(2)D(7) +
8
n2
D(3)3
+
48
n2
D(2)D(3)D(4) +
24
n2
D(2)2D(5) +
64
n3
D(2)3D(3),
J (9) = Tr[t(tt)tt(tt)(tt)]
=
2D(9,1) +
4
n
D(4)D(5) +
4
n
D(2)D(7) +
4
n
D(2)D(7,1) +
8
n
D(3)D(6,1)
+
32
n2
D(2)D(3)D(4) +
32
n2
D(2)2D(5) +
64
n3
D(2)3D(3),
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
2D(9,4) +
4
n
D(2)D(7) +
4
n
D(2)D(7,1) +
12
n
D(3)D(6,1)
+
32
n2
D(2)D(3)D(4) +
32
n2
D(2)2D(5) +
64
n3
D(2)3D(3),
J (9) = Tr[t(tt)t(tt)t(tt)]
=
2D(9,2) +
4
n
D(4)D(5) +
8
n
D(3)D(6) +
8
n
D(2)D(7) +
4
n
D(2)D(7,1)
+
8
n2
D(3)3 +
40
n2
D(2)D(3)D(4) +
32
n2
D(2)2D(5) +
64
n3
D(2)3D(3),
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
2D(9,3) +
8
n
D(2)D(7) +
4
n
D(2)D(71) +
12
n
D(3)D(6) +
8
n
D(3)3
+
40
n2
D(2)D(3)D(4) +
32
n2
D(2)2D(5) +
64
n3
D(2)3D(3),
where t = tµUµ and two variants of two last expressions for J (9)(U) were obtained from two
variants of expression for J (6)(U) during calculation J (9)(U). Because the result of calculation
does not depend on the order of calculation, we can obtain relations between new invariant
chiral currents and basic invariant currents D(M)(U)
D(6,1) = D(6) +
2
n
D(3)2 − 2
n
D(2)D(4),
D(7,1) = D(7) +
4
n
D(3)D(4) − 4
n
D(2)D(5),
D(81) = D(8) +
2
n
D(3)D(5) − 2
n
D(2)D(6),
D(8,2) = D(8) +
4
n
D(3)D(5) − 4
n
D(2)D(6) − 4
n2
D(2)D(3)2 +
4
n2
D(2)2D(4),
D(8,3) = D(8) +
2
n
D(4)2 − 2
n
D(2)D(6),
D(9,1) = D(9) +
2
n
D(4)D(5) − 4
n2
D(3)3 +
8
n2
D(2)D(3)D(4) +
4
n2
D(2)2D(5),
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP (n) Groups 13
D(9,2) = D(9) +
2
n
D(4)D(5) − 2
n
D(2)D(7) − 4
n2
D(2)D(3)D(4) +
4
n2
D(2)2D(5),
D(9,3) = D(9) +
4
n
D(4)D(5) − 2
n
D(2)D(7) − 2
n
D(3)D(6) − 4
n2
D(2)D(3)D(4) +
4
n2
D(2)2D(5),
D(9,4) = D(9) +
4
n
D(4)D(5) − 2
n
D(3)D(6) − 8
n2
D(3)3 +
12
n2
D(2)D(3)D(4) +
4
n2
D(2)2D(5).
Hence we can obtain the new relations for symmetric tensors
dk
(µνd
l
λρd
n
σϕ)d
kln = dk
(µνd
kl
λ dln
ρ dn
σϕ) +
2
n
d(µνλdρσϕ) −
2
n
δ(µνd
k
λρd
k
σϕ),
dk
(µνd
l
λρd
n
σϕdnm
τ) dklm = dk
(µνd
kl
λ dln
ρ dnm
σ dm
ϕτ) +
4
n
d(µνλdk
ρσdk
ϕτ) −
4
n
δ(µνd
k
λρd
kl
σ dl
ϕτ),
dk
(µνd
l
λρd
n
σϕdnm
τ dmp
τ) dklp = dk
(µνd
kl
λ dln
ρ dnm
σ dmp
ϕ dp
τθ) +
4
n
d(µνλdk
ρσdkl
ϕ dl
τθ) −
2
n
δ(µνd
k
λρd
kl
σ dln
ϕ dn
τθ),
dk
(µνd
l
λρd
n
σϕdm
τθ)d
klpdnmp = dk
(µνd
kl
λ dln
ρ dnm
σ dmp
ϕ dp
τθ) +
4
n
d(µνλdk
ρσdkl
ϕ dl
τθ)
− 4
n
δ(µνd
k
λρd
kl
σ dln
ϕ dn
τθ) −
4
n2
δ(µνdλρσdϕτθ) +
4
n2
δ(µνδλρd
k
σϕdk
τθ).
It is possible to obtain similar relations for invariant symmetric tensors of ninth order. The
commutation relations of chiral currents in terms of the basic invariant currents are as follows
{D(5)(x), D(3)(y)} = −
[
15D(6)(x) +
6
n
D(3)2(x)− 6
n
D(2)(x)D(4)(x)
]
∂
∂x
δ(x− y)
− 1
3
∂
∂x
[
15D(6)(x) +
6
n
D(3)2(x)− 6
n
D(2)(x)D(4)(x)
]
δ(x− y),
{D(5)(x), D(4)(y)} = −
[
20D(7)(x) +
16
n
D(3)(x)D(4)(x)− 16
n
D(2)(x)D(5)(x)
]
∂
∂x
δ(x− y)
− 3
7
∂
∂x
[
20D(7)(x)+
16
n
D(3)(x)D(4)(x)− 16
n
D(2)(x)D(5)(x)
]
δ(x− y),
{D(5)(x), D(5)(y)} = −
[
25D(8)(x) +
36
n
D(3)(x)D(5)(x)− 20
n
D(2)(x)D(6)(x)
− 4
n
D(2)(x)D(3)2(x) +
4
n2
D(2)2(x)D(4)(x)
]
∂
∂x
δ(x− y)
− 1
2
∂
∂x
[
25D(8)(x) +
36
n
D(3)(x)D(5)(x)− 20
n
D(2)(x)D(6)(x)
− 4
n
D(2)(x)D(3)2(x) +
4
n2
D(2)2(x)D(4)(x)
]
,
{D(6)(x), D(4)(y)} = −
[
24D(8) +
12
n
D(4)2 − 12
n
D(2)D(6)
]
∂
∂x
δ(x− y)
− 3
8
∂
∂x
[
24D(8) +
12
n
D(4)2 − 12
n
D(2)D(6)
]
δ(x− y),
{D(7)(x), D(3)(y)} = −
[
21D(8) +
6
n
D(4)2 +
12
n
D(3)D(5) − 18
n
D(2)D(6)
]
∂
∂x
δ(x− y)
− 1
4
∂
∂x
[
21D(8) +
6
n
D(4)2 +
12
n
D(3)D(5) − 18
n
D(2)D(6)
]
δ(x− y),
{D(8)(x), D(3)(y)} = −
[
24D(9) − 12
n
D(2)D(7) +
24
n
D(4)D(5) − 24
n2
D(3)3 +
24
n2
D(2)D(3)D(4)
+
48
n2
D(2)2D(5)
]
∂
∂x
δ(x− y)− 2
9
∂
∂x
[
24D(9) − 12
n
D(2)D(7)
14 V.D. Gershun
+
24
n
D(4)D(5)− 24
n2
D(3)3+
24
n2
D(2)D(3)D(4)+
48
n2
D(2)2D(5)
]
δ(x− y),
{D(7)(x), D(4)(y)} = −
[
28D(9) − 8
n
D(3)D(6) − 24
n
D(2)D(7) +
32
n
D(4)D(5)
− 48
n2
D(2)D(3)D(4) +
48
n2
D(2)2D(5)
]
∂
∂x
δ(x− y)
− 1
3
∂
∂x
[
28D(9) − 8
n
D(3)D(6) − 24
n
D(2)D(7) +
32
n
D(4)D(5)
− 48
n2
D(2)D(3)D(4) +
48
n2
D(2)2D(5)
]
δ(x− y),
{D(6)(x), D(5)(y)} = −
[
30D(9) − 4
n
D(3)D(6) − 12
n
D(2)D(7) +
32
n
D(4)D(5) − 32
n2
D(3)3
+
24
n2
D(2)D(3)D(4) +
56
n2
D(2)2D(5)
]
∂
∂x
δ(x− y)
− 4
9
∂
∂x
[
30D(9) − 4
n
D(3)D(6) − 12
n
D(2)D(7) +
32
n
D(4)D(5)
− 32
n2
D(3)3 +
24
n2
D(2)D(3)D(4) +
56
n2
D(2)2D(5)
]
δ(x− y).
Appendix B
The invariant chiral currents J (2N) and V (2N) and the new dependent totally symmetric invariant
tensors for SO(2l + 1), SP (2l) groups can be obtained under different order of calculation of
trace of the product of the generators of corresponding algebras. Let us mark the matrix product
of three generators tµ in round brackets
(t(µtνtλ)) = vµνλρtρ.
A different position of this triplet inside of J2N produces different expressions for V 2N
J (10) = Tr[((t1t2t3)t4(t5t6t7)(t8t9t10))]U1 · · ·U10 = 2vk
123v
kl
45v
ln
67v
n
8910U1 · · ·U10 = 2V (10),
J (10) = Tr[((t1t2t3)(t4t5t6)(t7t8t9)t10)]U1 · · ·U10 = 2vk
123v
l
456v
n
789v
kln
10 U1 · · ·U10 = 2V (10,1),
J (12) = Tr[(t1(t2t3t4)t5(t6t7t8)t9(t10t11t12))]U1 · · ·U12
= 2vk
123v
kl
45v
ln
67v
nm
89 vm
101112U1 · · ·U12 = 2V (12),
J (12) = Tr[((t1t2t3)(t4t5t6)(t7t8t9)(t10t11t12))]U1 · · ·U12
= 2vk
123v
l
456v
n
789v
m
101112v
klnmU1 · · ·U12 = 2V (12,1),
J (14) = Tr[((t1t2t3)t4(t5t6t7)t8(t9t10t11)(t12t13t14))]U1 · · ·U14
= 2vk
123v
kl
45v
ln
67v
nm
89 vmp
1011v
p
121314U1 · · ·U14 = 2V (14),
J (14) = Tr[((t1t2t3)(t4t5t6)(t7t8t9)(t10t11t12)t13t14)]U1 · · ·U14
= 2vk
123v
l
456v
n
789v
m
101112v
klp
13 vnmp
14 U1 · · ·U14 = 2V (14,1).
Here we introduced the short notation tµk
= tk, Uµk = Uk and vk
µlµnµm
=vk
lnm. New invariant
chiral tensors do not lead to new invariant chiral currents.
Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP (n) Groups 15
Acknowledgments
The author would like to thank J.A. de Azcarraga for the stimulating discussion about non-
primitive invariant tensors on simple Lie algebras.
References
[1] Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156–1162.
[2] Okubo S., Das A., The integrability condition for dynamical systems, Phys. Lett. B 209 (1988), 311–314.
[3] Dubrovin B.A., Novikov S.P., Hamiltonian formalism of one-dimensional systems of hydrodynamic type and
the Bogolyubov–Whitham averaging method, Dokl. Akad. Nauk SSSR 270 (1983), 781–785 (in Russian).
[4] Dubrovin B.A., Novikov S.P., Hydrodynamics of weakly deformed soliton lattices. Differential geometry and
Hamiltonian theory, Russian Math. Surveys 44 (1989), no. 6, 35–124.
[5] Ferapontov E.V., Differential geometry of nonlocal Hamiltonian of hydrodynamic type, Funct. Anal. Appl.
25 (1991), 195–204.
[6] Mokhov O.I., Ferapontov E.V., Nonlocal Hamiltonian operators of hydrodynamic type that are connected
with metrics of constant curvature, Russian Math. Surveys 45 (1990), no. 3, 218–219.
[7] Mokhov O.I., Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems,
Russian Math. Surveys 53 (1998), no. 3, 515–622.
[8] Tsarev S.P., Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Dokl. Akad.
Nauk SSSR 282 (1985), 534–537.
[9] Maltsev A.Ya., On the compatible weakly nonlocal Poisson brackets of hydrodynamic type, Int. J. Math.
Math. Sci. 32 (2002), 587–614, nlin.SI/0111015.
[10] Ferapontov E.V., Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and appli-
cations, Amer. Math. Soc. Transl. (2) 170 (1995), 33–58.
[11] Mokhov O.I., The Liouville canonical form of compatible nonlocal Poisson brackets of hydrodynamic type,
and integrable hierarchies, Funct. Anal. Appl. 37 (2003), no 2, 103–113, math.DG/0201223.
[12] Mokhov O.I., Compatible Dubrovin–Novikov Hamiltonian operators, Lie derivative and integrable systems
of hydrodynamic type, Theoret. and Math. Phys. 133 (2002), 1557–1564, math.DG/0201281.
[13] Pavlov M.V., Multi-Hamiltonian structures of the Whitham equations, Russian Acad. Sci. Dokl. Math. 50
(1995), no 2, 220–223.
[14] Pavlov M.V., Integrability of Egorov systems of hydrodynamic type, Theoret. and Math. Phys. 150 (2007),
no. 2, 225–243, nlin.SI/0606017.
[15] Maltsev A.Ya., Novikov S.P., On the local systems Hamiltonian in the weakly non-local Poisson brackets,
Phys. D 156 (2001), 53–80, nlin.SI/0006030.
[16] Goldschmidt Y.Y., Witten E., Conservation laws in some two-dimensional models, Phys. Lett. B 91 (1980),
392–396.
[17] Aref’eva I.Ya., Kulish P.P., Nissimov E R., Pacheva S.J., Infinite set of conservation laws of the quantum
chiral field in 2D spacetime, preprint LOMI E-I-1978, unpublished.
[18] Gershun V.D., Integrable string models of hydrodynamic type, J. Kharkov University, Phys. Ser. Nucl.,
Part., Fields 657 (2005), no. 1 (26), 109–113.
[19] Gershun V.D., Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral
currents, Problems of Atomic Science and Technology 3(1) (2007), Part 1, 16–21.
[20] Dijkgraaf R., Witten E., Mean field theory, topological field theory, and multi-matrix models, Nuclear
Phys. B 342 (1990), 486–522.
[21] Dijkgraaf R., Verlinde H., Verlinde E., Topological strings in d < 1, Nuclear Phys. B 352 (1991), 59–86.
[22] Dubrovin B.A., Geometry of 2D topological field theories, Lecture Notes in Math. 1620 (1996), 120–348,
hep-th/9407018.
[23] Dubrovin B., Zhang Y., Bi-Hamiltonian hierarchies in 2D topological field theory at one-loop approximation,
Comm. Math. Phys. 198 (1998), 311–361, hep-th/9712232.
[24] Evans J.M., Hassan M., MacKay N.J., Mountain A.J., Local conserved charges in principal chiral models,
Nuclear Phys. B 561 (1999), 385–412, hep-th/9902008.
http://arxiv.org/abs/nlin.SI/0111015
http://arxiv.org/abs/math.DG/0201223
http://arxiv.org/abs/math.DG/0201281
http://arxiv.org/abs/nlin.SI/0606017
http://arxiv.org/abs/nlin.SI/0006030
http://arxiv.org/abs/hep-th/9407018
http://arxiv.org/abs/hep-th/9712232
http://arxiv.org/abs/hep-th/9902008
16 V.D. Gershun
[25] de Azcarraga J.A., Macfarlane A.J., Mountain A.J., Perez Bueno J.C., Invariant tensors for simple groups,
Nuclear Phys. B 510 (1998), 657–687, physics/9706006.
[26] Pohlmeyer K., A group-theoretical approach to the quantization of the free relativistic closed string, Phys.
Lett. B 119 (1982), 100–104.
[27] Meusburger C., Rehren K.-H., Algebraic quantization of the closed bosonic string, Comm. Math. Phys. 237
(2003), 69-85, math-ph/0202041.
[28] Thiemann T., The LQS string-loop quantum gravity quantization of string theory. I. Flat target space,
Classical Quantum Gravity 23 (2006), 1923–1970, math-ph/0401172.
[29] Klein A., Invariant operators of the unitary unimodular group in n dimensions, J. Math. Phys. 4 (1963),
1283–1284.
[30] Sudbery A., Some aspects of chiral [δ]u(3)× [δ]u(3) symmetry in hadron dynamics, PhD Thesis, Cambridge
University, 1970.
Sudbery A., Computer-friendly d-tensor identities for SU(n), J. Phys. A: Math. Gen. 23 (1990), L705–L710.
http://arxiv.org/abs/physics/9706006
http://arxiv.org/abs/math-ph/0202041
http://arxiv.org/abs/math-ph/0401172
1 Introduction
2 String model of principal chiral model type
3 Integrable string models of hydrodynamic type with null torsion
4 New string equation in terms of Pohlmeyer tensor nonlocal currents
5 Integrable string models with constant torsion
Appendix A
Appendix B
References
|