Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents
We considered two types of string models: on the Riemann space of string coordinates with null torsion and on the Riemann-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable systems and the Dubrovin solutions of the WDVV asso...
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| Published in: | Вопросы атомной науки и техники |
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| Date: | 2007 |
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| Format: | Article |
| Language: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Cite this: | Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents / V.D. Gershun // Вопросы атомной науки и техники. — 2007. — № 3. — С.16-21. — Бібліогр.: 27 назв. — англ. |
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| author | Gershun, V.D. |
| author_facet | Gershun, V.D. |
| citation_txt | Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents / V.D. Gershun // Вопросы атомной науки и техники. — 2007. — № 3. — С.16-21. — Бібліогр.: 27 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | We considered two types of string models: on the Riemann space of string coordinates with null torsion and on the Riemann-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable systems and the Dubrovin solutions of the WDVV associativity equation to construct new integrable string models of hydrodynamic type on the torsion less Riemann space of chiral currents in the 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 models of hydrodynamic type on the Riemann-Cartan space of invariant chiral currents and on the Casimir operators, considered as the Hamiltonians, in the second case.
Розглянуто два типу струнних моделей: на просторі Рімана струнних координат з нульовим скрутом та на просторі Рімана-Картана з постійним скрутом. В першому випадку, ми використали гідродинамічний підхід Дубровіна, Новікова до інтегрованих систем та розв’язок Дубровіна рівняння асоціативності ВДВВ, щоб побудувати нові інтегровані струнні моделі гідродинамічного типу на безскрутному просторі Рімана кіральних токів. У другому випадку використали інваріантні локальні кіральні токи SU(n), SO(n), SP(n)-моделі головного кірального поля, щоб побудувати нові інтегровані струнні моделі гідродинамічного типу на просторі Рімана-Картана інваріантних кіральних токів та на операторах Казіміра, розглянутих як гамільтоніани.
Рассмотрены два типа струнных моделей: на пространстве Римана струнных координат с нулевым кручением и на пространстве Римана-Картана с постоянным кручением. В первом случае использовали гидродинамический подход Дубровина, Новикова к интегрированным системам и Дубровина решения ВДВВ уравнения ассоциативности, чтобы построить новые итегрированные струнные модели гидродинамического типа на пространстве Римана киральных токов с нулевым кручением. Во втором случае использовали локальные инвариантные киральные токи в модели главного кирального поля для SU(n), SO(n), SP(n)-групп, чтобы построить новые интегрированные струнные модели гидродинамического типа на Римана-Картана-пространстве инвариантных киральных токов и на операторах Казимира, рассматриваемых как гамильтонианы.
|
| first_indexed | 2025-12-07T18:33:45Z |
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INTEGRABLE STRING MODELS AND SIGMA-MODELS
OF HYDRODYNAMIC TYPE IN TERMS
OF INVARIANT CHIRAL CURRENTS
V.D. Gershun
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
e-mail: mailto: gershun@kipt.kharkov.ua
We considered two types of string models: on the Riemann space of string coordinates with null torsion and on
the Riemann-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Du-
brovin, Novikov to integrable systems and the Dubrovin solutions of the WDVV associativity equation to construct
new integrable string models of hydrodynamic type on the torsion less Riemann space of chiral currents in the first
case. We used the invariant local chiral currents of principal chiral models for SU(n), SO(n), SP(n) groups to con-
struct new integrable string models of hydrodynamic type on the Riemann-Cartan space of invariant chiral currents
and on the Casimir operators, considered as the Hamiltonians, in the second case.
PACS: 74.50.+R, 74.80.FP
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 operator of charge algebra can be considered as
Hamiltonian in bi-Hamiltonian approach to integrable
models.
Magri [1] initiated the bi-Hamiltonian approach to
integrable systems. Two Poisson brackets (PB’s):
))(,()}(),({
),)(,()}(),({
11
00
ϕϕϕ
ϕϕϕ
yxPyx
yxPyx
abba
abba
=
= (1)
are called compatible, if any linear combination of these
PB’s is PB also for arbitrary constant
λ. The functions ϕ are local coordi-
nates on a certain given smooth n-dimensional manifold
. The Hamiltonian operators
10 *,*}{,*}{* λ+
ta ,( nax ,...,2,1), =
nM
))(,(),)(,( 10 ϕϕ yxPyxP abab
are the functions of local coordinates ϕ . It is pos-
sible to find such Hamiltonians and , which
satisfy the bi-Hamiltonian condition [2]:
)(xa
H0H 1
,}),({}),({)(
1100 HxHx
dt
xd aa
a
ϕϕ
ϕ
== (2)
where ∫ ==
π
ϕ
2
0
.1,0,))(( MdyyhH MM
Two branches of hierarchies arise under two equa-
tions of motion under two different parameters of
evolution t and [2]: M0 0Mt
;
)(
),(),(
)(
),(}),({)(
0
2
0
0
2
0
0
110
01
dy
y
h
yzPzxR
dy
y
h
yxPHx
dt
xd
b
cba
c
b
aba
a
ϕ
ϕ
ϕ
ϕ
π
π
∂
∂
=
∂
∂
==
∫
∫
.
)(
),(),()(
)(
),(}),({)(
0
2
0
0
1
2
0
1
001
10
dy
y
hyzPzxR
dy
y
hyxPHx
dt
xd
b
cbac
b
aba
a
ϕ
ϕ
ϕ
ϕ
π
π
∂
∂
=
∂
∂
==
∫
∫
−
),( yxRa
b ),()1 yxR a
b
− is recursion operator and ( is its
inverse:
∫ −=
π2
0
101 .),())(,(),( dzyzPzxPyxR bc
abac (3)
The first branch of the hierarchy of dynamical systems
has the following form:
2
1 2
0 0
( ) ( ( , )... ( , ))
a
a
M M
M
d x R x y R y y
dt
πϕ
− −= ∫ 1 c
× 0
0 1 1( , ) ...
( )
cb
M M Mb
M
h
P y y dy dy
yϕ
−
∂
∂
.
The second branch of the hierarchy can be obtained by
replacement → and t → . Dubrovin,
Novikov [3, 4] and Tsarev [5] introduced the local PB
of hydrodynamical type for Hamiltonian description of
equations of hydrodynamics. Ferapontov [6] and Mok-
hov, Ferapontov [7] generalized it on the non-local PB’s
of hydrodynamic type. Integrable systems of hydrody-
namic type are described by Hamiltonians of hydrody-
namic type, which are not depending of derivatives of
local coordinates. Integrable bi-Hamiltonian systems of
hydrodynamic type were considered by Maltsev [8],
Ferapontov [9], Mokhov [11], Pavlov [12], Maltsev,
Novikov [13]. Polynomials of local chiral currents were
considered by Goldshmidt and Witten [14] (see also
R 1−R M0 0Mt
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p. 16-21. 16
[15]). Local conserved chiral charges in principal chiral
models were considered by Evans, Hassan, MacKay,
Mountain [23]. Integrable string models of hydrody-
namic type were considered by author [16,17].
2. STRING MODEL OF SIGMA-MODEL
TYPE
String model is described by the Lagrangian
2
0
1 ( , )( ( , ))
2
a b
ab
t x t xL g t x
x x
π
αβ
α β
ϕ ϕ
η ϕ
∂ ∂
=
∂ ∂
∫
( , ) dx
and by two first kind constraints:
( ) ( )( ( ))[
( ) ( )] 0,
( ) ( )( ( )) 0 .
a b
ab
a b
a b
ab
x xg x
t t
x x
x x
x xg x
t x
ϕ ϕ
ϕ
ϕ ϕ
ϕ ϕ
ϕ
∂ ∂
∂ ∂
∂ ∂
+ ≈
∂ ∂
∂ ∂
≈
∂ ∂
− −
The target space of local coordinates ϕ
belong to certain given smooth n-
dimensional manifold with nongenerated metric
tensor where
are indexes of tangent space for the mani-
fold in some point ϕ The veilbein
and its inverse satisfy to the conditions:
( ),a x
)(ϕµ
ae
1,...,a =
,...,1, =νµ
nM
µ
µ = b
a
b
a ee
n
nM
aµν
)ϕ
.µν
( ( )) ( ( )) ( ( )) ,ab bg x e x e xµ νϕ η ϕ ϕ=
n
.)(xa
(µ
ae
, νµ ηδ =aa ee
The coordinates belong to the
world sheet with metric tensor in the conformal
gauge. String equations of motion have the form:
),( 10 xxtxx ==α
)(xgαβ
,0])([ =∂∂+∂ cba
bc
a Г ϕϕϕϕη βααβ
αβ
where ][
2
1)(
b
c
c
baa
bc
ee
eГ
ϕϕ
ϕ
µµ
µ
∂
∂
+
∂
∂
= is connection.
In terms of the canonical chiral currents
( ) ( ) ,a
aJ e
x
µ µ
α α α α
ϕ ϕ ϕ
∂
= ∂ ∂ =
∂
equations of motion
have the form:
( ( , )) 0 ,
( ) ( ) ( ) ( ) ( ) 0 ,
J t x
J J C J J
µαβ
α β
µ µµ ν λ
α β α α ββ νλ
η ϕ
ϕ ϕ ϕ ϕ ϕ
∂ =
∂ − ∂ − =
where ][
2
1)(
a
b
b
aba eeeeC
ϕϕ
ϕ
µµ
λν
µ
νλ ∂
∂
−
∂
∂
= is torsion. The
Hamiltonian has form:
,][
2
1
110
2
0
0 dxJJJJH νµ
µνν
π
µ
µν ηη += ∫ (4)
where
x
eJpeJ
a
aa
a
∂
∂
==
ϕ
ϕϕϕϕ µµ
µµ )()(,)()( 10
)x
and
is canonical momentum. The canonical com-
mutation relations of currents are the following:
,(tpa
0 0 0
0 1 1
11
{ ( ), ( )} ( ( )) ( ( )) ( ) ,
{ ( ), ( )} ( ( )) ( ( )) ( )
1 ( ) ,
2
{ ( ), ( )} 0 .
J x J y C x J x x y
J x J y C x J x x y
x y
x
J x J y
λ
µ ν µν λ
ν ν λ
µ µλ
ν
µ
µ ν
ϕ ϕ δ
ϕ ϕ δ
δ δ
= −
= −
∂
− −
∂
=
Let us introduce chiral currents:
., 1010
µ
ν
µνµµ
ν
µνµ ηη JJVJJU −=+=
The commutation relations of the chiral currents are the
following:
3 1{ ( ), ( )} [ ( ) ( )] ( )
2 2
( ) ,
{ ( ), ( )} [ ( ) ( )] ( ) ,
3 1{ ( ), ( )} [ ( ) ( )] ( )
2 2
( ) .
U x U y C U x V x x y
x y
x
U x V y C U x V x x y
V x V y C V x U x x y
x y
x
µνµ ν λ λ
λ
µν
µνµ ν λ λ
λ
µνµ ν λ λ
λ
µν
δ
η δ
δ
δ
η δ
= −
∂
= + −
= −
∂
+ −
∂
(5)
−
∂
−
The equations of motion in the light-cone coordinates
xtx
xtx
∂
∂
±
∂
∂
=
∂
∂
±=
±
± ,)(
2
1 have the form:
, .U C U V V C V Uµ µµ ν λ µ ν
νλ νλ− +∂ = ∂ = λ
In the case of null torsion:
0)(,0)(
,)(,0)(
2
==
∂∂
∂
=
∂
∂
==
ϕ
ϕϕ
ϕ
ϕ
ϕϕ
µ
νλρ
µ
µ
µµµ
νλ
ReeГ
eeC
cb
aa
bc
aa
string model is integrable one. The Hamiltonian (4)
describes two independent left and right movers:
and V )( xtU +µ .)( xt −µ
3. INTEGRABLE STRING MODELS
OF HYDRODYNAMIC TYPE
WITH NULL TORSION
We want to construct new integrable string models
with Hamiltonians, which are polynomials of the initial
chiral currents U . The PB of chiral currents
coincides with the flat PB of Dubrovin, No-
vikov
))(( xϕµ
)(xU µ
. )()}(),({ 0 yx
x
yUxU −
∂
∂
−= δη µννµ
Let us introduce local Dubrovin, Novikov PB [3,4]. It
has the form:
.)()())((
)())(()}(),({
yx
x
xUxUГ
yx
x
xUgyUxU
−
∂
∂
−
−
∂
∂
=
δ
δ
λµν
λ
µννµ
(6)
The PB is skew-symmetric if and it
satisfy to Jacobi identity [18] if
)()( UgUg νµµν =
17
.0)(,0)(,)()( === URUCUГUГ a
bcd
a
bc
a
cb
a
bc
In the case of non-zero curvature tensor we must in-
clude [6,7] Weingarten operators to right side of PB
with the step-function
. )()()()sgn( 1 yxyx
dx
dyx −=−=− − νδ
The PB’s {*,*} and {*,*} are compatible by Magri
[1] if pencil {*,*} is PB also. As a result,
Mokhov [11,10] have obtained compatible pair of PB’s:
0 1
1{*,*}λ0 +
;)(),)((0 yx
x
yxUP −
∂
∂
−= δηµνµν (7)
.)())((
)())((2),)((
3
2
1
yx
x
U
UUU
xUF
yx
xUU
xUFyxUP
−
∂
∂
∂∂∂
∂
+
−
∂
∂
∂∂
∂
=
δ
δ
λ
λνµ
νµµν
(8)
The function F(U) satisfies equation:
.)()(
)()(
33
33
λσµλρν
λσνλρµ
UUU
UF
UUU
UF
UUU
UF
UUU
UF
∂∂∂
∂
∂∂∂
∂
=
∂∂∂
∂
∂∂∂
∂
(9) 0∂
This equation is WDVV [19,20] associativity equa-
tion and it was obtained in 2D topological field theory.
Dubrovin [21,22] obtained a lot of solutions of WDVV
equation. He showed, that local fields U must
belong to Frobenius manifolds to solve the WDVV
equation and gave examples of Frobenius structures.
Associative Frobenius algebra may be written in the
following form:
)(xµ
.)(:*
λ
λ
µννµ U
Ud
UU ∂
∂
=
∂
∂
∂
∂
Totally symmetric structure function has form:
( )( ) , 1,...,F Ud U
U U U
µνλ µ ν λ
µ
∂
= =
∂ ∂ ∂
. n
The associativity condition
)*(**)*(
λνµλνµ UUUUUU ∂
∂
∂
∂
∂
∂
=
∂
∂
∂
∂
∂
∂
leads to the WDVV equation. Function F(U) is quasi-
homogeneous function of its variables:
.),...,(),...,( 111 ndndd UUFUUF Fn λλλ =
Frobenius manifolds can be realized as Coxeter
groups (group of reflections). Coxeter groups of corre-
spondent simple Lie algebras (SU(n),SO(n),SP(n)) are
Weyl groups. Dubrovin examples of certain solutions of
WDVV equation are:
23 2
1 1
11, ( ) ; 2, ( ) ,
2
3, ( ) , 1, 2,3.
Un F U U n F U U U e
n F U d U U Uµ ν λ
µνλ µ
= = = = +
= = =
2
(10)
We used local fields U with the low indexes there for
convenience. One of Dubrovin polynomial solution is:
µ
.
60
1
4
1)(
2
1)( 5
3
2
3
2
2
2
2132
1 UUUUUUUUF +++= (11)
In the bi-Hamiltonian approach to integrable string
model we must construct the recursion operator to gen-
erate hierarchy of PB’s and hierarchy of Hamiltonians:
.)(
)(
)(
)(
2
),())(,(),(
22
2
0
1
01
yx
UU
UF
dx
dyx
UU
UF
dzyzPzxPyxR
−
∂∂
∂
+−
∂∂
∂
=
= ∫ −
νδ
νµνµ
π
λν
µλµ
ν
The Hamiltonian equation of motion with Hamiltonian
is the following: 0H
. (12) ,)()(
2
0
0 ∫ ∂
∂
=
∂
∂
=
π µµ
νµ
µνη
x
U
t
UdxxUxUH
First of new equations of motion under new time t
has the form [11]:
01
).
)(
(
)(
),(
2
01 ν
µν
π νµ
ν
µ
η
U
xF
dx
ddy
y
yU
yxR
t
U
∂
∂
=
∂
∂
=
∂
∫ (13)
This equation of motion can be obtained as Hamiltonian
equation with the new Hamiltonian : 1H
,)())((2
0
1 dxxU
U
xUFH µ
π
µ∫ ∂
∂
= (14)
where F(U) is each of the Dubrovin solutions of
WDVV associativity equation (10), (11).
Any system of the following hierarchy [11]:
∏∫
=
− ∂
∂
=
∂
∂ M
k
k
M
MM
M
dy
y
UyyRyxR
t
U
1
2
0
,11
0
))()...,((
νπ
µ
ν
µ
is integrable system. As a result, we will obtain chiral
currents where
is solution of equation of motion. In the case of
Hamiltonian and the equation of motion (27) one
can introduce new currents:
,)),(()),(( 00 xtfxtU MM ϕϕ µµ =
1H
)(ϕµf
.),(),(
),,(),(
01
011
01010
ν
µνµ
µµ
η
U
xtFxtJ
xtUxtJ
∂
∂
=
=
Consequently, we can introduce new metric tensor
and new velbein depending on new time coordinate.
New string equation of motion has form:
.
)),((
))),(((
)),((),()),((
01
01
0101
01
xtf
xtfF
dx
xtde
x
xtxte
a
a
ϕ
ϕ
η
ϕϕ
ϕ
ν
µν
µµ
∂
∂
=
=
∂
∂
4. INTEGRABLE STRING MODELS
WITH CONSTANT TORSION
Let us come back to commutation relations of chiral
currents. Let torsion С and are struc-
ture constant of simple Lie algebra. We will consider
( ( )) 0xµ
νλ ϕ ≠ Cµ
νλ
18
string model with constant torsion in light-cone gauge
in target space. This model coincides to principal chiral
model on compact simple Lie group. We cannot divide
motion on right and left mover because the chiral cur-
rents , do not
conserve. The correspondent charges are not Casimirs.
Evans, Hassan, MacKay, Mountain ([27] and other ref-
erences therein) constructed local invariant chiral cur-
rents as polynomial of initial chiral currents of SU(n),
SO(n), SP(n) principal chiral models and they found
commutative combination of them. Correspondent
charges are Casimir operators of these dynamical sys-
tems. This paper was based on the paper of de Azcar-
raga, Macfarlane, MacKay, Perez Buena ([24] and other
references therein) about invariant tensors for simple
Lie group.
U C U Vµµ ν
νλ−∂ =
tµ
λ V C V Uµµ ν
νλ+∂ =
1( ) .
2
Tr t tµ ν µνδ= = −] ,t t C tλ
µ ν µν λ
1{ , }t t
nµ ν µν= − −
1 1
( ... )
M M
d STr t tµ µ= =
1... M
dµ µ
M n≥
( ) M
MJ U STr U d= =
U t U µ
µ= 1,...,µ =
λ
Let be a matrix representation of generators Lie
algebra: [ ,
There is additional relation for SU(n) algebra:
2, 1,..., 1.id t nλ
µν λδ µ = −
Invariant tensors have the following form:
1 2
31 2 3 1... )( ... ,
M M
k k
kkd d dµ µ µ µµ µ µ − −1 M
where STr means Tr of completely symmetrized prod-
uct of matrix and is totally symmetric tensor.
There are n-1 primitive tensors for SU(n). The invariant
tensors for are functions of primitive tensors.
De Azcarraga et al. gave some examples of these
functions and they gave general method to calculate
them. Evans et al. introduced local chiral currents based
on the symmetric tensors of simple Lie algebra:
1
1
,M
M
U Uµ µ
µ µ
where The commutation
relations of invariant currents SU(n) [23] are PB’s of
hydrodynamics type:
2 1.n −
2
1 1
{ ( ), ( )} [ (
( ) ( )] ( )
M N M N
M N
J x J x MN J x
MN J x J x x y
n x
+ −
− −
= −
∂
− δ
∂
)
−
2
1
1
( )( 1)[
2
( )
( ) ] ( ) .
M N
N
M
J xMN N
M N x
J xMN J x x y
n x
δ
+ −
−
−
∂−
−
+ − ∂
∂
− −
∂
Let us note, that ultra local term in commutation rela-
tion of chiral currents U (5) does not contribute to the
commutation relations of invariant chiral currents be-
cause of totally symmetric invariant tensors
µ
1... .
M
dµ µ
Therefore, chiral currents form closed
algebra under canonical PB. Evans et.al. found con-
served combination of invariant chiral currents:
( ( ))MJ U x
2 2 3 3
2
4 4 2
5 5 2 3
2
6 6 3
3
4 2 22
( ) ( ) , ( ) ( ) ,
3( ) ( ) ( ) ,
2
10( ) ( ) ( ) ( ) ,
3
5( ) ( ) ( )
3
15 25( ) ( ) ( ).
4 8
K U J U K U J U
K U J U J U
n
K U J U J U J U
n
K U J U J U
n
J U J U J U
n n
= =
= −
= −
= − −
+
Corresponding charges of chiral currents are
Casimir operators.
( )MK U
Let us apply the hydrodynamics approach to inte-
grable string models with constant torsion. In this case
we must consider the conserved chiral currents
as the local fields of Rie-
mann-Cartan manifold. The corresponding local charges
form the hierarchy of new Hamiltonians in bi-
Hamiltonian approach to integrable systems. The com-
mutation relations of invariant chiral currents are local
PB’s of hydrodynamics type. The metric tensor
for the SU(3) group has the following form:
( ( )) , 2,3,..., 1MK U x M n=
( )MNg K
−
2 3
2
3 0 2
4 6
( ) ( )
6MN
K K
g K
K c K
− −
= − −
.
The metric tensor for SU(4) group is the following:
24 | 6 | 8 32 3 4 2
45 126 | 9 |3 4 2 3 28 2( ) ( ) .12 28 3 | |4 2 3 2 1 3 2 4 22
3
3 2
MN
K K K K
K K K K K
g K
K K K K c K c K K
c K
− − − +
− − − − =
− + − +
+
The equations of motion for local fields are
the following:
( ( ))MK U x
2
0
( ( ))
{ ( ( )), ( ( )) ,
2,3,..., 1 ; 2 .
M
M P
P
K U x
K U x K U y dy
t
M n P
π∂
=
∂
= − ≥
∫
The similar method of construction of chiral currents
for SO(2n+1), SP(n) groups was used by Evans et al. on
the basis of symmetric invariant tensors of Azcarraga
et al. Symmetric product of three generators of these
algebras let to introduce symmetric structure tensor:
( ) .t t t V tρ
µ ν λ ρµνλ=
Invariant symmetric tensors have the form:
1 2
1 2 2 3 2 2 2 1 21 2 3 4 5 1... )( ... .
M M M
V V V Vν ν
µ µ ν µ µ µµ µ µ µ µ ν − − −
=
M M
The invariant chiral currents can be constructed
from invariant symmetric tensor and initial chiral cur-
rents U :
2MJ
µ
1 2
1 22 ... ... .M
MMJ V U Uµ µ
µ µ=
The commutation relations of invariant chiral currents
are PB’s of hydrodynamics type:
19
2
2
{ ( ), ( )} ( ) (
( )( 1) ( ) .
2
M N M N
M N
J x J x MN J x x y
x
J xMN N x y
M N x
δ
δ
+ −
+ −
∂
= − −
∂
∂−
− −
+ − ∂
)
Conserved densities of Casimir operators of SU(2n+1),
SP(n) group have the form:
2
2 2 4 4 2
2 3
6 6 4 2
2
8 8 6 2 4
2 2 3 4
4 2 2
1( ) , ( ) (3 ) ,
2
3 1( ) (5 ) (5 ) ,
4 8
2 1( ) (7 ) (7 )
3 4
1(7 ) (7 ) .
48
K U J K U J J
K U J J J J
K U J J J J
J J J
α
α α
α α
α α
= = −
= − +
= − − +
−
2
Constant parameter α is arbitrary one.
5. INTEGRABLE STRING MODELS
IN TERMS OF POLMEYER TENSOR
NONLOCAL CURRENTS
In the case of flat space C , there are nonlocal
tensor totally symmetric chiral currents, such called as
“Polmeyer” currents [25-27]:
0µ
νλ =
1
2
1 2
...( )
( )
1 1 1 1
0 0
( ( )) ( ( ))
( ) ( ) ... ( ) ,
M
M
M
M
xx
M M
R U x R U x
U x U x dx U x dx
µ µ
µ µ µ
−
− −
≡
= ∫ ∫
where round brackets mean totally symmetric product
of the chiral currents U New Hamiltonians have
the following form:
( ) .xµ
2
( ) ( ) ( )
0
1 ( ( )) ( ( )) ,
2
M M MH R U x M R U x dx
π
= ∫
where M is totally symmetric invariant constant tensor,
which can be constructed from Kronecker deltas. For
example:
2
(2)
1 1
0
2
1 1
0
1 [ ( ) ( )
2
( ) ( ) ] ,
R R U x U x dx
U x U x dx
π
µν µ ν
π
ν µ
≡ =
+
∫
∫
2 2 2
(2)
1 1 2 2
0 0 0
2 2
1 1 2 2
0 0
1 [ ( ) ( ) ( ) ( )
2
( ) ( ) ( ) ( ) ] .
H U x U x U x dx U x
U x U x U x dx U x dx dx
π π π
µ µ ν ν
π π
µ ν µ ν
=
+
∫ ∫ ∫
∫ ∫
dx
The Hamiltonian commutes with Hamiltonian (2)H
2
(1)
0
1 ( )
2
H U x
π
= ∫
2
0
( )U x d
π
µ∫
(2)H
( )x Uµ µ
.x
dx and it commutes with the
Casimir The equation of motion under the
Hamiltonian is the following:
2 2
1 1 2 2
0 0
2 2
1 1 2 2
0 0
2 2
1 1 1 1
0 0
( ) [ ( ) ( ) ( )
( ) ( ) ( ) ]
( ) ( ) ( ) ( ) ( ) ( ) .
U x U x U x dx U x dx
t x
U x U x dx U x dx
U x U x U x dx U x U x U x dx
π πµ
µ ν ν
π π
ν µ ν
π π
ν ν µ µ ν ν
∂ ∂
=
∂ ∂
+
− −
∫ ∫
∫ ∫
∫ ∫
In the variables the last equation
can be rewritten as the following:
1 1
0
( ) ( )
x
S x U x dxµ µ= ∫
2
12
10
[ ( )] ( )
1,..., .
xS SS S S S S dx
t x x
n
µ ν
µ ν ν µ ν
µ
∂ ∂ ∂
= +
∂ ∂ ∂
=
∫
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ИНТЕГРИРОВАННЫЕ СТРУННЫЕ МОДЕЛИ И СИГМА-МОДЕЛИ ГИДРОДИНАМИЧЕСКОГО
ТИПА В ТЕРМИНАХ ИНВАРИАНТНЫХ КИРАЛЬНЫХ ТОКОВ
В.Д. Гершун
Рассмотрены два типа струнных моделей: на пространстве Римана струнных координат с нулевым кру-
чением и на пространстве Римана-Картана с постоянным кручением. В первом случае использовали гидро-
динамический подход Дубровина, Новикова к интегрированным системам и Дубровина решения ВДВВ
уравнения ассоциативности, чтобы построить новые итегрированные струнные модели гидродинамического
типа на пространстве Римана киральных токов с нулевым кручением. Во втором случае использовали ло-
кальные инвариантные киральные токи в модели главного кирального поля для SU(n), SO(n), SP(n)-групп,
чтобы построить новые интегрированные струнные модели гидродинамического типа на Римана-Картана-
пространстве инвариантных киральных токов и на операторах Казимира, рассматриваемых как гамильто-
нианы.
ІНТЕГРОВАНІ СТРУННІ МОДЕЛІ ТА СИГМА-МОДЕЛІ ГІДРОДИНАМІЧНОГО ТИПУ
У ТЕРМІНАХ ІНВАРІАНТНИХ КІРАЛЬНИХ ТОКІВ
В.Д. Гершун
Розглянуто два типу струнних моделей: на просторі Рімана струнних координат з нульовим скрутом та
на просторі Рімана-Картана з постійним скрутом. В першому випадку, ми використали гідродинамічний
підхід Дубровіна-Новікова до інтегрованих систем та розв’язок Дубровіна рівняння асоціативності ВДВВ,
щоб побудувати нові інтегровані струнні моделі гідродинамічного типу на безскрутному просторі Рімана
кіральних токів. У другому випадку використали інваріантні локальні кіральні токи SU(n), SO(n), SP(n)-
моделі головного кірального поля, щоб побудувати нові інтегровані струнні моделі гідродинамічного типу
на просторі Рімана-Картана інваріантних кіральних токів та на операторах Казіміра, розглянутих як гаміль-
тоніани.
21
|
| id | nasplib_isofts_kiev_ua-123456789-110898 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:33:45Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Gershun, V.D. 2017-01-06T18:06:32Z 2017-01-06T18:06:32Z 2007 Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents / V.D. Gershun // Вопросы атомной науки и техники. — 2007. — № 3. — С.16-21. — Бібліогр.: 27 назв. — англ. 1562-6016 PACS: 74.50.+R, 74.80.FP https://nasplib.isofts.kiev.ua/handle/123456789/110898 We considered two types of string models: on the Riemann space of string coordinates with null torsion and on the Riemann-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable systems and the Dubrovin solutions of the WDVV associativity equation to construct new integrable string models of hydrodynamic type on the torsion less Riemann space of chiral currents in the 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 models of hydrodynamic type on the Riemann-Cartan space of invariant chiral currents and on the Casimir operators, considered as the Hamiltonians, in the second case. Розглянуто два типу струнних моделей: на просторі Рімана струнних координат з нульовим скрутом та на просторі Рімана-Картана з постійним скрутом. В першому випадку, ми використали гідродинамічний підхід Дубровіна, Новікова до інтегрованих систем та розв’язок Дубровіна рівняння асоціативності ВДВВ, щоб побудувати нові інтегровані струнні моделі гідродинамічного типу на безскрутному просторі Рімана кіральних токів. У другому випадку використали інваріантні локальні кіральні токи SU(n), SO(n), SP(n)-моделі головного кірального поля, щоб побудувати нові інтегровані струнні моделі гідродинамічного типу на просторі Рімана-Картана інваріантних кіральних токів та на операторах Казіміра, розглянутих як гамільтоніани. Рассмотрены два типа струнных моделей: на пространстве Римана струнных координат с нулевым кручением и на пространстве Римана-Картана с постоянным кручением. В первом случае использовали гидродинамический подход Дубровина, Новикова к интегрированным системам и Дубровина решения ВДВВ уравнения ассоциативности, чтобы построить новые итегрированные струнные модели гидродинамического типа на пространстве Римана киральных токов с нулевым кручением. Во втором случае использовали локальные инвариантные киральные токи в модели главного кирального поля для SU(n), SO(n), SP(n)-групп, чтобы построить новые интегрированные струнные модели гидродинамического типа на Римана-Картана-пространстве инвариантных киральных токов и на операторах Казимира, рассматриваемых как гамильтонианы. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum field theory Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents Інтегровані струнні моделі та сигма-моделі гідродинамічного типу у термінах інваріантних кіральних токів Интегрированные струнные модели и сигма-модели гидродинамического типа в терминах инвариантных киральных токов Article published earlier |
| spellingShingle | Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents Gershun, V.D. Quantum field theory |
| title | Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents |
| title_alt | Інтегровані струнні моделі та сигма-моделі гідродинамічного типу у термінах інваріантних кіральних токів Интегрированные струнные модели и сигма-модели гидродинамического типа в терминах инвариантных киральных токов |
| title_full | Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents |
| title_fullStr | Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents |
| title_full_unstemmed | Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents |
| title_short | Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents |
| title_sort | integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents |
| topic | Quantum field theory |
| topic_facet | Quantum field theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110898 |
| work_keys_str_mv | AT gershunvd integrablestringmodelsandsigmamodelsofhydrodynamictypeintermsofinvariantchiralcurrents AT gershunvd íntegrovanístrunnímodelítasigmamodelígídrodinamíčnogotipuutermínahínvaríantnihkíralʹnihtokív AT gershunvd integrirovannyestrunnyemodeliisigmamodeligidrodinamičeskogotipavterminahinvariantnyhkiralʹnyhtokov |