On integrability of 2-dimensional σ-models of Poisson-Lie type
HJE
On integrability of 2-dimensional -models of Poisson-Lie type
Pavol Severa 0
0 Section of Mathematics, University of Geneva
We describe a simple procedure for constructing a Lax pair for suitable 2dimensional -models appearing in Poisson-Lie T-duality 1Supported in part by the grant MODFLAT of the European Research Council and the NCCR SwissMAP of the Swiss National Science Foundation.
Integrable Field Theories; Sigma Models; String Duality
1 Introduction
2
3
4
5
terms of d-valued 1-forms A 2
1
( ; d) satisfying
There is a class of 2-dimensional -models, introduced in the context of Poisson-Lie
Tduality [5], whose solutions are naturally described in terms of certain at connections. The
target space of such a -model is D=H, where D is a Lie group and H
D a subgroup. The
-model is de ned by the following data: an invariant symmetric non-degenerate pairing
h; i on the Lie algebra d such that the Lie subalgebra h
d is Lagrangian, i.e. h? = h, and
a subspace V+
d such that dim V+ = (dim d)=2 and such that h; ijV+ is positive de nite.
The construction and properties of these
-models are recalled in section 2 (including
the Poisson-Lie T-duality, which says that the -model, seen as a Hamiltonian system, is
essentially independent of H). Let us call them
-models of Poisson-Lie type.
The solutions
! D=H of equations of motion of such a -model can be encoded in
dA + [A; A]=2 = 0
A 2
1;0( ; V+)
0;1( ; V );
(1.1a)
(1.1b)
where V
:= (V+)?
d. Namely, the atness (1.1a) of A implies that there is a map
` : ~ ! D (where ~ is the universal cover of ) such that A =
of A is in H then ` gives us a well-de ned map
in this way are exactly the solutions of equations of motion.
d` ` 1. If the holonomy
! D=H. The maps
! D=H obtained
As rst observed by Klimc k [3], and later by Sfetsos [12], and Delduc, Magro, and
Vicedo [2], some -models of Poisson-Lie type are integrable. Their integrability is proven
by nding a Lax pair, i.e. a 1-parameter family of at connections (with parameter )
A
2
1
( ; g)
dA + [A ; A ]=2 = 0
of the phase space, i.e. for every A 2
1
( ; d) satisfying (1.1).
{ 1 {
The aim of this note is to make the construction of A transparent. We simply observe
1
that if A 2
( ; d) satis es (1.1) and if p : d ! g is a linear map such that
A suitable family p : d ! g will then give us a family of at connections
leave this question open.
2
-models of Poisson-Lie type and Poisson-Lie T-duality
In this section we review the properties of the \2-dimensional -models of Poisson-Lie type"
introduced in [5] (together with their Hamiltonian picture from [6] and using the target
spaces of the form D=H, as introduced in [7]).
Let d be a Lie algebra with an invariant non-degenerate symmetric bilinear form h; i
of symmetric signature and let V+
d be a linear subspace with dim V+ = (dim d)=2, such
that h; ijV+ is positive-de nite.
Let M = D=H where D is a connected Lie group integrating d and H
D is a closed
connected subgroup such that its Lie algebra h
d is Lagrangian in d.
This data de nes a Riemannian metric g and a closed 3-form
on M . They are
given by
g( (X); (Y )) =
p
=
1
2
hX; Y i
1
2 D +
8X; Y 2 V+
1
2 dhA; Li
right-translates of V+.
1
Here
is the action of d on M = D=H, p : D ! D=H is the projection, D 2
the Cartan 3-form (given by
D(XL; Y L; ZL) = h[X; Y ]; Zi (8X; Y; Z 2 d)), L 2
is the left-invariant Maurer-Cartan form on D (i.e. L(XL) = X), and A 2
the connection on the principal H-bundle p : D ! D=H whose horizontal spaces are the
3(D) is
1(D; d)
1(D; h) is
1The conceptual de nition of g and
is as follows: the trivial vector bundle d
M ! M is naturally an
exact Courant algebroid, with the anchor given by
and the Courant bracket of its constant sections being
the Lie bracket on d. Then V+
M
d
M is a generalized metric, which is equivalent to the metric g
then de ne a -model with the standard action
functional
Z
S(f ) =
Z
Y
f
where
is (say) the cylinder with the usual metric d 2
d 2 and f :
extended to the solid cylinder Y with boundary .
! M is a map
For our purposes, the main properties of these -models are the following:
The solutions of the equations of motion are in (almost) 1-1 correspondence with
1-forms A 2
1
i it admits a lift ` : ~ ! D such that A :=
( ; d) satisfying (1.1). More precisely, a map f :
uniquely speci ed by f (the lift ` is not unique | it can be multiplied by an element
d` ` 1 satis es (1.1). Notice that A is
inverse of the matrix of hea; ebi). The Hamiltonian of the -model is
(written using a basis ea of d, with facb being the structure constants of d and tab the
H =
of maps ` : R ! D which are quasi-periodic in the sense that for some h 2 H we have
`(
+ 2 ) = `( )h, modulo the action of H by right multiplication. The reduced phase
space is (LD)=D (i.e. periodic maps modulo the action of D); it is the subspace of 1(S1; d)
given by the unit holonomy constraint.)
3
Constructing new
at connections
As we have seen, the solutions of our -model give rise to (...truncated)