Yang Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices
HJE
Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices
Calan Appadu 0 1
Timothy J. Hollowood 0 1
Dafydd Price 0 1
Daniel C. Thompson 0 1
0 Swansea , SA2 8PP , U.K
1 Department of Physics, Swansea University
Integrable deformation of SU(2) sigma and lambda models are considered at the classical and quantum levels. These are the Yang-Baxter and XXZ-type anisotropic deformations. The XXZ type deformations are UV safe in one regime, while in another regime, like the Yang-Baxter deformations, they exhibit cyclic RG behaviour. The associated a ne quantum group symmetry, realized classically at the Poisson bracket level, has q a complex phase in the UV safe regime and q real in the cyclic RG regime, where q is an RG invariant. Based on the symmetries and RG S-matrices to describe the scattering of states in the lambda models, from which the sigma models follow by taking a limit and non-abelian T-duality. In the cyclic RG regimes, the S-matrices are periodic functions of rapidity, at large rapidity, and in the Yang-Baxter case violate parity.
Integrable Field Theories; Quantum Groups; Sigma Models
-
Yang
1 Introduction
2
3
2.1
2.2
3.1
3.2
3.3
3.4
4.1
4.2
4.3
5.1
5.2
5.3
5.4
5.5
Classical SU(2) sigma models
Lax connection and Poisson brackets
Non-local charges and in nite symmetries
Classical SU(2) lambda models
Target spaces
Lax formalism Poisson structure and symmetries Non-local charges and in nite symmetries
4
Renormalization group ow
ow
The sigma model RG
Yang Baxter lambda model RG
ow
Anisotropic XXZ lambda model RG
ow
5
Quantum group S-matrix: q complex phase
Quantum group S-matrix: q real
The RSOS S-matrix
High energy limit
The S-matrix proposals
6
Discussion
A Lambda spacetimes
1
Introduction
Sigma models are fascinating because they are the building blocks of string worldsheet
theories but also they share many of the features of QFTs in higher dimensions in a
simpler context. And within the space of sigma models, the ones that are integrable have
the additional lure of tractability.
The key examples are the Principal Chiral Models (PCM), whose target spaces are
group manifolds G. There is a G-valued eld f and the action can be written1
S =
2
(1.1)
1We take x = t x and so for vectors A = A0
A1 along with A = 12 (A0
A1).
{ 1 {
outgoing particles as well as the internal quantum numbers i; j; k; l.
The PCM can appear as a bosonic sub-sector of a consistent string theory CFT background,
e.g. the D1-D5 near horizon geometry, providing a modern holographic motivation for
studying this theory. The more prosaic view, which we adopt here, is that PCMs are an
exceptionally informative 1 + 1-dimensional QFTs exhibiting asymptotic freedom in the
running coupling ( ) and a dynamically generated mass gap.
The action given in eq. (1.1) manifests a GL
GR global symmetry, f ! U f V . A
feature that makes the PCM tractable is that it is classically integrable and the GL
GR
symmetry is part of a much larger classical Yangian Y (gL)
Y (gR) symmetry generated
by non-local charges.2
At the quantum level this integrability persists leading to the factorization of its
Smatrix [2, 3]. This means that it is completely determined by 2 ! 2 body processes which
preserve the individual momenta, as illustrated in gure 1. The states are labelled by their
rapidity
and by internal quantum numbers i; j; : : :. For example, in the SU(N ) PCM,
there are N
1 particle multiplets with mass ma = m sin( a=N ), a =; 1; 2; : : : ; N
1,
and each multiplet transforms in the [!a]
[!a] representation of the GL
GR symmetry,
where !a are the highest weight vectors of the ath fundamental representation.3 The 2-body
S-matrix has the characteristic product form [4]:
S( ) = SGL ( )
SGR ( ) ;
(1.2)
where
= 1
2. The product form re ects the fact that the states transform in a product
of representations of GL and GR. The S-matrix building block SG( ) is G-invariant, in fact
Yangian invariant, and is built from a rational solution of the Yang-Baxter Equation.4
Since the PCM is asymptotically free and its spectrum is massive and dynamically
generated, directly connecting the conjectured quantum S-matrix picture to the Lagrangian
description in eq. (1.1) is subtle. Nonetheless, consistency checks can be made by studying
the theory in a regime in which perturbation theory can be employed and compared against
the factorized S-matrix. The study of the exact solution of the model was initiated in the
classic works [4{7]. As a byproduct of the successful comparison of Thermodynamic Bethe
2A concise introduction to this symmetry can be found in [1]. There are also an in nite number of local
conserved charges which include and energy and momentum.
3For the groups SO(N ) the representations are actually reducible combinations.
4For the higher rank groups, the product form of the S-matrix must be multiplied by a scalar factor to
provide the bound state poles.
{ 2 {
Ansatz and perturbative calculations o (...truncated)