Quantum group symmetries and non-local currents in 2D QFT

We construct and study the implications of some new non-local conserved currents that exist is a wide variety of massive integrable quantum field theories in 2 dimensions, including the sine-Gordon theory and its generalization to affine Toda theory. These non-local currents provide a non-perturbative formulation of the theories. The symmetry algebras correspond to the quantum affine Kac-Moody algebras. TheS-matrices are completely characterized by these symmetries. FormalS-matrices for the imaginary-coupling affine Toda theories are thereby derived. The application of theseS-matrices to perturbed coset conformal field theory is studied. Non-local charges generating the finite dimensional Quantum Group in the Liouville theory are briefly presented. The formalism based on non-local charges we describe provides an algernative to the quantum inverse scattering method for solving integrable quantum field theories in 2d.     

Quantum Group Symmetry in sine-Gordon and Affine Toda Field Theories on the Half-Line

 We consider the sine-Gordon and affine Toda field theories on the half-line with classically integrable boundary conditions, and show that in the quantum theory a remnant survives of the bulk quantized affine algebra symmetry generated by non-local charges. The paper also develops a general framework for obtaining solutions of the reflection equation by solving an intertwining property for representations of certain coideal subalgebras of U _q (ĝ). Received: 10 December 2001 / Accepted: 7 October 2002 Published online: 19 December 2002     

Affine Toda Field Theory as a 3-Dimensional Integrable System

The affine Toda field theory is studied as a 2+1-dimensional system. The third dimension appears as the discrete space dimension, corresponding to the simple roots in the A _N affine root system, enumerated according to the cyclic order on the A _N affine Dynkin diagram. We show that there exists a natural discretization of the affine Toda theory. The quantum analog of the τ-variables is found. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit L,N→∞. It is shown that the free energy of the systems grows proportionally to the volume. Received: 23 May 1996 / Accepted: 22 August 1996     

Yangians,integrable quantum systems and Dorey's rule

It was pointed out by P. Dorey that the three-point couplings between the quantum particles in affine Toda field theories have a remarkable Lie-theoretic interpretation. It is also well known that such theories admit quantum affine algebras as quantum symmetry groups, and widely believed that the quantum particles correspond to the so-called fundamental representations of these algebras. This led to the conjecture that Dorey's rule should describe when a fundamental representation occurs with non-zero multiplicity in a tensor product of two other fundamental representations. The purpose of this paper is to prove this conjecture, both for quantum affine algebras and for Yangians. The result reveals a hitherto unsuspected role played by Coxeter elements (and their twisted analogues) in the representation theory of these algebras.     

Non-unitarity in quantum affine Toda theory and perturbed conformal field theory

There has been some debate about the validity of quantum affine Toda field theory at imaginary coupling, owing to the non-unitarity of the action, and consequently of its usefulness as a model of perturbed conformal field theory. Drawing on our recent work, we investigate the two simplest affine Toda theories for which this is an issue –a₂⁽¹⁾ and a₂⁽²⁾. By investigating the S-matrices of these theories before RSOS restriction, we show that quantum Toda theory (with or without RSOS restriction) indeed has some fundamental problems, but that these problems are of two different sorts. For a₂⁽¹⁾, the scattering of solitons and breathers is flawed in both classical and quantum theories, and RSOS restriction cannot solve this problem. For a₂⁽²⁾ however, while there are no problems with breather-soliton scattering there are instead difficulties with soliton-excited soliton scattering in the unrestricted theory. After RSOS restriction, the problems with kink-excited kink may be cured or may remain, depending in part on the choice of gradation, as we found earlier [Nucl. Phys. B 489 [FS] (1997) 557]. We comment on the importance of regradations, and also on the survival of R-matrix unitarity and the S-matrix bootstrap in these circumstances.     

Solitons and vertex operators in twisted affine Toda field theories

Affine Toda field theories in two dimensions constitute families of integrable, relativistically invariant field theories in correspondence with the affine Kac-Moody algebras. The particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary. Here these results are extended from the untwisted to the twisted algebras. The new soliton solutions and their masses are found by a folding procedure which can be applied to the affine Kac-Moody algebras themselves to provide new insights into their structures. The relevant foldings are related to inner automorphisms of the associated finite dimensional Lie group which are calculated explicitly and related to what is known as the twisted Coxeter element. The fact that the twisted affine Kac-Moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions.     

Probing integrable perturbations of conformal theories using singular vectors (II). N = 1 superconformal theories

In this work we pursue the singular-vector analysis of the integrable perturbations of conformal theories that was initiated in our earlier paper [Nucl. Phys. B 475 (1996) 361]. Here we consider the detailed study of the N = 1 superconformal theory and show that all integrable perturbations can be identified from a simple singular-vector argument. We identify these perturbations as theories based on affine Lie superalgebras and show that the results we obtain relating two perturbations can be understood by the extension of affine Toda duality to these theories with fermions. We also discuss how this duality is broken in specific cases.     

Wave function renormalization constants and one-particle form factors in Dl(1) Toda field theories

We apply the method of angular quantization to the calculation of the wave function renormalization constants in D₁⁽¹⁾ affine Toda quantum field theories. A general formula for the wave function renormalization constants in ADE Toda field theories is proposed. We also calculate all one-particle form factors and some of the two-particle form factors of an exponential field.     

Lax connections in T■-deformed integrable field theories

In this work,we attempt to construct the Lax connections of TT-deformed integrable field theories in two different ways.With reasonable assumptions,we make an ansatz and find the Lax pairs in the TT-deformed affine Toda theories and the principal chiral model by solving the Lax equations directly.This method is straightforward,but it may be difficult to apply for general models.We then make use of a dynamic coordinate transformation to read the Lax connection in the deformed theory from the undeformed one.We find that once the inverse of the transformation is available,the Lax connection can be read easily.We show the construction explicitly for a few classes of scalar models and find consistency with those determined using the first method.     

Gravitation,Electromagnetism and Cosmological Constant in Purely Affine Gravity

The Ferraris-Kijowski purely affine Lagrangian for the electromagnetic field, that has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the metric Einstein-Maxwell Lagrangian, except the zero-field limit, for which the metric tensor is not well-defined. This feature indicates that, for the Ferraris-Kijowski model to be physical, there must exist a background field that depends on the Ricci tensor. The simplest possibility, supported by recent astronomical observations, is the cosmological constant, generated in the purely affine formulation of gravity by the Eddington Lagrangian. In this paper we combine the electromagnetic field and the cosmological constant in the purely affine formulation. We show that the sum of the two affine (Eddington and Ferraris-Kijowski) Lagrangians is dynamically inequivalent to the sum of the analogous (ΛCDM and Einstein-Maxwell) Lagrangians in the metric-affine/metric formulation. We also show that such a construction is valid, like the affine Einstein-Born-Infeld formulation, only for weak electromagnetic fields, on the order of the magnetic field in outer space of the Solar System. Therefore the purely affine formulation that combines gravity, electromagnetism and cosmological constant cannot be a simple sum of affine terms corresponding separately to these fields. A quite complicated form of the affine equivalent of the metric Einstein-Maxwell-Λ Lagrangian suggests that Nature can be described by a simpler affine Lagrangian, leading to modifications of the Einstein-Maxwell-ΛCDM theory for electromagnetic fields that contribute to the spacetime curvature on the same order as the cosmological constant.     

Common structures of quantum field theories and lattice systems through boundary symmetry

The sine-Gordon model and affine Toda field theories on the half-line, on the one hand, the XXZ spin chain with nondiagoual boundary terms, and interacting many-body lattice systems with a flow, on the other, have a common characteristic. They possess nonlocal conserved boundary charges, generating the Askey-Wilson algebra, a coideal subalgebra of the bulk quantized affine symmetry. We argue that the boundary Askey-Wilson symmetry is the deep algebraic property allowing for integrability of the physical system in consideration.     

Balanced Topological Field Theories

We describe a class of topological field theories called “balanced topological field theories”. These theories are associated to moduli problems with vanishing virtual dimension and calculate the Euler character of various moduli spaces. We show that these theories are closely related to the geometry and equivariant cohomology of “iterated superspaces” that carry two differentials. We find the most general action for these theories, which turns out to define Morse theory on field space. We illustrate the constructions with numerous examples. Finally, we relate these theories to topological sigma-models twisted using an isometry of the target space. Received: 2 September 1996 / Accepted: 3 October 1996     

On a2(1) reflection matrices and affine Toda theories

We construct new non-diagonal solutions to the boundary Yang-Baxter equation corresponding to a two-dimensional field theory with U_q(a₂⁽¹⁾) quantum affine symmetry on a half-line. The requirements of boundary unitarity and boundary crossing symmetry are then used to find overall scalar factors which lead to consistent reflection matrices. Using the boundary bootstrap equations we also compute the reflection factors for scalar bound states (breathers). These breathers are expected to be identified with the fundamental quantum particles in a₂⁽¹⁾ affine Toda field theory and we therefore obtain a conjecture for the affine Toda reflection factors. We compare these factors with known classical results and discuss their duality properties and their connections with particular boundary conditions.     

Quantum corrections to the classical reflection factor in a2(1) Toda field theory

The O(β²) quantum correction to the classical reflection factor is calculated for one of the integrable boundary conditions of a₂⁽¹⁾ affine Toda field theory. This is found to agree with the conjectured exact reflection factor of the quantum theory. We consider the existence of other exact reflection factors consistent with our perturbative answer and examine the question of how duality transformations might relate theories with different boundary conditions.     

Applications of Reflection Amplitudes in Toda-Type Theories

Reflection amplitudes are defined as two-point functions of certain class of conformal field theories where primary fields are given by vertex operators with real couplings. Among these, we consider (Super-) Liouville theory and simply and non-simply laced Toda theories. In this paper we show how to compute the scaling functions of effective central charge for the models perturbed by some primary fields which maintains integrability. This new derivation of the scaling functions are compared with the results from conventional TBA approach and confirms our approach along with other non-perturbative results such as exact expressions of the on-shell masses in terms of the parameters in the action, exact free energies. Another important application of the reflection amplitudes is a computation of one-point functions for the integrable models. Introducing functional relations between the one-point functions in terms of the reflection amplitudes, we obtain explicit expressions for simply-laced and non-simply-laced affine Toda theories. These nonperturbative results are confirmed numerically by comparing the free energies from the scaling functions with exact expressions we obtain from the one-point functions.     

Polynomial Recursion Equations in Form Factors of ADE-Toda Field Theories

It is shown that the problem of calculating form factors in ADE affine Toda field theories can be reduced to the nonperturbative recursive calculation of polynomials symmetric in each sort of variables. We determine these recursion equations explicitly for the ADE series and characterize the polynomial solutions by an interplay between the weight space of the underlying Lie algebra and representations of the symmetric group.     

Inverse scattering and solitons in An − 1 affine Toda field theories II

New single soliton solutions to the affine Toda field theories are constructed, exhibiting previously unobserved topological charges. This goes some of the way in filling the weights of the fundamental representations, but nevertheless holes in the representations remain. We use the group double-cross product form of the inverse scattering method, and restrict ourselves to the rank-one solutions.     

Moduli Space of IIB Superstring and SUYM in AdS5 × S5

This paper consists of two parts. In part I, we interpret the hidden symmetry of the moduli space of IIB superstring on AdS₅×S⁵ in terms of the chiral embedding in AdS₅, which turns out to be the CP³ conformal affine Toda model. We review how the position μ of poles in the Riemann-Hilbert formulation of dressing transformation and the value of loop parameter μ in the vertex operator of affine algebra determine the moduli space of the soliton solutions, which describes the moduli space of the Green-Schwarz superstring. We show also how this affine SU(4) symmetry affinizes the conformal symmetry in the twistor space, and how a soliton string corresponds to a Robinson congruence with twist and dilation spin coefficients μ of twistor. In part II, by extending the dressing symmetric action of IIB string in AdS₅×S⁵ to the D₃ brane, we find a gauged WZW action of Higgs Yang-Mills field including the 2-cocycle of axially anomaly. The left and right twistor structures of left and right α-planes glue into an ambitwistor. The symmetry group of Nahm equations is centrally extended to an affine group, thus we explain why the spectral curve is given by affine Toda.