Exact conserved quantities on the cylinder II: off-critical case

Journal of High Energy Physics, 2003

The nonlinear integral equations describing the spectra of the left and right (continuous) quantum KdV equations on the cylinder are derived from integrable lattice field theories, which turn out to allow the Bethe Ansatz equations of a twisted "spin −1/2" chain. A very useful mapping to the more common nonlinear integral equation of the twisted continuous spin +1/2 chain is found. The diagonalization of the transfer matrix is performed. The vacua sector is analysed in detail detecting the primary states of the minimal conformal models and giving integral expressions for the eigenvalues of the transfer matrix. Contact with the seminal papers [1, 2] by Bazhanov, Lukyanov and Zamolodchikov is realised. General expressions for the eigenvalues of the infinite-dimensional abelian algebra of local integrals of motion are given and explicitly calculated at the free fermion point.

Journal of Physics A-mathematical and General, 2001

We study integrable and conformal boundary conditions for s (2) Z k parafermions on a cylinder. These conformal field theories are realized as the continuum scaling limit of critical A-D-E lattice models with negative spectral parameter. The conformal boundary conditions labelled by (a, m) ∈ (G, Z 2k ) are identified with associated integrable lattice boundary conditions labelled by (r, a) ∈ (A g−2 , G) where g is the Coxeter number of the A-D-E graph G. We obtain analytically the boundary free energies, present general expressions for the parafermion cylinder partition functions and confirm these results by numerical calculations.

We consider the quantum group invariant XXZ-model. In infrared limit it describes Conformal Field Theory (CFT) with modified energy -momentum tensor. The correlation functions are related to solutions of level -4 of qKZ equations. We describe these solutions relating them to level 0 solutions. We further consider general matrix elements (form factors) containing local operators and asymptotic states. We explain that the formulae for solutions of qKZ equations suggest a decomposition of these matrix elements with respect to states of corresponding CFT.

Journal of High Energy Physics

We show that any 2D scalar field theory compactified on a cylinder and with a Fourier expandable potential V is equivalent, in the small coupling limit, to a 1D theory involving a massless particle in a potential V and an infinite tower of free massive Kaluza-Klein (KK) modes. Moving slightly away from the deep IR region has the effect of switching on interactions between the zero mode and the KK modes, whose strength is controlled by powers of the coupling, hence making the interactions increasingly suppressed. We take the notable example of Liouville field theory and, starting from its worldline version, we compute the torus (one-loop) partition function perturbatively in the coupling constant. The partition function at leading order is invariant under a T-duality transformation that maps the radius of the cylinder to its inverse and rescales it by the square of the Schwinger parameter of the cylinder. We show that this behavior is a universal feature of cylinder QFTs.

Annals of Physics, 2004

We consider the Lagrangian particle model introduced in [1] for zero mass but nonvanishing second central charge of the planar Galilei group. Extended by a magnetic vortex or a Coulomb potential the model exibits conformal symmetry. In the former case we observe an additional SO(2, 1) hidden symmetry. By either a canonical transformation with constraints or by freezing scale and special conformal transformations at t = 0 we reduce the six-dimensional phasespace to the physically required four dimensions. Then we discuss bound states (bounded solutions) in quantum dynamics (classical mechanics). We show that the Schrödinger equation for the pure vortex case may be transformed into the Morse potential problem thus providing us with an explanation of the hidden SO(2, 1) symmetry.

Journal of High Energy Physics, 2021

In this paper we consider systems of quantum particles in the 4d Euclidean space which enjoy conformal symmetry. The algebraic relations for conformal-invariant combinations of positions and momenta are used to construct a solution of the Yang-Baxter equation in the unitary irreducibile representations of the principal series ∆ = 2 + iν for any left/right spins ℓ,$$ \dot{\ell} $$ ℓ ̇ of the particles. Such relations are interpreted in the language of Feynman diagrams as integral star-triangle identites between propagators of a conformal field theory. We prove the quantum integrability of a spin chain whose k-th site hosts a particle in the representation (∆k, ℓk,$$ \dot{\ell} $$ ℓ ̇ k) of the conformal group, realizing a spinning and inhomogeneous version of the quantum magnet used to describe the spectrum of the bi-scalar Fishnet theories [1]. For the special choice of particles in the scalar (1, 0, 0) and fermionic (3/2, 1, 0) representation the transfer matrices of the model are ...

Nuclear Physics B, 2004

The integrals of motion of the tricritical Ising model are obtained by Thermodynamic Bethe Ansatz (TBA) equations derived from the A 4 integrable lattice model. They are compared with those given by the conformal field theory leading to a unique one-to-one lattice-conformal correspondence. They can also be followed along the renormalization group flows generated by the action of the boundary field ϕ 1,3 on conformal boundary conditions in close analogy to the usual TBA description of energies.

Journal of Geometry and Physics, 1997

We consider lattice analogues of some conformal theories, including WZW and Toda models. We describe discrete versions of Drinfeld-Sokolov reduction and Sugawara construction for the WZW model. We formulate perturbation theory in chiral sector. We describe the Spaces of Integrals of Motion in the perturbed theories. We i n terpret the perturbed WZW model in terms of NLS-hierarchy and obtain an embedding of this model into the lattice KP-hierarchy.

2011

We study Stokes phenomena of the k × k isomonodromy systems with an arbitrary Poincaré index r, especially which correspond to the fractional-superstring (or parafermionic-string) multi-critical points (p,q) = (1, r − 1) in the k-cut two-matrix models. Investigation of this system is important for the purpose of figuring out the non-critical version of M theory which was proposed to be the strong-coupling dual of fractional superstring theory as a two-matrix model with an infinite number of cuts. Surprisingly the multi-cut boundary-condition recursion equations have a universal form among the various multi-cut critical points, and this enables us to show explicit solutions of Stokes multipliers in quite wide classes of (k, r). Although these critical points almost break the intrinsic Z k symmetry of the multi-cut twomatrix models, this feature makes manifest a connection between the multi-cut boundary-condition recursion equations and the structures of quantum integrable systems. In particular, it is uncovered that the Stokes multipliers satisfy multiple Hirota equations (i.e. multiple T-systems). Therefore our result provides a large extension of the ODE/IM correspondence to the general isomonodromy ODE systems endowed with the multi-cut boundary conditions. We also comment about a possibility that N = 2 QFT of Cecotti-Vafa would be "topological series" in non-critical M theory equipped with a single quantum integrability.

Physical Review D

In this note, we extend the striking connections between quantum integrable systems and conformal blocks recently found in [1] in several directions. First, we explicitly demonstrate that the action of quartic conformal Casimir operator on general d-dimensional scalar conformal blocks, can be expressed in terms of certain combinations of commuting integrals of motions of two particle hyperbolic BC 2 Calogero-Sutherland system. The permutation and reflection properties of the underlying Dunkl operators play crucial roles in establishing such a connection. Next, we show that the scalar superconformal blocks in SCFTs with four and eight supercharges and suitable chirality constraints can also be identified with the eigenfunctions of the same Calogero-Sutherland system, this demonstrates the universality of such a connection. Finally, we observe that the so-called "seed" conformal blocks for constructing four point functions for operators with arbitrary space-time spins in four dimensional CFTs can also be linearly expanded in terms of Calogero-Sutherland eigenfunctions.