arXiv subjects
Publications and source records attributed to Gor Sarkissian.
In this paper we show how the Baxter Q-operator and the product formula for eigenfunctions of two-particle hyperbolic Ruijsenaars system can be derived from the genus one Moore-Seiberg duality identity in two-dimensional Liouville conformal field theory. We expect that this relation would reveal genuine role of the Moore-Seiberg identity in integrable systems.
In this paper we derive new symmetry and new expression for $6j$-symbols of the unitary principal series representations of the $SL(2,\mathbb{C})$ group. This allowed us to derive for them the analogue of the Regge symmetry.
In this paper we calculate matrix of modular transformations of the one-point toric conformal blocks in the Neveu-Schwarz sector of $N=1$ super Liouville field theory. For this purpose we use explicit expression for this matrix as integral of product of certain elements of fusion matrix. This integral is computed using the chain of integral identities for supersymmetric hyperbolic gamma functions derived by the degeneration of the integrals of parafermionic elliptic gamma functions.
We study properties of a parafermionic generalization of the hyperbolic hypergeometric function appearing as the most important part in the fusion matrix for Liouville field theory and the Racah-Wigner symbols for the Faddeev modular double. We show that this generalized hypergeometric function is a limiting form of the rarefied elliptic hypergeometric function $V^{(r)}$ and derive its transformation properties and a mixed difference-recurrence equation satisfied by it. At the intermediate level we describe symmetries of a more general rarefied hyperbolic hypergeometric function. An important $r=2$ case corresponds to the supersymmetric hypergeometric function given by the integral appearing in the fusion matrix of $N=1$ super Liouville field theory and the Racah-Wigner symbols of the quantum algebra ${\rm U}_q({\rm osp}(1|2))$. We indicate relations to the standard Regge symmetry and prove some previous conjectures for the supersymmetric Racah-Wigner symbols by establishing their different parametrizations.
We consider a univariate beta integral composed from general modular quantum dilogarithm functions and prove its exact evaluation formula. It represents the partition function of a particular $3d$ supersymmetric field theory on the general squashed lens space. Its possible applications to $2d$ conformal field theory are briefly discussed as well.
We consider the rarefied elliptic beta integral in various limiting forms. In particular, we obtain an integral identity for parafermionic hyperbolic gamma functions which describes the star-triangle relation for parafermionic Liouville theory.
We construct topological defects in the Liouville field theory producing jump in the value of cosmological constant. We construct them using the Cardy-Lewellen equation for the two-point function with defect. We show that there are continuous and discrete families of such kind of defects. For the continuous family of defects we also find the Lagrangian description and check its agreement with the solution of the Cardy-Lewellen equation using the heavy asymptotic semiclasscial limit.
We study mini-superspace semiclassical limit of the boundary three-point function in the Liouville field theory. We compute also matrix elements for the Morse potential quantum mechanics. An exact agreement between the former and the latter is found. We show that both of them are given by the generalized hypergeometric functions.
We study several aspects of the $N=1$ super Liouville theory. We show that certain elements of the fusion matrix in the Neveu-Schwarz sector related to the structure constants according to the same rules which we observe in rational conformal field theory. We collect some evidences that these relations should hold also in the Ramond sector. Using them the Cardy-Lewellen equation for defects is studied, and defects are constructed.
We compute the light asymptotic limit of $A_{n-1}$ Toda conformal blocks by using the AGT correspondence. We show that for certain class of CFT blocks the corresponding Nekrasov partition functions in this limit are simplified drastically being represented as a sum of a restricted class of Young diagrams. In the particular case of $A_{2}$ Toda we also compute the corresponding conformal blocks using conventional CFT techniques finding a perfect agreement with the results obtained from the Nekrasov partition functions.
The Lagrangian of the Liouville theory with topological defects is analyzed in detail and general solution of the corresponding defect equations of motion is found. We study the heavy and light semiclassical limits of the defect two-point function found before via the bootstrap program. We show that the heavy asymptotic limit is given by the exponential of the Liouville action with defects, evaluated on the solutions with two singular points. We demonstrate that the light asymptotic limit is given by the finite dimensional path integral over solutions of the defect equations of motion with a vanishing energy-momentum tensor.
We construct topological defects generating non-abelian T-duality for isometry groups acting without isotropy. We find that these defects are given by line bundles on the correspondence space with curvature which can be considered as a non-abelian generalization of the curvature of the Poincarè bundle. We show that the defect equations of motion encode the non-abelian T-duality transformation. The Fourier-Mukai transform of the Ramond-Ramond fields generated by the gauge invariant flux of these defects is studied. We show that it provides elegant and compact way of computation of the transformation of the Ramond-Ramond fields under the non-abelian T-duality.
Topological defects are interfaces joining two conformal field theories, for which the energy momentum tensor is continuous across the interface. A class of the topological defects is provided by the interfaces separating two bulk systems each described by its own Lagrangian, where the two descriptions are related by a discrete symmetry. In this paper we elaborate on the cases in which the discrete symmetry is a bosonic or a fermionic T- duality. We review how the equations of motion imposed by the defect encode the general bosonic T- duality transformations for toroidal compactifications. We generalize this analysis in some detail to the case of topological defects allowed in coset CFTs, in particular to those cosets where the gauged group is either an axial or vector U(1). This is discussed in both the operator and Lagrangian approaches. We proceed to construct a defect encoding a fermionic T-duality. We show that the fermionic T-duality is implemented by the Super-Poincaré line bundle. The observation that the exponent of the gauge invariant flux on a defect is a kernel of the Fourier-Mukai transform of the Ramond-Ramond fields, is generalized to a fermionic T-duality. This is done via a fiberwise integration on supermanifolds.
In this paper we analyze the Cardy-Lewellen equation in general diagonal model. We show that in these models it takes simple form due to some general properties of conformal field theories, like pentagon equations and OPE associativity. This implies, that the Cardy-Lewellen equation has simple form also in non-rational diagonal models. We specialize our finding to the Liouville and Toda field theories. In particular we prove, that conjectured recently defects in Toda field theory indeed satisfy the cluster equation. We also derive the Cardy-Lewellen equation in all $sl(n)$ Toda field theories and prove that the forms of boundary states found recently in $sl(3)$ Toda field theory hold in all $sl(n)$ theories as well.
In this paper we perform canonical quantization of the product of the gauged WZW models on a strip with boundary conditions specified by permutation branes. We show that the phase space of the $N$-fold product of the gauged WZW model $G/H$ on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of the double Chern-Simons theory on a sphere with $N$ holes times the time-line with $G$ and $H$ gauge fields both coupled to two Wilson lines. For the special case of the topological coset $G/G$ we arrive at the conclusion that the phase space of the $N$-fold product of the topological coset $G/G$ on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of Chern-Simons theory on a Riemann surface of the genus $N-1$ times the time-line with four Wilson lines.
In this paper we construct defects in coset $G/H$ theory. Canonical quantization of the gauged WZW model $G/H$ with $N$ defects on a cylinder and a strip is performed and the symplectomorphisms between the corresponding phase spaces and those of double Chern-Simons theory on an annulus and a disc with Wilson lines are established. Special attention to topological coset $G/G$ has been paid. We prove that a $G/G$ theory on a cylinder with $N$ defects coincides with Chern-Simons theory on a torus times the time-line $R$ with 2N Wilson lines. We have shown also that a $G/G$ theory on a strip with $N$ defects coincides with Chern-Simons theory on a sphere times the time-line $R$ with $2N+4$ Wilson lines. This particular example of topological field theory enables us to penetrate into a general picture of defects in semisimple 2D topological field theory. We conjecture that defects in this case described by a 2-category of matrices of vector spaces and that the action of defects on boundary states is given by the discrete Fourier-Mukai transform.
We perform canonical quantization of the WZW model with defects and permutation branes. We establish symplectomorphism between phase space of WZW model with $N$ defects on cylinder and phase space of Chern-Simons theory on annulus times $R$ with $N$ Wilson lines, and between phase space of WZW model with $N$ defects on strip and Chern-Simons theory on disc times $R$ with $N+2$ Wilson lines. We obtained also symplectomorphism between phase space of the $N$-fold product of the WZW model with boundary conditions specified by permutation branes, and phase space of Chern-Simons theory on sphere with $N$ holes and two Wilson lines.
The defects and permutation branes for the Liouville field theory are considered. By exploiting cluster condition, equations satisfied by permutation branes and defects reflection amplitudes are obtained. It is shown that two types of solutions exist, discrete and continuous families.