This paper represents a continuation of our previous work, where the Boltzmann weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed using their relation to rational conformal field theories. Here, we focus on deriving solutions for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically, the unrestricted face model, based on the $$ \mathfrak{su}{(3)}_k $$ affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum matrix of the $$ {U}_q\left(\mathfrak{sl}(3)\right) $$ quantum algebra in the adjoint representation and then applying the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs of IRF models in terms of matrix elements.

Israel

Vladimir Belavin

Doron Gepner

J. Cabezas

Boris Runov

On different approaches to IRF lattice models. Part II

We study 𝔰𝔩2 and 𝔰𝔩3 global conformal blocks on a sphere and a torus, using the shadow formalism. These blocks arise in the context of Virasoro and 𝒲3 conformal field theories in the large central charge limit. In the 𝔰𝔩2 case, we demonstrate that the shadow formalism yields the known expressions in terms of conformal partial waves. Then, we extend this approach to the 𝔰𝔩3 case and show that it allows to build simple integral representations for 𝔰𝔩3 global blocks. We demonstrate this construction on two examples: the four-point block on the sphere and the one-point torus block.

Global conformal blocks via shadow formalism

We study <math><msub><mrow><mi>W</mi></mrow><mrow><mn>3</mn></mrow></msub></math> toroidal conformal blocks for degenerate primary fields in AdS/CFT context. In the large central charge limit <math><msub><mrow><mi>W</mi></mrow><mrow><mn>3</mn></mrow></msub></math> algebra reduces to <math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub></math> algebra and <math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub></math> blocks are defined as contributions to <math><msub><mrow><mi>W</mi></mrow><mrow><mn>3</mn></mrow></msub></math> blocks coming from the generators of <math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub></math> subalgebra. We consider the construction of <math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub></math> toroidal blocks in terms of Wilson lines operators of 3d Chern-Simons gravity in the thermal AdS$_{3}$ space-time. According to the correspondence, degenerate primary fields are associated with Wilson lines operators acting in the corresponding finite-dimensional <math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub></math> representations. We verify this dual construction for one-point toroidal block using <math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub></math> tensor technique in the bulk theory and an algorithm based on AGT correspondence in the boundary CFT.

Pietro Oreglia

J. Ramos Cabezas

Wilson lines construction of Toroidal Conformal Blocks

We study N=1 superconformal theory in the context of AdS/CFT correspondence in the large central charge limit using Chern-Simons formulation of 3d gravity. In this limit conformal dimensions of a subclass of so-called light primary superfields remain finite and are governed by <math><mrow><mi>osp</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo></math> subalgebra of N=1 super-Virasoro algebra. We describe the construction of <math><mrow><mi>osp</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo></math> conformal blocks in terms of Wilson lines of the Chern-Simons 3d gravity. We consider examples of two and three-point blocks on the sphere and one-point torus blocks of light superprimary fields, which belong to finite-dimensional representations of <math><mrow><mi>osp</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo></math>. We study the correlation function for lower and upper components of the primary <math><mrow><mi>osp</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo></math> doublets and show that the associated conformal blocks are obtained via Wilson line construction in Chern-Simons theory.

Wilson lines construction of Conformal Blocks

In this work we present a new approach to constructing Calabi-Yau orbifold models required for compactification in superstring theory. We use the connection of CY orbifolds with the class of exactly solvable N=2 SCFT models to explicitly construct a complete set of fields in these models using the twisting of the spectral flow and the requirement of mutual locality of the fields.

Russia

Alexander Belavin

Sergey Parkhomenko

Explicit construction of SCFT orbifold models. Spectral flow and mutual locality

We consider the Wilson line networks of the Chern-Simons 3d gravity theory with toroidal boundary conditions which calculate global conformal blocks of degenerate quasi-primary operators in torus 2d CFT. After general discussion that summarizes and further extends results known in the literature we explicitly obtain the one-point torus block and two-point torus blocks through particular matrix elements of toroidal Wilson network operators in irreducible finite-dimensional representations of sl(2, ℝ) algebra. The resulting expressions are given in two alternative forms using different ways to treat multiple tensor products of sl(2, ℝ) representations: (1) 3mj Wigner symbols and intertwiners of higher valence, (2) totally symmetric tensor products of the fundamental sl(2, ℝ) representation.

Konstantin Alkalaev

More on Wilson toroidal networks and torus blocks

We describe the Boltzmann weights of the <math><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msub></math> algebra spin vertex models. Thus, we find the <math><mi>S</mi><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> spin vertex models, for any N, completing the <math><msub><mrow><mi>B</mi></mrow><mrow><mi>k</mi></mrow></msub></math> case found earlier. We further check that the real (self–dual) SO(N) models obey quantum algebras, which are the Birman–Murakami–Wenzl (BMW) algebra for three blocks, and certain generalizations, which include the BMW algebra as a sub–algebra, for four and five blocks. In the case of five blocks, the <math><msub><mrow><mi>B</mi></mrow><mrow><mn>4</mn></mrow></msub></math> model is shown to satisfy additional twenty new relations, which are given. The <math><msub><mrow><mi>D</mi></mrow><mrow><mn>6</mn></mrow></msub></math> model is shown to obey two additional relations.

United States

Hans Wenzl

On SO( ) Spin Vertex Models

AGT allows one to compute conformal blocks of d = 2 CFT for a large class of chiral CFT algebras. This is related to the existence of a certain orthogonal basis in the module of the (extended) chiral algebra. The elements of the basis are eigenvectors of a certain integrable model, labeled in general by N-tuples of Young diagrams. In particular, it was found that in the Virasoro case these vectors are expressed in terms of Jack polynomials, labeled by 2-tuples of ordinary Young diagrams, and for the super-Virasoro case they are related to Uglov polynomials, labeled by two colored Young diagrams. In the case of a generic central charge this statement was checked in the case when one of the Young diagrams is empty. In this note we study the N = 1 SCFT and construct 4 point correlation function using the basis. To this end we need to clarify the connection between basis elements and Uglov polynomials, we also need to use two bosonizations and their connection to the reflection operator. For the central charge <math><mi>c</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math> we checked that there is a connection with the Uglov polynomials for the whole set of diagrams.

Abay Zhakenov

AGT basis in SCFT for c = 3/2 and Uglov polynomials

We study the algebras underlying solvable lattice models of the type fusion interaction round the face (IRF). We propose that the algebras are universal, depending only on the number of blocks, which is the degree of polynomial equation obeyed by the Boltzmann weights. Using the Yang-Baxter equation and the ansatz for the Baxterization of the models, we show that the three blocks models obey a version of Birman-Murakami­Wenzl (BMW) algebra. For four blocks, we conjecture that the algebra, which is termed 4-CB (Conformal Braiding) algebra, is the BMW algebra with a different skein relation, along with one additional relation, and we provide evidence for this conjecture. We connect these algebras to knot theory by conjecturing new link invariants. The link invariants, in the case of four blocks, depend on three arbitrary parameters. We check our result for G 2 model with the seven dimensional representation and for SU(2) with the isospin 3/2 representation, which are both four blocks theories.

Austria

Jian-Rong Li

Ran Tessler

The 4-CB algebra and solvable lattice models

We study the connection between minimal Liouville string theory and generalized open KdV hierarchies. We are interested in generalizing Douglas string equation formalism to the open topology case. We show that combining the results of the closed topology, based on the Frobenius manifold structure and resonance transformations, with the appropriate open case modification, which requires the insertion of macroscopic loop operators, we reproduce the well-known result for the expectation value of a bulk operator for the FZZT brane coupled to the general (q, p) minimal model. The matching of the results of the two setups gives new evidence of the connection between minimal Liouville gravity and the theory of Topological Gravity.