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Abstract 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.

Abstract 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.

Abstract We treat here interaction round the face (IRF) solvable lattice models. We study the algebraic structures underlining such models. For the three block case, we show that the Yang Baxter equation is obeyed, if and only if, the Birman-Murakami-Wenzl (BMW) algebra is obeyed. We prove this by an algebraic expansion of the Yang Baxter equation (YBE). For four blocks IRF models, we show that the BMW algebra is also obeyed, apart from the skein relation, which is different. This indicates that the BMW algebra is a sub-algebra for all models with three or more blocks. We find additional relations for the four block algebra using the expansion of the YBE. The four blocks result, that is the BMW algebra and the four blocks skein relation, is enough to define new knot invariant, which depends on three arbitrary parameters, important in knot theory.

Abstract 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.

Abstract We study large-c SCFT2 on a torus specializing to one-point superblocks in the N $$ \mathcal{N} $$ = 1 Neveu-Schwarz sector. Considering different contractions of the Neveu-Schwarz superalgebra related to the large central charge limit we explicitly calculate three superblocks, osp(1|2) global, light, and heavy-light superblocks, and show that they are related to each other. We formulate the osp(1|2) superCasimir eigenvalue equations and identify their particular solutions as the global superblocks. It is shown that the resulting differential equations are the Heun equations. We study exponentiated global superblocks arising at large conformal dimensions and demonstrate that in the leading approximation the osp(1|2) superblocks are equal to the non-supersymmetric sl(2) block.

Abstract We develop a recursive approach to computing Neveu-Schwarz conformal blocks associated with n-punctured Riemann surfaces. This work generalizes the results of [1] obtained recently for the Virasoro algebra. The method is based on the analysis of the analytic properties of the superconformal blocks considered as functions of the central charge c. It consists of two main ingredients: the study of the singular behavior of the conformal blocks and the analysis of their asymptotic properties when c tends to infinity. The proposed construction is applicable for computing multi-point blocks in different topologies. We consider some examples for genus zero and one with different numbers of punctures. As a by-product, we propose a new way to solve the recursion relations, which gives more efficient computational procedure and can be applied to SCFT case as well as to pure Virasoro blocks.

Abstract There are two alternative approaches to the minimal gravity — direct Liouville approach and matrix models. Recently there has been a certain progress in the matrix model approach, growing out of presence of a Frobenius manifold (FM) structure embedded in the theory. The previous studies were mainly focused on the spherical topology. Essentially, it was shown that the action principle of Douglas equation allows to define the free energy and to compute the correlation numbers if the resonance transformations are properly incorporated. The FM structure allows to find the explicit form of the resonance transformation as well as the closed expression for the partition function. In this paper we elaborate on the case of gravitating disk. We focus on the bulk correlators and show that in the similar way as in the closed topology the generating function can be formulated using the set of flat coordinates on the corresponding FM. Moreover, the resonance transformations, which follow from the spherical topology consideration, are exactly those needed to reproduce FZZ result of the Liouville gravity approach.

Abstract We study CFT2 conformal blocks on a torus and their holographic realization. The classical conformal blocks arising in the regime where conformal dimensions grow linearly with the large central charge are shown to be holographically dual to the geodesic networks stretched in the thermal AdS bulk space. We discuss the n-point conformal blocks and their duals, the 2-point case is elaborated in full detail. We develop various techniques to calculate both quantum and classical conformal block functions. In particular, we show that exponentiated global torus blocks reproduce classical torus blocks in the specific perturbative regimes of the conformal parameter space.

Abstract We continue to investigate the dual description of the Virasoro conformal blocks arising in the framework of the classical limit of the AdS3/CFT2 correspondence. To give such an interpretation in previous studies, certain restrictions were necessary. Our goal here is to consider a more general situation available through the worldline approximation to the dual AdS gravity. Namely, we are interested in computing the spherical conformal blocks without the previously imposed restrictions on the conformal dimensions of the internal channels. The duality is realised as an equality of the so-called heavy-light limit of the n-point conformal block and the action of n−2 particles propagating in some AdS-like background with either a conical singularity or a BTZ black hole. We describe a procedure that allows relaxing the constraint on the intermediate channels. To obtain an explicit expression for the conformal block on the CFT side, we use a recently proposed recursion procedure and find full agreement between the results of the boundary and bulk computations.