arXiv subjects

Vladimir Belavin

Publications and source records attributed to Vladimir Belavin.

28 records · Page 2Linked to original sources

arXiv2019

On the Algebraic Approach to Solvable Lattice Models

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.

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arXiv2018

Open minimal strings and open Gelfand-Dickey hierarchies

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.

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arXiv2018

Three Blocks Solvable Lattice Models and Birman--Murakami--Wenzl Algebra

Birman--Murakami--Wenzl (BMW) algebra was introduced in connection with knot theory. We treat here interaction round the face solvable (IRF) lattice models. We assume that the face transfer matrix obeys a cubic polynomial equation, which is called the three block case. We prove that the three block theories all obey the BMW algebra. We exemplify this result by treating in detail the $SU(2)$ $2\times 2$ fused models, and showing explicitly the BMW structure. We use the connection between the construction of solvable lattice models and conformal field theory. This result is important to the solution of IRF lattice models and the development of new models, as well as to knot theory.

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arXiv2017

Rigid Fuchsian systems in 2-dimensional conformal field theories

We investigate Fuchsian equations arising in the context of 2-dimensional conformal field theory (CFT) and we apply the Katz theory of Fucshian rigid systems to solve some of these equations. We show that the Katz theory provides a precise mathematical framework to answer the question whether the fusion rules of degenerate primary fields are enough for determining the differential equations satisfied by their correlation functions. We focus on the case of W3 Toda CFT: we argue that the differential equations arising for four-point conformal blocks with one n-th level semi-degenerate field and a fully-degenerate one in the fundamental sl3 representation are associated to Fuchsian rigid systems. We show how to apply Katz theory to determine the explicit form of the differential equations, the integral expression of solutions and the monodromy group representation. The theory of twisted homology is also used in the analysis of the integral expression. This approach allows to construct the corresponding fusion matrices and to perform the whole bootstrap program: new explicit factorization of W3 correlation functions as well shift relations between structure constants are also provided.

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arXiv2017

Minimal gravity and Frobenius manifolds: bulk correlation on sphere and disk

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.

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arXiv2017

Second level semi-degenerate fields in W3 Toda theory: matrix element and differential equation

In a recent study we considered W3 Toda 4-point functions that involve matrix elements of a primary field with the highest-weight in the adjoint representation of sl3. We generalize this result by considering a semi-degenerate primary field, which has one null vector at level two. We obtain a sixth-order Fuchsian differential equation for the conformal blocks. We discuss the presence of multiplicities, the matrix elements and the fusion rules.

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arXiv2016

On the construction of the correlation numbers in Minimal Liouville Gravity

The computation of the correlation numbers in Minimal Liouville Gravity involves an integration over moduli spaces of complex curves. There are two independent approaches to the calculation: the direct one, based on the CFT methods and Liouville higher equations of motion, and the alternative one, motivated by discrete description of 2D gravity and based on the Douglas string equation. However these two approaches give rise to the results that are not always consistent among themselves. In this paper we explore this problem. We show that in order to reconcile two methods the so-called discrete terms in the operator product expansion in the underlying Liouville theory must be properly taken into account. In this way we propose modified version of the expression for four-point correlation number and find full agreement between direct and alternative approaches. Our result allows to consider correlators without any restrictions on the number of conformal blocks contributing to the matter sector correlation function.

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arXiv2016

Correlation functions with fusion-channel multiplicity in W3 Toda field theory

Current studies of WN Toda field theory focus on correlation functions such that the WN highest-weight representations in the fusion channels are multiplicity-free. In this work, we study W3 Toda 4-point functions with multiplicity in the fusion channel. The conformal blocks of these 4-point functions involve matrix elements of a fully-degenerate primary field with a highest-weight in the adjoint representation of sl3, and a semi-degenerate primary field with a highest-weight in the fundamental representation of sl3. We show that, when the fusion rules are obeyed, the matrix elements of the fully-degenerate adjoint field, between two arbitrary descendant states, can be computed explicitly, on equal footing with the matrix elements of the semi-degenerate fundamental field. Using null-state conditions, we obtain a fourth-order Fuchsian differential equation for the conformal blocks. Using Okubo theory, we show that, due to the presence of multiplicities, this differential equation belongs to a class of Fuchsian equations that is different from those that have appeared so far in WN theories. We solve this equation, compute its monodromy group, and construct the monodromy-invariant correlation functions.

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arXiv2016

On exact solution of topological CFT models based on Kazama-Suzuki cosets

We compute the flat coordinates on the Frobenius manifolds arising on the deformation space of Gepner $\hat{SU}(3)_k$ chiral rings. The explicit form of the flat coordinates is important for exact solutions of models of topological CFT and 2d Liouville gravity. We describe the case k=3, which is of particular interest because apart from the relevant chiral fields it contains a marginal one. Whereas marginal perturbations are relevant in different contexts, their analysis requires additional care compared to the relevant perturbations.

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arXiv2015

AGT, N-Burge partitions and W_N minimal models

Let ${\mathcal B}^{\, p, \, p^{\prime}, \, {\mathcal H}}_{N, n}$ be a conformal block, with $n$ consecutive channels $χ_ı$, $ı= 1, \cdots, n$, in the conformal field theory $\mathcal{M}^{\, p, \, p^{\prime}}_N \! \times \! \mathcal{M}^{\mathcal{H}}$, where $\mathcal{M}^{\, p, \, p^{\prime}}_N$ is a $\mathcal{W}_N$ minimal model, generated by chiral fields of spin $1, \cdots, N$, and labeled by two co-prime integers $p$ and $p^{\prime}$, $1 < p < p^{\prime}$, while $\mathcal{M}^{\mathcal{H}}$ is a free boson conformal field theory. $\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$ is the expectation value of vertex operators between an initial and a final state. Each vertex operator is labelled by a charge vector that lives in the weight lattice of the Lie algebra $A_{N-1}$, spanned by weight vectors $ω_1, \cdots, ω_{N-1}$. We restrict our attention to conformal blocks with vertex operators whose charge vectors point along $ω_1$. The charge vectors that label the initial and final states can point in any direction. Following the $\mathcal{W}_N$ AGT correspondence, and using Nekrasov's instanton partition functions without modification, to compute $\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$, leads to ill-defined expressions. We show that restricting the states that flow in the channels $χ_ı$, $ı= 1, \cdots, n$, to states labeled by $N$ partitions that satisfy conditions that we call $N$-Burge partitions, leads to well-defined expressions that we identify with $\mathcal{B}^{\, p, \, p^{\prime}, \, \mathcal{H}}_{N, n}$. We check our identification by showing that a specific non-trivial conformal block that we compute, using the $N$-Burge conditions satisfies the expected differential equation.

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