arXiv subjects

Vladimir Belavin

Publications and source records attributed to Vladimir Belavin.

At least 19 recordsLinked to original sources

arXiv2026

The complete massless singlet spectrum in the free-field construction of heterotic strings on Calabi--Yau orbifolds

We apply the free-field construction of the heterotic string compactified on Berglund--H\"ubsch Calabi--Yau orbifolds by computing the full spectrum of massless $E_6$ singlets. The contribution of the descendant vertices is obtained by combining the exact content of the irreducible $N=2$ minimal model representations, encoded in the ranks of the Shapovalov matrices, with the twisted sector structure of the orbifold. The method reproduces the known spectrum of the quintic orbifold with Hodge numbers $(17,21)$, namely $17$ generations, $21$ antigenerations and $234$ singlets. For the quintic itself we obtain $330$ singlets. We show that this number, rather than the frequently quoted value $326$, obtained from the geometric description, is the correct one at the Gepner point, in agreement with the Landau--Ginzburg computation of Kachru and Witten. We also provide an explicit construction of the general vertices. We then compute the new spectra of the two remaining quintic orbifolds, $Z_5[0,1,2,3,4]$ with $(21,1,210)$ and $Z_5[0,0,0,1,4]$ with $(49,5,258)$, where the exceptional Hodge number $h^{2,1}=49$ arises from the twisted sectors. All four examples satisfy exact mirror-symmetry checks.

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arXiv2026

Monodromy and geometry of heavy-light Virasoro blocks

The AdS/CFT correspondence relates gravity in anti-de Sitter space to a boundary conformal field theory, and in its AdS$_3$/CFT$_2$ instance the Virasoro symmetry of the boundary theory organizes correlation functions into conformal blocks. In the semiclassical limit these blocks are computed by lengths of geodesic networks in the bulk, most sharply in the heavy-light regime, where heavy operators source a background probed by light ones. We relate the classical monodromy method to this bulk geometry in holographic coordinates, showing that the eigenvectors of the monodromy matrix encode the endpoints of bulk geodesics. This yields the light action and the equations determining the internal geodesic network; crucially, the internal network equations are independent of the heavy background. For two heavy operators we rederive the same equations from elementary Euclidean geometry, which provides an independent geometric check. As an application we compute the full non-vacuum 5-point HHLLL block, so far known only in the superlight approximation. More broadly, our construction gives a general framework for computing heavy-light blocks from the bulk, while at the same time fixing its threshold of computability.

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arXiv2026

Four-point correlation numbers in super Minimal Liouville Gravity in the Ramond sector

In this work, we continue the investigation of correlation numbers in $\mathcal{N}=1$ super Minimal Liouville Gravity (SMLG), with physical fields in the Ramond sector. Building upon our previous construction of physical operators and the evaluation of three-point correlation functions involving Ramond and Neveu-Schwarz (NS) insertions, we now turn to the analytic computation of four-point correlation numbers. This development is motivated by the framework established for the bosonic Minimal Liouville Gravity and its supersymmetric NS analog, where the integration over moduli space in correlation functions can be performed explicitly using the higher equations of motion (HEM) in Liouville theory. In particular, if one of the insertions corresponds to a degenerate field, the four-point amplitude can be expressed in terms of boundary contributions obtained from the OPE structure of logarithmic counterparts of ground ring elements. We aim to adapt and generalize this approach to the Ramond sector.Our result is a closed-form analytic expression for four-point correlation numbers involving Ramond fields.

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arXiv2025

On construction of correlation numbers in super Minimal Liouville Gravity in the Ramond sector

We study the construction of correlation numbers in super minimal Liouville gravity. In particular, we construct the fundamental physical fields in the Ramond sector and compute the three-point correlation number involving two physical fields in the Ramond sector and one in the NS sector. Furthermore, we establish the relation between Ramond physical fields and the elements of the ground ring. Using the higher equations of motion of super Liouville theory, this relation leads to a new representation of the Ramond physical fields. This formulation enables a direct analytic computation of correlation numbers involving Ramond field insertions. As an application, we demonstrate the method in the simplest case of a three-point correlation function.

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arXiv2024

On different approaches to integrable lattice models II

This paper represents a continuation of our previous work, where the Bolzmann 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 $R$ matrix of the $U_q(\mathfrak{sl} (3))$ 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 $R$ matrix elements.

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arXiv2024

Shadow formalism for supersymmetric conformal blocks

Shadow formalism is a technique in two-dimensional CFT allowing straightforward computation of conformal blocks in the limit of infinitely large central charge. We generalize the construction of shadow operator for superconformal field theories. We demonstrate that shadow formalism yields known expressions for the large-c limit of the four-point superconformal block on a plane and of the one-point superconformal block on a torus. We also explicitly find the two-point global torus superconformal block in the necklace channel and check it against the Casimir differential equation.

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arXiv2024

Global conformal blocks via Shadow formalism

We study $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ global conformal blocks on a sphere and a torus, using the shadow formalism. These blocks arise in the context of Virasoro and $\mathcal{W}_3$ conformal field theories in the large central charge limit. In the $\mathfrak{sl}_2$ case, we demonstrate that the shadow formalism yields the known expressions in terms of conformal partial waves. Then, we extend this approach to the $\mathfrak{sl}_3$ case and show that it allows to build simple integral representations for $\mathfrak{sl}_3$ global blocks. We demonstrate this construction on two examples: the four-point block on the sphere and the one-point torus block.

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arXiv2023

On different approaches to integrable lattice models

Interaction-Round the Face (IRF) models are two-dimensional lattice models of statistical mechanics defined by an affine Lie algebra and admissibility conditions depending on a choice of representation of that affine Lie algebra. Integrable IRF models, i.e., the models the Boltzmann weights of which satisfy the quantum Yang-Baxter equation, are of particular interest. In this paper, we investigate trigonometric Boltzmann weights of integrable IRF models. By using an ansatz proposed by one of the authors in some previous works, the Boltzmann weights of the restricted IRF models based on the affine Lie algebras $\mathfrak{su}(2)_k$ and $\mathfrak{su}(3)_k$ are computed for fundamental and adjoint representations for some fixed levels $k$. New solutions for the Boltzmann weights are obtained. We also study the vertex-IRF correspondence in the context of an unrestricted IRF model based on $\mathfrak {su}(3)_k$ (for general $k$) and discuss how it can be used to find Boltzmann weights in terms of the quantum $\hat{R}$ matrix when the adjoint representation defines the admissibility conditions.

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arXiv2023

Wilson lines construction of $\mathfrak{sl}_3$ toroidal conformal blocks

We study $\mathcal{W}_3$ toroidal conformal blocks for degenerate primary fields in AdS/CFT context. In the large central charge limit $\mathcal{W}_3$ algebra reduces to $\mathfrak{sl}_3$ algebra and $\mathfrak{sl}_3$ blocks are defined as contributions to $\mathcal{W}_3$ blocks coming from the generators of $\mathfrak{sl}_3$ subalgebra. We consider the construction of $\mathfrak{sl}_3$ 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 $\mathfrak{sl}_3$ representations. We verify this dual construction for one-point toroidal block using $\mathfrak{sl}_3$ tensor technique in the bulk theory and an algorithm based on AGT correspondence in the boundary CFT.

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arXiv2022

Torus one-point correlation numbers in minimal Liouville gravity

We present a method for the first principles calculation of tachyon one-point amplitudes in $(2,2p+1)$ minimal Liouville gravity defined on a torus. The method is based on the higher equations of motion in the Liouville CFT. These equations were earlier successfully applied for analytic calculations of the amplitudes in the spherical topology. We show that this approach allows to reduce the moduli integrals entering the definition of the torus amplitudes to certain boundary contributions, which can be calculated explicitly. The results agree with the calculations performed in the matrix models approach.

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arXiv2022

Explicit construction of $N = 2$ SCFT orbifold models. Spectral flow and mutual locality

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.

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arXiv2022

Wilson lines construction of $\mathfrak{osp}(1|2)$ 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 $\mathfrak{osp}(1|2)$ subalgebra of N=1 super-Virasoro algebra. We describe the construction of $\mathfrak{osp}(1|2)$ 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 $\mathfrak{osp}(1|2)$. We study the correlation function for lower and upper components of the primary $\mathfrak{osp}(1|2)$ doublets and show that the associated conformal blocks are obtained via Wilson line construction in Chern-Simons theory.

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arXiv2021

The crossing multiplier for solvable lattice models

We study the large class of solvable lattice models, based on the data of conformal field theory. These models are constructed from any conformal field theory. We consider the lattice models based on affine algebras described by Jimbo et al., for the algebras $ABCD$ and by Kuniba et al. for $G_2$. We find a general formula for the crossing multipliers of these models. It is shown that these crossing multipliers are also given by the principally specialized characters of the model in question. Therefore we conjecture that the crossing multipliers in this large class of solvable interaction round the face lattice models are given by the characters of the conformal field theory on which they are based. We use this result to study the local state probabilities of these models and show that they are given by the branching rule, in regime III.

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arXiv2021

Periods of the multiple Berglund-Huebsch-Krawitz mirrors

We consider the multiple Calaby-Yau (CY) mirror phenomenon which appears in Berglund-Hübsch-Krawitz (BHK) mirror symmetry. We show that for any pair of Calabi--Yau orbifolds that are BHK mirrors of a loop--chain type pair of Calabi--Yau manifolds in the same weighted projective space the periods of the holomorphic nonvanishing form coincide.

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arXiv2020

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

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 $c=3/2$ we checked that there is a connection with the Uglov polynomials for the whole set of diagrams.

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arXiv2020

On SO$(N)$ spin vertex models

We describe the Boltzmann weights of the $D_k$ algebra spin vertex models. Thus, we find the $SO(N)$ spin vertex models, for any $N$, completing the $B_k$ 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 $B_4$ model is shown to satisfy additional twenty new relations, which are given. The $D_6$ model is shown to obey two additional relations.

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arXiv2020

The 5-CB Algebra and Fused $SU(2)$ Lattice Models

We study the fused $SU(2)$ models put forward by Date et al., that are a series of models with arbitrary number of blocks, which is the degree of the polynomial equation obeyed by the Boltzmann weights. We demonstrate by a direct calculation that a version of BMW (Birman--Murakami--Wenzl) algebra is obeyed by five, six and seven blocks models, conjecturing that it is part of the algebra valid for any model with more than two blocks. To establish this conjecture, we assume that a certain ansatz holds for the baxterization of the models. We use the Yang--Baxter equation to describe explicitly the algebra for five blocks, obtaining $19$ additional non--trivial relations. We name this algebra 5--CB (Conformal Braiding) algebra. Our method can be utilized to describe the algebra for any solvable model of this type and for any number of blocks.

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arXiv2019

The 4--CB Algebra and Solvable Lattice Models

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--Murakami--Wenzl (BMW) algebra. For four blocks, we conjecture that the 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.

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