Search

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.

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., Comment: 17 pages, no figures. Major revision. Added an appendix for the proof for WZW models of the character formula

We consider the multiple Calaby-Yau (CY) mirror phenomenon which appears in Berglund-H\"ubsch-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., Comment: Some important changes of the content and title, typos fixed

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., 20 pages, no figures, Mathematica notebook attached. Added references and corrected some erroneous 5--CB relations

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., Comment: 15 pages, no figures

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.

We study large-c SCFT2 on a torus specializing to one-point superblocks in the 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., 21 pages, v2: minor corrections, references added, journal version

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., 51 pages, 2 figures

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., Comment: 23 pages, 10 figures

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., 37 pages, v2: more comments, Appendix E --> Section 7.1, refs added, journal version

In a recent study we considered W 3 $$ {\mathcal{W}}_3 $$ Toda 4-point functions that involve matrix elements of a primary field with the highest-weight in the adjoint representation of s l 3 $$ \mathfrak{s}{\mathfrak{l}}_3 $$ . 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.

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., Comment: 49 pages

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.

We propose a simple method for the computation of the flat coordinates and Saito primitive forms on Frobenius manifolds of the deformations of Jacobi rings associated with isolated singularities. The method is based on using a conjecture about integral representations for the flat coordinates and on the Saito cohomology theory. This reduces the computation to a simple linear problem. We consider the case of the deformed Gepner chiral rings. The knowledge of the flat structures of Frobenius manifolds can be used for exact solution of the models of the topological conformal field theories corresponding to these chiral rings., Comment: 17 pages

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., 9 pages

We consider Virasoro conformal blocks in the large central charge limit. There are different regimes depending on the behavior of the conformal dimensions. The most simple regime is reduced to the global sl(2, C) conformal blocks while the most complicated one is known as the classical conformal blocks. Recently, Fitzpatrick, Kaplan, and Walters showed that the two regimes are related through the intermediate stage of the so-called heavy-light semiclassical limit. We study this idea in the particular case of the 5-point conformal block. To find the 5-point global block we use the projector technique and the Casimir operator approach. Furthermore, we discuss the relation between the global and the heavy-light limits and construct the heavy-light block from the global block. In this way we reproduce our previous results for the 5-point perturbative classical block obtained by means of the monodromy method., Comment: 19 pages, v2: minor corrections, typos removed; v3: more comments, extended Introduction and Conclusion, refs added, journal version

We propose the holographic interpretation of the 1-point conformal block on a torus in the semiclassical regime. To this end we consider the linearized version of the block and find its coefficients by means of the perturbation procedure around natural seed configuration corresponding to the zero-point block. From the AdS/CFT perspective the linearized block is given by the geodesic length of the tadpole graph embedded into the thermal AdS plus the holomorphic part of the thermal AdS action., 15 pages, v2: more comments, typos corrected, acknowledgments added, journal version

We compute 5-point classical conformal blocks with two heavy, two light, and one superlight operator using the monodromy approach up to third order in the superlight expansion. By virtue of the AdS/CFT correspondence we show the equivalence of the resulting expressions to those obtained in the bulk computation for the corresponding geodesic configuration., Comment: 20 pages, v2: more comments, refs added, NPB version

Current studies of $$ {\mathcal{W}}_N $$ Toda field theory focus on correlation functions such that the $$ {\mathcal{W}}_N $$ highest-weight representations in the fusion channels are multiplicity-free. In this work, we study $$ {\mathcal{W}}_3 $$ 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 $$ \mathfrak{s}{\mathfrak{l}}_3 $$ , and a fully-degenerate primary field with a highest-weight in the fundamental representation of $$ \mathfrak{s}{\mathfrak{l}}_3 $$ . We show that, when the fusion rules do not involve multiplicities, 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 $$ {\mathcal{W}}_N $$ theories. We solve this equation, compute its monodromy group, and construct the monodromy-invariant correlation functions. This computation shows in detail how the ambiguities that are caused by the presence of multiplicities are fixed by requiring monodromy-invariance.

AGT correspondence gives an explicit expressions for the conformal blocks of d = 2 conformal field theory. Recently an explanation of this representation inside the CFT framework was given through the assumption about the existence of the special orthogonal basis in the module of algebra A = Vir ⊗ H . The basis vectors are the eigenvectors of the infinite set of commuting integrals of motion. It was also proven that some of these vectors take form of Jack polynomials. In this note we conjecture and verify by explicit computations that in the case of the Virasoro central charge c = 1 all basis vectors are just the products of two Jack polynomials. Each of the commuting integrals of motion becomes the sum of two integrals of motion of two noninteracting Calogero models. We also show that in the case c ≠ 1 it is necessary to use two different Feigin–Fuks bosonizations of the Virasoro algebra for the construction of all basis vectors which take form of one Jack polynomial.

The dynamics of non-Abelian gauge theory can be described not only in terms of local gauge fields but also in terms of nonlocal gauge-invariant variables known as Wilson loops. In Wilson loop space, specific trajectories (defects) are considered on which Wilson loop operators take values in the center of the underlying gauge group. It is shown that, at finite temperature, the density of static (thermal) defects in the Euclidean formulation of Yang-Mills theory is sensitive to the thermodynamic phase transition: numerical calculations reveal that, in contrast to the gluon-plasma phase, where the defect density is high, the density of static defects is very low in the confining phase.

We construct the four-point correlation functions containing the top component of the supermultiplet in the Neveu–Schwarz sector of the N = 1 SUSY Liouville field theory. The construction is based on the recursive representation for the NS conformal blocks. We test our results in the case where one of the fields is degenerate with a singular vector on the level 3/2. In this case, the correlation function satisfies a third-order ordinary differential equation, which we derive. We numerically verify the crossing symmetry relations for the constructed correlation functions in the nondegenerate case.

We present explicit recursion relations for the four-point superconformal block functions that are essentially particular contributions of the given conformal class to the four-point correlation function. The approach is based on the analytic properties of the superconformal blocks as functions of the conformal dimensions and the central charge of the superconformal algebra. We compare the results with the explicit analytic expressions obtained for special parameter values corresponding to the truncated operator product expansion. These recursion relations are an efficient tool for numerically studying the four-point correlation function in superconformal field theory in the framework of the bootstrap approach, similar to that in the case of the purely conformal symmetry.

We continue to develop the holographic interpretation of classical conformal blocks in terms of particles propagating in an asymptotically $AdS_3$ geometry. We study $n$-point block with two heavy and $n-2$ light fields. Using the worldline approach we propose and explicitly describe the corresponding bulk configuration, which consists of $n-3$ particles propagating in the conical defect background produced by the heavy fields. We test this general picture in the case of five points. Using the special combinatorial representation of the Virasoro conformal block we compute $5$-point classical block and find the exact correspondence with the bulk worldline action. In particular, the bulk analysis relies upon the special perturbative procedure which treats the $5$-point case as a deformation of the $4$-pt case., 28 pages, v2: more comments, journal version

Using the connection with the Frobenius manifold structure, we study the matrix model description of minimal Liouville gravity (MLG) based on the Douglas string equation. Our goal is to find an exact discrete formulation of the (q,p) MLG model that intrinsically contains information about the conformal selection rules. We discuss how to modify the Frobenius manifold structure appropriately for this purposes. We propose a modification of the construction for Lee-Yang series involving the $A_{p-1}$ algebra instead of the previously used $A_1$ algebra. With the new prescription, we calculate correlators on the sphere up to four points and find full agreement with the continuous approach without using resonance transformations., 11 pages

We study the temperature dependence of the monopole condensate in different Abelian projections of the SU(2) gauge theory on the lattice. Using the Frohlich-Marchetti monopole creation operator, we show numerically that the monopole condensate depends on the choice of the Abelian projection.

Two-point correlation functions of spin operators in the minimal models M p,p′ perturbed by the field Φ13 are studied in the framework of conformal perturbation theory [Nucl. Phys. B 348 (1991) 619]. The first-order corrections for the structure functions are derived analytically in terms of gamma functions. Together with the exact vacuum expectation values of local operators, this gives the short-distance expansion of the correlation functions. The long-distance behaviors of these correlation functions in the case M 2,2n+1 have been worked out using a form-factor bootstrap approach. The results of numerical calculations demonstrate that the short- and long-distance expansions match at the intermediate distances. Including the descendent operators in the OPE drastically improves the convergency region. The combination of the two methods thus describes the correlation functions at all length scales with good precision.

We study gauge dependence of the recently suggested definition of the singlet and adjoint potentials in SU(2) lattice gauge theory. We find that in the (time local) maximal tree axial gauge the singlet potential obtained from the gauge dependent correlator ${Tr} L(x)L^\dagger(y)$ differs from that computed in the Coulomb gauge. In the generalized Coulomb gauge we find the range of the parameter values in which the singlet potential differs from that in the Coulomb gauge., Comment: 9 pages, 3 figures, Latex

The monopole creation operator proposed recently by Frohlich and Marchetti is investigated in the Abelian Higgs model with compact gauge field. We show numerically that the creation operator detects the condensation of monopoles in the presence of the dynamical matter field.

We use the connection between the Frobrenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensure the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has explicit and simple form in the flat coordinates on the Frobenious manifold in the general case of (p,q) Minimal Liouville gravity., Comment: 17 pages; v2: typos removed, some comments added, minor corrections

We study the AGT correspondence between four-dimensional supersymmetric gauge field theory and two-dimensional conformal field theories in the context of $$ {\mathcal{W}}_N $$ minimal models. The origin of the AGT correspondence is in a special integrable structure which appears in the properly extended conformal theory. One of the basic manifestations of this integrability is the special orthogonal basis which arises in the extended theory. We propose modification of the AGT representation for the $$ {\mathcal{W}}_N $$ conformal blocks in the minimal models. The necessary modification is related to the reduction of the orthogonal basis. This leads to the explicit combinatorial representation for the conformal blocks of minimal models and employs the sum over N-tupels of Young diagrams with additional restrictions.

We use the connection between the Frobenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search for a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensures the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has an explicit and simple form in the flat coordinates on the Frobenius manifold in the general case of (p,q) Minimal Liouville gravity.

We consider the problem of computing N=2 superconformal block functions. We argue that the Kazama-Suzuki coset realization of N=2 superconformal algebra in terms of the affine sl(2) algebra provides relations between N=2 and affine sl(2) conformal blocks. We show that for N=2 chiral fields the corresponding sl(2) construction of the conformal blocks is based on the ordinary highest weight representation. We use an AGT-type correspondence to relate the four-point sl(2) conformal block with Nekrasov's instanton partition functions of a four-dimensional N=2 SU(2) gauge theory in the presence of a surface operator. Since the previous relation proposed by Alday and Tachikawa requires some special modification of the conformal block function, we revisit this problem and find direct correspondence for the four-point conformal block. We thus find an explicit representation for the affine sl(2) four-point conformal block and hence obtain an explicit combinatorial representation for the N=2 chiral four-point conformal block., 15 pages