arXiv subjects

A. Marshakov

Publications and source records attributed to A. Marshakov.

At least 19 recordsLinked to original sources

arXiv2026

Hadrons in $\mathcal{N}=2$ supersymmetric QCD from non-Abelian string on 2D black hole

We continue the study of non-Abelian vortex string in 4D $\mathcal{N}=2$ supersymmetric QCD as critical superstring, and extend this analysis to $U(N)$ gauge theory with arbitrary even $N$ and $N_f=2N$ number of quarks. We introduce a special mass deformation and show that the SQCD hadron spectrum is still given by the string spectrum on the 2D $\mathcal{N}=2$ supersymmetric black hole. We perform a cross-check by computing the multiplicity of hadronic states of the high-energy part of the spectrum both from string and field theory pictures. We also clarify the spontaneous breaking of the global flavor symmetry by VEV of the massless baryon. We finally claim, that phase diagram of $\mathcal{N}=2$ SQCD with $N_f=2N$ consists of the Higgs phase at weak coupling and string/hadronic phase at strong coupling, separated by phase transition, and is seen as a conifold transition from string theory point of view.

hep-th↗

arXiv2025

Critical Non-Abelian Vortex String and 2D N=2 Black Hole

It has been shown that the non-Abelian vortex string in 4D $\mathcal{N}=2$ SQCD with the U(2) gauge group and $N_f=4$ flavors becomes a critical superstring. Its 10D target space is a product of the flat 4D space and an internal noncompact Calabi-Yau threefold, namely, the conifold. It was also shown that the Coulomb branch of the associated string sigma model, which opens up at strong coupling, can be described by $\mathcal{N}=2$ Liouville theory. We continue here the study of the recently proposed mass deformation of the U(2) theory with $N_f=4$, interpolating to SQCD with the U(4) gauge group and $N_f=8$ quarks, by analyzing the mass-deformed $\mathcal{N}=2$ Liouville theory on the string world sheet, and show that it is always described by the trumpet geometry of the target space, which is $T$-dual to the 2D $\mathcal{N}=2$ supersymmetric black hole. We use this correspondence to find the low-lying hadron spectrum in the deformed SQCD, and explain the expected increase in the number of hadronic states in the theory with more gauge fields and quarks by considering the near-Hagedorn behavior of the 2D black hole.

hep-th↗

arXiv2024

On Krichever tau-function and Verlinde formula

We remind the definition and main properties of the Krichever quasiclassical tau-function, and turn to the application of these formulas for recent studies of two-dimensional quantum gravity. We show, that in the case of minimal gravity it turns to be directly related with the Verlinde formula for minimal models, giving in particular case its one more direct proof. Generalizations for continuous "complex Liouville" theory are also briefly discussed.

hep-th↗

arXiv2019

Cluster integrable systems and spin chains

We discuss relation between the cluster integrable systems and spin chains in the context of their correspondence with 5d supersymmetric gauge theories. It is shown that $\mathfrak{gl}_N$ XXZ-type spin chain on $M$ sites is isomorphic to a cluster integrable system with $N \times M$ rectangular Newton polygon and $N \times M$ fundamental domain of a 'fence net' bipartite graph. The Casimir functions of the Poisson bracket, labeled by the zig-zag paths on the graph, correspond to the inhomogeneities, on-site Casimirs and twists of the chain, supplemented by total spin. The symmetricity of cluster formulation implies natural spectral duality, relating $\mathfrak{gl}_N$-chain on $M$ sites with the $\mathfrak{gl}_M$-chain on $N$ sites. For these systems we construct explicitly a subgroup of the cluster mapping class group $\mathcal{G}_\mathcal{Q}$ and show that it acts by permutations of zig-zags and, as a consequence, by permutations of twists and inhomogeneities. Finally, we derive Hirota bilinear equations, describing dynamics of the tau-functions or A-cluster variables under the action of some generators of $\mathcal{G}_\mathcal{Q}$.

hep-th↗

arXiv2018

Cluster Toda chains and Nekrasov functions

In this paper the relation between the cluster integrable systems and $q$-difference equations is extended beyond the Painlevé case. We consider the class of hyperelliptic curves when the Newton polygons contain only four boundary points. The corresponding cluster integrable Toda systems are presented, and their discrete automorphisms are identified with certain reductions of the Hirota difference equation. We also construct non-autonomous versions of these equations and find that their solutions are expressed in terms of 5d Nekrasov functions with the Chern-Simons contributions, while in the autonomous case these equations are solved in terms of the Riemann theta-functions.

math-ph↗

arXiv2018

Twist-field representations of W-algebras, exact conformal blocks and character identities

We study the twist-field representations of W-algebras and generalize construction of the corresponding vertex operators to D- and B-series. It is shown, how the computation of characters of these representations leads to nontrivial identities involving lattice theta-functions. We also propose a way to calculate their exact conformal blocks, expressing them for D-series in terms of geometric data of the corresponding Prym variety for covering curve with involution.

hep-th↗

arXiv2018

Cluster integrable systems, q-Painleve equations and their quantization

We discuss the relation between the cluster integrable systems and $q$-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices. We also define quantum $q$-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using $q$-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.

math-ph↗

arXiv2016

Free fermions, W-algebras and isomonodromic deformations

We consider the theory of multicomponent free massless fermions in two dimensions and use it for construction of representations of W-algebras at integer Virasoro central charges. We define the vertex operators in this theory in terms of solutions of the corresponding isomonodromy problem. We use this construction to get some new insights on tau-functions of the multicomponent Toda type hierarchies for the class of solutions, given by the isomonodromy vertex operators and get useful representation for the tau-function of isomonodromic deformations.

hep-th↗

arXiv2016

Residue Formulas for Prepotentials, Instanton Expansions and Conformal Blocks

We study the extended prepotentials for the S-duality class of quiver gauge theories, considering them as quasiclassical tau-functions, depending on gauge theory condensates and bare couplings. The residue formulas for the third derivatives of extended prepotentials are proven, which lead to effective way of their computation, as expansion in the weak-coupling regime. We discuss also the differential equations, following from the residue formulas, including the WDVV equations, proven to be valid for the $SU(2)$ quiver gauge theories. As a particular example we consider the constrained conformal quiver gauge theory, corresponding to the Zamolodchikov conformal blocks by 4d/2d duality. In this case part of the found differential equations turn into nontrivial relations for the period matrices of hyperelliptic curves.

hep-th↗

arXiv2016

Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations

We consider the conformal blocks in the theories with extended conformal W-symmetry for the integer Virasoro central charges. We show that these blocks for the generalized twist fields on sphere can be computed exactly in terms of the free field theory on the covering Riemann surface, even for a non-abelian monodromy group. The generalized twist fields are identified with particular primary fields of the W-algebra, and we propose a straightforward way to compute their W-charges. We demonstrate how these exact conformal blocks can be effectively computed using the technique arisen from the gauge theory/CFT correspondence. We discuss also their direct relation with the isomonodromic tau-function for the quasipermutation monodromy data, which can be an encouraging step on the way of definition of generic conformal blocks for W-algebra using the isomonodromy/CFT correspondence.

hep-th↗

arXiv2014

On Lie Groups and Toda Lattices

We extend the construction of the relativistic Toda chains as integrable systems on the Poisson submanifolds in Lie groups beyond the case of A-series. For the simply-laced case this is just a direct generalization of the well-known relativistic Toda chains, and we construct explicitly the set of the Ad-invariant integrals of motion on symplectic leaves, which can be described by the Poisson quivers being just the blown up Dynkin diagrams. We also demonstrate how to get the set of "minimal" integrals of motion, using the co-multiplication rules for the corresponding Lie algebras. In the non simply-laced case the corresponding Bogoyavlensky-Coxeter-Toda systems are constructed using the Fock-Goncharov folding of the corresponding Poisson submanifolds. We discuss also how this procedure can be extended for the affine case beyond A-series, and consider explicitly an example from the affine D-series.

hep-th↗

arXiv2014

Loop groups, Clusters, Dimers and Integrable systems

We describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups $\widehat{PGL}(N)$, which can be enumerated by cyclically irreducible elements the co-extended affine Weyl groups $(\widehat{W}\times \widehat{W})^\sharp$. Their phase spaces admit cluster coordinates, whereas the integrals of motion are cluster functions. We show, that this class of integrable systems coincides with the constructed by Goncharov and Kenyon out of dimer models on a two-dimensional torus and classified by the Newton polygons. We construct the correspondence between the Weyl group elements and polygons, demonstrating that each particular integrable model admits infinitely many realisations on the Poisson-Lie groups. We also discuss the particular examples, including the relativistic Toda chains and the Schwartz-Ovsienko-Tabachnikov pentagram map.

math.AG↗

arXiv2013

Tau-functions for Quiver Gauge Theories

The prepotentials for the quiver supersymmetric gauge theories are defined as quasiclassical tau-functions, depending on two different sets of variables: the parameters of the UV gauge theory or the bare compexified couplings, and the vacuum condensates of the theory in IR. The bare couplings are introduced as periods on the UV base curve, and the consistency of corresponding gradient formulas for the tau-functions is proven using the Riemann bilinear relations. It is shown, that dependence of generalised prepotentials for the quiver gauge theories upon the bare couplings turns to coincide with the corresponding formulas for the derivatives of tau-functions for the isomonodromic deformations. Computations for the SU(2) quiver gauge theories with bi- and tri-fundamental matter are performed explicitly and analysed in the context of 4d/2d correspondence.

hep-th↗

arXiv2012

Lie Groups, Cluster Variables and Integrable Systems

We discuss the Poisson structures on Lie groups and propose an explicit construction of the integrable models on their appropriate Poisson submanifolds. The integrals of motion for the SL(N)-series are computed in cluster variables via the Lax map. This construction, when generalised to the co-extended loop groups, gives rise not only to several alternative descriptions of relativistic Toda systems, but allows to formulate in general terms some new class of integrable models.

hep-th↗

arXiv2012

On Gauge Theories as Matrix Models

The relation between the Seiberg-Witten prepotentials, Nekrasov functions and matrix models is discussed. We derive quasiclassically the matrix models of Eguchi-Yang type, describing the instantonic contribution to the deformed partition functions of supersymmetric gauge theories. The exact quasiclassical solution for the case of conformal four-dimensional theory is studied in detail, and some aspects of its relation with the recently proposed logarithmic beta-ensembles are considered. We discuss also the "quantization" of this picture in terms of two-dimensional conformal theory with extended symmetry, and stress its difference from common picture of perturbative expansion a la matrix models. Instead, the representation for Nekrasov functions in terms of conformal blocks or Whittaker vector suggests some nontrivial relation with Teichmueller spaces and quantum integrable systems.

hep-th↗

arXiv2010

On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles

We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block through the beta-ensemble resolvents and to its asymptotics in the limit of large dimensions (both external and intermediate) taken asymmetrically in terms of the deformation epsilon-parameters. The next-to-leading term in the asymptotics defines the generating differential in the Bohr-Sommerfeld representation of the one-parameter deformed Seiberg-Witten prepotentials (whose full two-parameter deformation leads to Nekrasov functions). This generating differential is also shown to be the one-parameter version of the single-point resolvent for the corresponding beta-ensemble, and its periods in the perturbative limit of the gauge theory are expressed through the ratios of the Harish-Chandra function. The Shrödinger/Baxter equations, considered earlier in this context, directly follow from the differential equations for the degenerate conformal block. This provides a powerful method for evaluation of the single-deformed prepotentials, and even for the Seiberg-Witten prepotentials themselves. We mostly concentrate on the representative case of the insertion into the four-point block on sphere and one-point block on torus.

hep-th↗

arXiv2010

Period Integrals, Quantum Numbers and Confinement in SUSY QCD

We present a direct computation of the period integrals on degenerate Seiberg-Witten curves for supersymmetric QCD, and show how these periods determine the changes in the quantum numbers of the states, when passing from the weak to the strong-coupling domains in the mass moduli space of the theory. The confinement of monopoles at strong coupling is discussed, and we demonstrate that the ambiguities in choosing the way in the moduli space do not influence to the physical conclusions on confinement of monopoles in the phase with the condensed light dyons.

hep-th↗

arXiv2010

On Combinatorial Expansions of Conformal Blocks

In a recent paper (arXiv:0906.3219) the representation of Nekrasov partition function in terms of nontrivial two-dimensional conformal field theory has been suggested. For non-vanishing value of the deformation parameter ε=ε_1+ε_2 the instanton partition function is identified with a conformal block of Liouville theory with the central charge c = 1+ 6ε^2/ε_1ε_2. If reversed, this observation means that the universal part of conformal blocks, which is the same for all two-dimensional conformal theories with non-degenerate Virasoro representations, possesses a non-trivial decomposition into sum over sets of the Young diagrams, different from the natural decomposition studied in conformal field theory. We provide some details about this intriguing new development in the simplest case of the four-point correlation functions.

hep-th↗