Publications (95)257.42 Total impact
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ABSTRACT: We consider the conformal blocks in the theories with extended conformal Wsymmetry 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 nonabelian monodromy group. The generalized twist fields are identified with particular primary fields of the Walgebra, and we propose a straightforward way to compute their Wcharges. 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 taufunction for the quasipermutation monodromy data, which can be an encouraging step on the way of definition of generic conformal blocks for Walgebra using the isomonodromy/CFT correspondence. 
Article: On Lie Groups and Toda Lattices
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ABSTRACT: We extend the construction of the relativistic Toda chains as integrable systems on the Poisson submanifolds in Lie groups beyond the case of Aseries. For the simplylaced case this is just a direct generalization, and we construct explicitly the set of Adinvariant integrals of motion on symplectic leaves, whose Poisson quivers can be presented as blown up Dynkin diagrams. We also demonstrate how to get the set of "minimal" integrals of motion, using the comultiplication rules. In the non simplylaced case the corresponding Toda systems are constructed using the folding of the corresponding Poisson submanifolds. We discuss how this procedure can be extended for the affine case beyond Aseries, and consider explicitly an example from the affine Dseries.  [Show abstract] [Hide abstract]
ABSTRACT: We describe a class of integrable systems on Poisson submanifolds of the affine PoissonLie groups $\widehat{PGL}(N)$, which can be enumerated by cyclically irreducible elements the coextended 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 twodimensional 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 PoissonLie groups. We also discuss the particular examples, including the relativistic Toda chains and the SchwartzOvsienkoTabachnikov pentagram map.  [Show abstract] [Hide abstract]
ABSTRACT: We study the extended prepotentials for the Sduality class of quiver gauge theories, considering them as quasiclassical taufunctions, 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 weakcoupling 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.  [Show abstract] [Hide abstract]
ABSTRACT: The prepotentials for the quiver supersymmetric gauge theories are defined as quasiclassical taufunctions, 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 taufunctions 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 taufunctions for the isomonodromic deformations. Computations for the SU(2) quiver gauge theories with bi and trifundamental matter are performed explicitly and analysed in the context of 4d/2d correspondence.  [Show abstract] [Hide abstract]
ABSTRACT: We discuss Poisson structures on Lie groups and propose an explicit construction of integrable models on their appropriate Poisson submanifolds. The integrals of motion for the S L (N) series are computed in cluster variables via the Lax map. This construction, when generalised to coextended loop groups, not only gives rise to several alternative descriptions of relativistic Toda systems but also allows to formulate in general terms some new class of integrable models. We discuss the subtleties of this Lax map in relation to the ambiguity in projection to the trivial coextension and propose a way to write the spectral curve equation, which fixes this ambiguity, for the periodic Toda chain and its generalisations.  [Show abstract] [Hide abstract]
ABSTRACT: The free field representation or "bosonization" rule1 for WessZuminoWitten model (WZWM) with arbitrary KacMoody algebra and arbitrary central charge is discussed. Energymomentum tensor, arising from Sugawara construction, is quadratic in the fields. In this way, all known formulae for conformal blocks and correlators may be easily reproduced as certain linear combinations of correlators of these free fields. Generalization to conformal blocks on arbitrary Riemann surfaces is straightforward. However, projection rules in the spirit of Ref. 2 are not specified. The special role of βγ systems is emphasized. From the mathematical point of view, the construction involved represents generators of KacMoody (KM) algebra in terms of generators of a Heisenberg one. If WZW Lagrangian is considered as d−1 of Kirillov form on an orbit of KM algebra,3 then the free fields of interest (i.e. generators of the Heisenberg algebra) diagonalize Kirillov form and the action. Reduction of KM algebra within the same construction should naturally lead to arbitrary coset models.  [Show abstract] [Hide abstract]
ABSTRACT: A review of the appearance of integrable structures in the matrix model description of 2D gravity is presented. Most of the ideas are demonstrated with technically simple but ideologically important examples. Matrix models are considered as a sort of “effective” description of continuum 2D field theory formulation. The main physical role in such a description is played by the VirasoroW conditions, which can be interpreted as certain unitarity or factorization constraints. Both discrete and continuum (generalized Kontsevich) models are formulated as the solutions to those discrete (continuous) VirasoroW constraints. Their integrability properties are proved, using mostly the determinant technique highly related to the representation in terms of free fields. The paper also contains some new observations connected with formulation of moregeneralthanGKM solutions and deeper understanding of their relation to 2D gravity.  [Show abstract] [Hide abstract]
ABSTRACT: In this series of papers we represent the "Whittaker" wave functional of the (d + 1)dimensional Liouville model as a correlator in (d + 0)dimensional theory of the sine–Gordon type (for d = 0 and 1). The asymptotics of this wave function is characterized by the HarishChandra function, which is shown to be a product of simple Γ function factors over all positive roots of the corresponding algebras (finitedimensional for d = 0 and affine for d = 1). This is in nice correspondence with the recent results on two and threepoint correlators in the 1+1 Liouville model, where emergence of peculiar double periodicity is observed. The Whittaker wave functions of (d + 1)dimensional nonaffine ("conformal") Toda type models are given by simple averages in the (d + 0)dimensional theories of the affine Toda type. This phenomenon is in obvious parallel with representation of the free field wave functional, which was originally a Gaussian integral over the interior of a (d + 1)dimensional disk with given boundary conditions, as a (nonlocal) quadratic integral over the ddimensional boundary itself. In this paper we concentrate on the finitedimensional case. The results for finitedimensional "Iwasawa" Whittaker functions are known, and we present a survey. We also construct new "Gauss" Whittaker functions.  [Show abstract] [Hide abstract]
ABSTRACT: I consider main features of the formulation of the finitegap solutions to integrable equations in terms of complex curves and generating 1differential. The example of periodic Toda chain solutions is considered in detail. Recently found exact nonperturbative solutions to SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to the integrable systems are discussed.  [Show abstract] [Hide abstract]
ABSTRACT: We consider 4D and 5D supersymmetric theories and demonstrate that in general their Seiberg–Witten prepotentials satisfy the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations. General proof for the Yang–Mills models (with matter in the first fundamental representation) makes use of the hyperelliptic curves and underlying integrable systems. A wide class of examples is discussed; it contains few understandable exceptions. In particular, in the perturbative regime of 5D theories, in addition to naive field theory expectations some extra terms appear, as happens in heterotic string models. We consider also the example of the Yang–Mills theory with matter hypermultiplet in the adjoint representation (related to the elliptic Calogero–Moser system) when the standard WDVV equations do not hold.  [Show abstract] [Hide abstract]
ABSTRACT: The exact solutions to quantum string and gauge field theories are discussed and their formulation in the framework of integrable systems is presented. In particular we consider in detail several examples of the appearance of solutions to the firstorder integrable equations of hydrodynamical type and stress that all known examples can be treated as partial solutions to the same problem in the theory of integrable systems.  [Show abstract] [Hide abstract]
ABSTRACT: Matrix models are equivalent to certain integrable theories, partition functions being equal to certain τfunctions, i.e., the section of determinant bundles over infinitedimensional Grassmannian. These τfunctions are evaluated at the points of Grassmannian, where high symmetry arises. In the case of onematrix models the symmetry is isomorphic to Borel subgroup of a Virasoro group. The orbits of the group are in onetoone correspondence with the types of "multicritical" behavior in the continuum limit. Interrelation between τfunctions in different models and their continuum limit is discussed in some details. We also discuss the implications for dynamical interpolation between various string models (CFT's), to be described in terms of geometrical quantization of Fairlielike algebras. 
Article: WDVV Equations from Algebra of Forms
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ABSTRACT: A class of solutions to the WDVV equations is provided by period matrices of hyperelliptic Riemann surfaces, with or without punctures. The equations themselves reflect associativity of explicitly described multiplicative algebra of (possibly meromorphic) onedifferentials, which holds at least in the hyperelliptic case. This construction is direct generalization of the old one, involving the ring of polynomials factorized over an ideal, and is inspired by the study of the Seiberg–Witten theory. It has potential to be further extended to reveal algebraic structures underlying the theory of quantum cohomologies and the prepotentials in string models with N=2 supersymmetry. 
Article: FROM VIRASORO CONSTRAINTS IN KONTSEVICH’S MODEL TO ${\mathcal W}$CONSTRAINTS IN TWOMATRIX MODELS
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ABSTRACT: The Ward identities in Kontsevichlike onematrix models are used to prove at the level of discrete matrix models the suggestion of Gava and Narain, which relates the degree of potential in asymmetric twomatrix model to the form of constraints imposed on its partition function.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the deformations of "monomial solutions" to generalized Kontsevich. model1,2 and establish the relation between the flows generated by these deformations with those of N = 2 LandauGinzburg topological theories. We prove that the partition function of a generic generalized Kontsevich model can be presented as a product of some "quasiclassical" factor and nondeformed partition function which depends only on the sum of Miwa transformed and flat times. This result is important for the restoration of explicit p − q symmetry in the interpolation pattern between all the (p, q)minimal string models with c < 1 and for revealing its integrable structure in pdirection, determined by deformations of the potential. It also implies the way in which supersymmetric LandauGinzburg models are embedded into the general context of GKM. From the point of view of integrable theory these deformations present a particular case of what is called equivalent hierarchies.  [Show abstract] [Hide abstract]
ABSTRACT: 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 hypergeometrictype differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block through the betaensemble resolvents and to its asymptotics in the limit of large dimensions (both external and intermediate) taken asymmetrically in terms of the deformation epsilonparameters. The nexttoleading term in the asymptotics defines the generating differential in the BohrSommerfeld representation of the oneparameter deformed SeibergWitten prepotentials (whose full twoparameter deformation leads to Nekrasov functions). This generating differential is also shown to be the oneparameter version of the singlepoint resolvent for the corresponding betaensemble, and its periods in the perturbative limit of the gauge theory are expressed through the ratios of the HarishChandra function. The Shr\"odinger/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 singledeformed prepotentials, and even for the SeibergWitten prepotentials themselves. We mostly concentrate on the representative case of the insertion into the fourpoint block on sphere and onepoint block on torus.  [Show abstract] [Hide abstract]
ABSTRACT: We consider N=2 supersymmetric QCD with the gauge group SU(Nc)=SU(N+1) and Nf number of quark matter multiplets, being perturbed by a small mass term for the adjoint matter, so that its Coulomb branch shrinks to a number of isolated vacua. We discuss the vacuum where r=N quarks develop VEV's for Nf⩾2N=2Nc−2 (in particular, we focus on the Nf=2N and Nf=2N+1 cases). In the equal quark mass limit at large masses this vacuum stays at weak coupling, the lowenergy theory has U(N) gauge symmetry and one observes the nonAbelian confinement of monopoles. As we reduce the average quark mass and enter the strong coupling regime the quark condensate transforms into the condensate of dyons. We show that the low energy description in the stronglycoupled domain for the original theory is given by U(N) dual gauge theory of Nf⩾2N light nonAbelian dyons, where the condensed dyons still cause the confinement of monopoles, and not of the quarks, as can be thought by naive duality.  [Show abstract] [Hide abstract]
ABSTRACT: I consider quasiclassical integrable systems, starting from the wellknown dispersionless KdV and Toda hierarchies, which can be totally understood in terms of jet spaces over the rational curves with one or two punctures. For the nontrivial geometry of the higher genus curves, the same approach leads to construction of quasiclassical taufunctions or prepotentials, using the period integrals for Abelian differentials. I discuss also some physical applications of this construction.  [Show abstract] [Hide abstract]
ABSTRACT: The AGT relations allow to convert the Zamolodchikov asymptotic formula for conformal block into the instanton expansion of the SeibergWitten prepotential for the theory with two colors and four fundamental flavors. This provides an explicit formula for the instanton corrections in this model. The answer is especially elegant for vanishing matter masses, then the bare charge of gauge theory q0 = eiπτ0 plays the role of a branch point on the spectral elliptic curve. The exact prepotential at this point is = (1/2πi)a2log q with q≠q0, unlike the case of another conformal theory, with massless adjoint field. Instead, 16q0 = θ104/θ004(q) = 16q(1+O(q)), i.e. the Zamolodchikov asymptotic formula gives rise, in particular, to an exact nonperturbative betafunction so that the effective coupling differs from the bare charge by infinite number of instantonic corrections.
Publication Stats
3k  Citations  
257.42  Total Impact Points  
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Institutions

19902013

Institute for Theoretical and Experimental Physics
 Laboratory of Theoretical Physics
Moskva, Moscow, Russia


2012

National Research University Higher School of Economics
 Department of Mathematics, MIEM
Moskva, Moscow, Russia


19882012

Russian Academy of Sciences
 Division of Theoretical Physics
Moskva, Moscow, Russia


19962002

Instituto de Tecnologia de Pernambuco
Arrecife, Pernambuco, Brazil


1995

Uppsala University
Uppsala, Uppsala, Sweden


1991

CERN
Genève, Geneva, Switzerland
