[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: 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, and we
construct explicitly the set of Ad-invariant 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
co-multiplication rules. In the non simply-laced 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 A-series, and consider explicitly an example from the affine D-series.
Journal of Physics A Mathematical and Theoretical 04/2014; 48(12). DOI:10.1088/1751-8113/48/12/125201 · 1.58 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: 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.
Journal of High Energy Physics 12/2013; 2014(5). DOI:10.1007/JHEP05(2014)097 · 6.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Journal of High Energy Physics 03/2013; 2013(7). DOI:10.1007/JHEP07(2013)068 · 6.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Journal of Geometry and Physics 07/2012; 67(03n04). DOI:10.1016/j.geomphys.2012.12.003 · 0.87 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The free field representation or "bosonization" rule1 for Wess-Zumino-Witten model (WZWM) with arbitrary Kac-Moody algebra and arbitrary central charge is discussed. Energy-momentum 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 Kac-Moody (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.
International Journal of Modern Physics A 04/2012; 05(13). DOI:10.1142/S0217751X9000115X · 1.70 Impact Factor
[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 Virasoro-W 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) Virasoro-W 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 more-general-than-GKM solutions and deeper understanding of their relation to 2D gravity.
International Journal of Modern Physics A 04/2012; 08(22). DOI:10.1142/S0217751X93001569 · 1.70 Impact Factor
[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 Harish-Chandra function, which is shown to be a product of simple Γ function factors over all positive roots of the corresponding algebras (finite-dimensional for d = 0 and affine for d = 1). This is in nice correspondence with the recent results on two- and three-point 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 d-dimensional boundary itself. In this paper we concentrate on the finite-dimensional case. The results for finite-dimensional "Iwasawa" Whittaker functions are known, and we present a survey. We also construct new "Gauss" Whittaker functions.
International Journal of Modern Physics A 01/2012; 12(14). DOI:10.1142/S0217751X97001444 · 1.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: I consider main features of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential. 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.
International Journal of Modern Physics B 01/2012; 11(26n27). DOI:10.1142/S0217979297001519 · 0.94 Impact Factor
[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.
International Journal of Modern Physics A 01/2012; 15(08). DOI:10.1142/S0217751X00000537 · 1.70 Impact Factor
[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 first-order 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.
Modern Physics Letters A 11/2011; 11(14). DOI:10.1142/S021773239600120X · 1.20 Impact Factor
[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 Landau-Ginzburg 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 non-deformed 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 p-direction, 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.
Modern Physics Letters A 11/2011; 08(11). DOI:10.1142/S0217732393002531 · 1.20 Impact Factor
[Show abstract][Hide abstract] 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) one-differentials, 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.
Modern Physics Letters A 11/2011; 12(11). DOI:10.1142/S0217732397000807 · 1.20 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The Ward identities in Kontsevich-like one-matrix 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 two-matrix model to the form of -constraints imposed on its partition function.
Modern Physics Letters A 11/2011; 07(15). DOI:10.1142/S0217732392001014 · 1.20 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Matrix models are equivalent to certain integrable theories, partition functions being equal to certain τ-functions, i.e., the section of determinant bundles over infinite-dimensional Grassmannian. These τ-functions are evaluated at the points of Grassmannian, where high symmetry arises. In the case of one-matrix models the symmetry is isomorphic to Borel subgroup of a Virasoro group. The orbits of the group are in one-to-one correspondence with the types of "multicritical" behavior in the continuum limit. Interrelation between τ-functions 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 Fairlie-like -algebras.
Modern Physics Letters A 11/2011; 06(33). DOI:10.1142/S0217732391003572 · 1.20 Impact Factor
[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 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 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 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.
Journal of Geometry and Physics 11/2010; 61(7). DOI:10.1016/j.geomphys.2011.01.012 · 0.87 Impact Factor
[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 low-energy theory has U(N) gauge symmetry and one observes the non-Abelian 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 strongly-coupled domain for the original theory is given by U(N) dual gauge theory of Nf⩾2N light non-Abelian dyons, where the condensed dyons still cause the confinement of monopoles, and not of the quarks, as can be thought by naive duality.
Nuclear Physics B 05/2010; DOI:10.1016/j.nuclphysb.2009.12.037 · 3.93 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: I consider quasiclassical integrable systems, starting from the well-known 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 tau-functions 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 Seiberg-Witten 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 non-perturbative beta-function so that the effective coupling differs from the bare charge by infinite number of instantonic corrections.
Journal of High Energy Physics 11/2009; 2009(11):048. DOI:10.1088/1126-6708/2009/11/048 · 6.11 Impact Factor