arXiv subjects

A. Mironov

Publications and source records attributed to A. Mironov.

At least 181 records · Page 10Linked to original sources

arXiv2011

Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations

Correlators in topological theories are given by the values of a linear form on the products of operators from a commutative associative algebra (CAA). As a corollary, partition functions of topological theory always satisfy the generalized WDVV equations. We consider the Hurwitz partition functions, associated in this way with the CAA of cut-and-join operators. The ordinary Hurwitz numbers for a given number of sheets in the covering provide trivial (sums of exponentials) solutions to the WDVV equations, with finite number of time-variables. The generalized Hurwitz numbers from arXiv:0904.4227 provide a non-trivial solution with infinite number of times. The simplest solution of this type is associated with a subring, generated by the dilatation operators tr X(d/dX).

hep-th↗

arXiv2011

On 3d extensions of AGT relation

An extension of the AGT relation from two to three dimensions begins from connecting the theory on domain wall between some two S-dual SYM models with the 3d Chern-Simons theory. The simplest kind of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the both quantities are very similar, especially if represented as integrals of the products of quantum dilogarithm functions. However, there are also various differences, especially in the "conservation laws" for integration variables, which hold for the monodromy traces, but not for the knot invariants. We also discuss another possibility: interpretation of knot invariants as solutions to the Baxter equations for the relativistic Toda system. This implies another AGT like relation: between 3d Chern-Simons theory and the Nekrasov-Shatashvili limit of the 5d SYM.

hep-th↗

arXiv2011

Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble

The exact free energy of matrix model always obeys the Seiberg-Witten (SW) equations on a complex curve defined by singularities of the quasiclassical resolvent. The role of SW differential is played by the exact one-point resolvent. We show that these properties are preserved in generalization of matrix models to beta-ensembles. However, since the integrability and Harer-Zagier topological recursion are still unavailable for beta-ensembles, we need to rely upon the ordinary AMM/EO recursion to evaluate the first terms of the genus expansion. Consideration in this paper is restricted to the Gaussian model.

hep-th↗

arXiv2011

Proving AGT conjecture as HS duality: extension to five dimensions

We extend the proof from arXiv:1012.3137, which interprets the AGT relation as the Hubbard-Stratonovich duality relation to the case of 5d gauge theories. This involves an additional q-deformation. Not surprisingly, the extension turns out to be trivial: it is enough to substitute all relevant numbers by q-numbers in all the formulas, Dotsenko-Fateev integrals by the Jackson sums and the Jack polynomials by the MacDonald ones. The problem with extra poles in individual Nekrasov functions continues to exist, therefore, such a proof works only for β= 1, i.e. for q=t in MacDonald's notation. For β\ne 1 the conformal blocks are related in this way to a non-Nekrasov decomposition of the LMNS partition function into a double sum over Young diagrams.

hep-th↗

arXiv2011

Integrability of Hurwitz Partition Functions. I. Summary

Partition functions often become τ-functions of integrable hierarchies, if they are considered dependent on infinite sets of parameters called time variables. The Hurwitz partition functions Z = \sum_R d_R^{2-k}χ_R(t^{(1)})...χ_R(t^{(k)})\exp(\sum_n ξ_nC_R(n)) depend on two types of such time variables, t and ξ. KP/Toda integrability in t requires that k\leq 2 and also that C_R(n) are selected in a rather special way, in particular the naive cut-and-join operators are not allowed for n>2. Integrability in ξfurther restricts the choice of C_R(n), forbidding, for example, the free cumulants. It also requires that k\leq 1. The quasiclassical integrability (the WDVV equations) is naturally present in ξvariables, but also requires a careful definition of the generating function.

hep-th↗

arXiv2010

A direct proof of AGT conjecture at beta = 1

The AGT conjecture claims an equivalence of conformal blocks in 2d CFT and sums of Nekrasov functions (instantonic sums in 4d SUSY gauge theory). The conformal blocks can be presented as Dotsenko-Fateev beta-ensembles, hence, the AGT conjecture implies the equality between Dotsenko-Fateev beta-ensembles and the Nekrasov functions. In this paper, we prove it in a particular case of beta=1 (which corresponds to c = 1 at the conformal side and to epsilon_1 + epsilon_2 = 0 at the gauge theory side) in a very direct way. The central role is played by representation of the Nekrasov functions through correlators of characters (Schur polynomials) in the Selberg matrix models. We mostly concentrate on the case of SU(2) with 4 fundamentals, the extension to other cases being straightforward. The most obscure part is extending to an arbitrary beta: for beta \neq 1, the Selberg integrals that we use do not reproduce single Nekrasov functions, but only sums of them.

hep-th↗

arXiv2010

Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals

The AGT relation identifies the Nekrasov functions for various N=2 SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (beta-ensemble) representations the latter being polylinear combinations of Selberg integrals. The "pure gauge" limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.

hep-th↗

arXiv2010

Algebra of differential operators associated with Young diagrams

We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups. The structure constants of the algebra are expressed through the Hurwitz numbers.

math.GT↗

arXiv2010

Towards a proof of AGT conjecture by methods of matrix models

A matrix model approach to proof of the AGT relation is briefly reviewed. It starts from the substitution of conformal blocks by the Dotsenko-Fateev beta-ensemble averages and Nekrasov functions by a double deformation of the exponentiated Seiberg-Witten prepotential in beta \neq 1 and g_s \neq 0 directions. Establishing the equality of these two quantities is a typical matrix model problem, and it presumably can be ascertained by investigation of integrability properties and developing an associated Harer-Zagier technique for evaluation of the exact resolvent.

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

On the Ground Level of Purely Magnetic Algebro-Geometric 2D Pauli Operator (spin 1/2)

Full manifold of the complex Bloch-Floquet eigenfunctions is investigated for the ground level of the purely magnetic 2D Pauli operators (equal to zero because of supersymmetry). Deep connection of it with the 2D analog of the "Burgers Nonlinear Hierarchy" plays fundamental role here. Everything is completely calculated for the broad class of Algebro-Geometric operators found in this work for this case. For the case of nonzero flux the ground states were found by Aharonov-Casher (1979) for the rapidly decreasing fields, and by Dubrovin-Novikov (1980) for the periodic fields. No Algebro-Geometric operators where known in the case of nonzero flux. For genus $g=1$ we found periodic operators with zero flux, singular magnetic fields and Bohm-Aharonov phenomenon. Our arguments imply that the delta-term really does not affect seriously the spectrum nearby of the ground state. For $g>1$ our theory requires to use only algebraic curves with selected point leading to the solutions elliptic in the variable $x$ for KdV and KP in order to get periodic magnetic fields. The algebro-geometric case of genus zero leads, in particular, to the slowly decreasing lump-like magnetic fields with especially interesting variety of ground states in the Hilbert Space $\cL_2(\bR^2)$.

math-ph↗

arXiv2010

On "Dotsenko-Fateev" representation of the toric conformal blocks

We demonstrate that the recent ansatz of arXiv:1009.5553, inspired by the original remark due to R.Dijkgraaf and C.Vafa, reproduces the toric conformal blocks in the same sense that the spherical blocks are given by the integral representation of arXiv:1001.0563 with a peculiar choice of open integration contours for screening insertions. In other words, we provide some evidence that the toric conformal blocks are reproduced by appropriate beta-ensembles not only in the large-N limit, but also at finite N. The check is explicitly performed at the first two levels for the 1-point toric functions. Generalizations to higher genera are briefly discussed.

hep-th↗

arXiv2010

2D Schrodinger Operator, (2+1) Systems and New Reductions. The 2D Burgers Hierarchy and Inverse Problem Data

The Theory of (2+1) Systems based on 2D Schrodinger Operator was started by S.Manakov, B.Dubrovin, I.Krichever and S.Novikov in 1976. The Analog of Lax Pairs introduced by Manakov, has a form $L_t=[L,H]-fL$ ("The $L,H,f$-triples") where $L=\partial_x\partial_y+G\partial_y+S$ and $H,f$-some linear PDEs. Their Algebro-Geometric Solutions and therefore the full higher order hierarchies were constructed by B.Dubrovin, I.Krichever and S.Novikov. The Theory of 2D Inverse Spectral Problems for the Elliptic Operator $L$ with $x,y$ replaced by $z,\bar{z}$, was started by B.Dubrovin, I.Krichever and S.Novikov: The Inverse Spectral Problem Data are taken from the complex "Fermi-Curve" consisting of all Bloch-Floquet Eigenfunctions $Lψ=const$. Many interesting systems were found later. However, specific properties of the very first system, offered by Manakov for the verification of new method only, were not studied more than 10 years until B.Konopelchenko found in 1988 analogs of Backund Transformations for it. He pointed out on the "Burgers-Type Reduction". Indeed, the present authors quite recently found very interesting extensions, reductions and applications of that system both in the theory of nonlinear evolution systems (The Self-Adjoint and 2D Burgers Hierarhies were invented, and corresponding reductions of Inverse Problem Data found) and in the Spectral Theory of Important Physical Operators ("The Purely Magnetic 2D Pauli Operators"). We call this system GKMMN by the names of authors who studied it.

math-ph↗

arXiv2010

Matrix model version of AGT conjecture and generalized Selberg integrals

Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special "conservation" relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko-Fateev integrals are inserted between the positions of the two original operators in the product. If such coefficients are combined to form an $n$-point conformal block on Riemann sphere, one reproduces the earlier conjectured $β$-ensemble representation of conformal blocks, thus proving this (matrix model) version of the celebrated AGT relation. The statement can also be regarded as a relation between the $3j$-symbols of the Virasoro algebra and the slightly generalized Selberg integrals $I_Y$, associated with arbitrary Young diagrams. The conformal blocks are multilinear combinations of such integrals and the remaining part of the original AGT conjecture relates them to the Nekrasov functions which have exactly the same structure.

hep-th↗

arXiv2010

New Reductions and Nonlinear Systems for 2D Schrodinger Operators

New Completely Integrable (2+1)-System is studied. It is based on the so-called L-A-B-triples $L_t=[H,L]-fL$ where L is a 2D Schrodinger Operator. This approach was invented by S.Manakov and B.Dubrovin, I.Krichever, S.Novikov(DKN) in the works published in 1976. A nonstandard reduction for the 2D Schrodinger Operator (completely different from the one found by S.Novikov and A.Veselov in 1984) compatible with time dynamics of the new Nonlinear System, is studied here. It can be naturally treated as a 2D extension of the famous Burgers System. The Algebro-Geometric (AG) Periodic Solutions here are very specific and unusual (for general and reduced cases). The reduced system is linearizable like Burgers. However, the general one (and probably the reduced one also) certainly lead in the stationary AG case to the nonstandard examples of algebraic curves $Γ\subset W$ in the full complex 2D manifold of Bloch-Floquet functions W for the periodic elliptic 2D operator H where $Hψ(x,y,P)=λ(P)ψ(x,y,P),P\in Γ$. However, in the nontrivial case the operators are nonselfadjoint. A Conjecture is formulated that for the nontrivial selfadjoint elliptic 2D Schrodinger operators H with periodic coefficients The Whole 2D Complex Manifolds W cannot not contain any Zariski open part of algebraic curve $Γ$ except maybe one selected level $Hψ=const$ found in 1976 by DKN. Version 2 contains new results. It also corrects some non-accurate claims; in particular, the non-reduced system is non-linearizable in any trivial sense. In version 3 a misprint in the formulation of Theorem 1 was corrected.

nlin.SI↗

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↗

arXiv2010

Conformal blocks as Dotsenko-Fateev Integral Discriminants

As anticipated in [1], elaborated in [2-4], and explicitly formulated in [5], the Dotsenko-Fateev integral discriminant coincides with conformal blocks, thus providing an elegant approach to the AGT conjecture, without any reference to an auxiliary subject of Nekrasov functions. Internal dimensions of conformal blocks in this identification are associated with the choice of contours: parameters of the DV phase of the corresponding matrix models. In this paper we provide further evidence in support of this identity for the 6-parametric family of the 4-point spherical conformal blocks, up to level 3 and for arbitrary values of external dimensions and central charges. We also extend this result to multi-point spherical functions and comment on a similar description of the 1-point function on a torus.

hep-th↗

arXiv2009

BGWM as Second Constituent of Complex Matrix Model

Earlier we explained that partition functions of various matrix models can be constructed from that of the cubic Kontsevich model, which, therefore, becomes a basic elementary building block in "M-theory" of matrix models. However, the less topical complex matrix model appeared to be an exception: its decomposition involved not only the Kontsevich tau-function but also another constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition function. The BGW tau-function can be represented either as a generating function of all unitary-matrix integrals or as a Kontsevich-Penner model with potential 1/X (instead of X^3 in the cubic Kontsevich model).

hep-th↗