arXiv subjects

A. Morozov

Publications and source records attributed to A. Morozov.

At least 271 records · Page 15Linked to original sources

arXiv2011

Geodesic deviation and particle creation in curved spacetimes

A quantum mechanical picture, relating accelerated geodesic deviation to creation of massive particles via quantum tunneling in curved background spacetimes, is presented. The effect is analogous to pair production by an electric field and leads naturally to production of massive particles in de Sitter and superluminal FRW spacetimes. The probability of particle production in de Sitter space per unit volume and time is computed in a leading semiclassical approximation and shown to coincide with the previously obtained expression.

gr-qc↗

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

Unitary Integrals and Related Matrix Models

Concise review of the basic properties of unitary matrix integrals. They are studied with the help of the three matrix models: the ordinary unitary model, Brezin-Gross-Witten model and the Harish-Charndra-Itzykson-Zuber model. Especial attention is paid to the tricky sides of the story, from De Wit-t'Hooft anomaly in unitary integrals to the problem of correlators with Itzykson-Zuber measure. Of technical tools emphasized is the method of character expansions. The subject of unitary integrals remains highly under-investigated and a lot of new results are expected in this field when it attracts sufficient attention.

hep-th↗

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

From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators

Brezin-Hikami contour-integral representation of exponential multidensities in finite N Hermitian matrix model is a remarkable implication of the old Hermitian-Kontsevich duality. It is also a simplified version of Okounkov's formulas for the same multidensities in the cubic Kontsevich model and of Nekrasov calculus for LMNS integrals, a central piece of the modern studies of AGT relations. In this paper we use Brezin-Hikami representation to derive explicit expressions for the Harer-Zagier multidensities (from arXiv:0906.0036): the only known exhaustive generating functions of all-genera Gaussian correlators which are fully calculable and expressed in terms of elementary functions. Using the Brezin-Hikami contour integrals, we rederive the 1-point function of Harer and Zagier and the 2-point arctangent function of arXiv:0906.0036. We also present (without a proof) the explicit expression for the 3-point function in terms of arctangents. Derivation of the 3-point and higher Harer-Zagier functions remains a challenging problem.

hep-th↗

arXiv2010

The matrix model version of AGT conjecture and CIV-DV prepotential

Recently exact formulas were provided for partition function of conformal (multi-Penner) beta-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted as Dotsenko-Fateev correlator of screenings and analytically continued in the number of screening insertions, represents generic Virasoro conformal blocks. Actually these formulas describe the lowest terms of the q_a-expansion, where q_a parameterize the shape of the Penner potential, and are exact in the filling numbers N_a. At the same time, the older theory of CIV-DV prepotential, straightforwardly extended to arbitrary beta and to non-polynomial potentials, provides an alternative expansion: in powers of N_a and exact in q_a. We check that the two expansions coincide in the overlapping region, i.e. for the lowest terms of expansions in both q_a and N_a. This coincidence is somewhat non-trivial, since the two methods use different integration contours: integrals in one case are of the B-function (Euler-Selberg) type, while in the other case they are Gaussian integrals.

hep-th↗

arXiv2010

Black Hole Motion in Entropic Reformulation of General Relativity

We consider a system of black holes -- a simplest substitute of a system of point particles in the mechanics of general relativity -- and try to describe their motion with the help of entropic action: a sum of the areas of black hole horizons. We demonstrate that such description is indeed consistent with the Newton's laws of motion and gravity, modulo numerical coefficients, which coincide but seem different from unity. Since a large part of the modern discussion of entropic reformulation of general relativity is actually based on dimensional considerations, for making a next step it is crucially important to modify the argument, so that these dimensionless parameters acquire correct values.

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↗

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↗