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A. A. Bytsenko

Publications and source records attributed to A. A. Bytsenko.

At least 19 recordsLinked to original sources
arXiv2020

Partition functions for supersymmetric gauge theories on spheres

In this paper we briefly review the main idea of the localization technique and its extension suitable in supersymmetric gauge field theory. We analyze the partition function of the vector multiplets with supercharges and its blocks on the even- and odd-dimensional spheres and squashed spheres. We exploit so-called Faà di Bruno's formula and show that multipartite partition functions can be written in the form of expansion series of the Bell polynomials. Applying the restricted specialization argument we show that $q$-infinite-product representation of partition functions admits presentation in terms of the Patterson-Selberg (or the Ruelle-type) spectral functions.

hep-th↗
arXiv2020

The Duistermaat-Heckman formula with application to circle actions and Poincaré $q$-polynomials in twisted equivariant K-theory

In this paper we deduce the sketch of proof of the Duistermaat-Heckman formula and investigate how the known Duistermaat-Heckman result could be specialized to the symplectic structure on the orbit space. The theorems of localization in equivariant cohomology not only provide us with beautiful mathematical formulas and stimulate achievements in algorithmic computations, but also promote progress in theoretical and mathematical physics. We present the elliptic genera and the characteristic $q$-series for the circle actions and twisted equivariant K-theory, with the case of the symmetric group of $n$ symbols separately analyzed. We show that the Poincaré $q$-polynomials admit presentation in terms of the Patterson-Selberg (or the Ruelle-type) spectral functions.

math.KT↗
arXiv2020

Hilbert schemes, Verma modules and spectral functions of hyperbolic geometry with application to quantum invariants

In this article we exploit Ruelle-type spectral functions and analyze the Verma module over Virasoro algebra, boson-fermion correspondence, the analytic torsion, the Chern-Simons and $η$ invariants, as well as the generation function associated to dimensions of the Hochschild homology of the crossed product $\mathbb{C}[S_n]\ltimes \mathcal{A}^{\otimes n}$ ($\mathcal{A}$ is the $q$-Weyl algebra). After analysing the Chern-Simons and $η$ invariants of Dirac operators by using irreducible $SU(n)$-flat connections on locally symmetric manifolds of non-positive section curvature, we describe the exponential action for the Chern-Simons theory.

hep-th↗
arXiv2019

Elliptic Genera and q-Series Development in Analysis, String Theory, and N=2 Superconformal Field Theory

In this article we examine the Ruelle type spectral functions $\cR(s)$,which define an overall description of the content of the work. We investigate the Gopakumar-Vafa reformulation of the string partition functions, describe the N=2 Landau-Ginzburg model in terms of Ruelle type spectral functions. Furthermore, we discuss the basic properties satisfied by elliptic genera in N=2 theories, construct the functional equations for $\cR(s)$, and analyze the modular transformation laws for the elliptic genus of the Landau-Ginzburg model and study their properties in details.

hep-th↗
arXiv2017

Multipartite Generating Functions and Infinite Products for Quantum Invariants

We show that multipartite generation functions can be written in terms of the Bell polynomials (known as Faà di Bruno's formula) and the Ruelle spectral functions, whose spectrum is encoded in the Patterson-Selberg function of the hyperbolic three-geometry. We derive an infinite-product formula for the Chern-Simons partition functions and analyze appropriate q-series which leads to the construction of knot invariants. With the help of the Ruelle spectral functions symmetric and modular properties in infinite-product structure can be described.

math-ph↗
arXiv2016

$S$-Functions, Spectral Functions of Hyperbolic Geometry, and Vertex Operators with Applications to Structure for Weyl and Orthogonal Group Invariants

In this paper we analyze the quantum homological invariants (the Poincaré polynomials of the $\mathfrak{sl}_N$ link homology). In the case when the dimensions of homologies of appropriate topological spaces are precisely known, the procedure of the calculation of the Kovanov-Rozansky type homology, based on the Euler-Poincaré formula can be appreciably simplified. We express the formal character of the irreducible tensor representation of the classical groups in terms of the symmetric and spectral functions of hyperbolic geometry. On the basis of Labastida-Mariño-Ooguri-Vafa conjecture, we derive a representation of the Chern-Simons partition function in the form of an infinite product in terms of the Ruelle spectral functions (the cases of a knot, unknot, and links have been considered). We also derive an infinite-product formula for the orthogonal Chern-Simons partition functions and analyze the singularities and the symmetry properties of the infinite-product structures.

hep-th↗
arXiv2015

Elliptic Genera and Characteristic $q$-Series of Superconformal Field Theory

We analyze the characteristic series, the $KO$ series and the series associated with the Witten genus, and their analytic forms as the $q$-analogs of classical special functions (in particular $q$-analog of the beta integral and the gamma function). $q$-series admit an analytic interpretation in terms of the spectral Ruelle functions, and their relations to appropriate elliptic modular forms can be described. We show that there is a deep correspondence between the characteristic series of the Witten genus and $KO$ characteristic series, on one side, and the denominator identities and characters of $N=2$ superconformal algebras, and the affine Lie (super)algebras on the other. We represent the characteristic series in the form of double series using the Hecke-Rogers modular identity.

hep-th↗
arXiv2014

D-Branes on Spaces Stratified Fibered Over Hyperbolic Orbifolds

We apply the methods of homology and K-theory for branes wrapping spaces stratified fibered over hyperbolic orbifolds. In addition, we discuss the algebraic K-theory of any discrete co-compact Lie group in terms of appropriate homology and Atiyah-Hirzebruch type spectral sequence with its non-trivial lift to K-homology. We emphasize the fact that the physical D-branes properties are completely transparent within the mathematical framework of K-theory. We derive criteria for D-brane stability in the case of strongly virtually negatively curved groups. We show that branes wrapping spaces stratified fibered over hyperbolic orbifolds carry charge structure and change the additive structural properties in K-homology.

hep-th↗
arXiv2014

Generalized q-deformed Correlation Functions as Spectral Functions of Hyperbolic Geometry

We analyse the role of vertex operator algebra and 2d amplitudes from the point of view of the representation theory of infinite dimensional Lie algebras, MacMahon and Ruelle functions. A p-dimensional MacMahon function is the generating function of p-dimensional partitions of integers. These functions can be represented as amplitudes of a two-dimensional c=1 CFT. In this paper we show that p-dimensional MacMahon functions can be rewritten in terms of Ruelle spectral functions, whose spectrum is encoded in the Patterson-Selberg function of three dimensional hyperbolic geometry.

hep-th↗
arXiv2013

Quantum Black Holes, Elliptic Genera and Spectral Partition Functions

We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lie in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi-Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.

hep-th↗
arXiv2013

BRST-Invariant Deformations of Geometric Structures in Topological Field Theories

We study a Lie algebra of formal vector fields $W_n$ with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. A relevant concept in the vertex operator algebra and the BRST cohomology is that of the elliptic genera (the one-loop string partition function). We show that the elliptic genera can be written in terms of spectral functions of the hyperbolic three-geometry (which inherits the cohomology structure of BRST-like operator). We show that equivalence classes of deformations are described by a Hochschild cohomology theory of the DG-algebra ${\mathfrak A} = (A, Q)$, $Q =\bar{\partial}+\partial_{\rm deform}$, which is defined to be the cohomology of $(-1)^n Q +d_{\rm Hoch}$. Here $\bar{\partial}$ is the initial non-deformed BRST operator while $\partial_{\rm deform}$ is the deformed part whose algebra is a Lie algebra of linear vector fields ${\rm gl}_n$. We discuss the identification of the harmonic structure $(HT^\bullet(X); HΩ_\bullet(X))$ of affine space $X$ and the group ${\rm Ext}_{X}^n({\cal O}_{\triangle}, {\cal O}_{\triangle})$ (the HKR isomorphism), and bulk-boundary deformation pairing.

math-ph↗
arXiv2013

Quantum Corrections to Bekenstein-Hawking Black Hole Entropy and Gravity Partition Functions

Algebraic aspects of the computation of partition functions for quantum gravity and black holes in $AdS_3$ are discussed. We compute the sub-leading quantum corrections to the Bekenstein-Hawking entropy. It is shown that the quantum corrections to the classical result can be included systematically by making use of the comparison with conformal field theory partition functions, via the $AdS_3/CFT_2$ correspondence. This leads to a better understanding of the role of modular and spectral functions, from the point of view of the representation theory of infinite-dimensional Lie algebras. Besides, the sum of known quantum contributions to the partition function can be presented in a closed form, involving the Patterson-Selberg spectral function. These contributions can be reproduced in a holomorphically factorized theory whose partition functions are associated with the formal characters of the Virasoro modules. We propose a spectral function formulation for quantum corrections to the elliptic genus from supergravity states.

hep-th↗
arXiv2013

On Partition Functions of Hyperbolic Three-Geometry and Associated Hilbert Schemes

Highest-weight representations of infinite dimensional Lie algebras and Hilbert schemes of points are considered, together with the applications of these concepts to partition functions, which are most useful in physics. Partition functions (elliptic genera) are conveniently transformed into product expressions, which may inherit the homology properties of appropriate (poly)graded Lie algebras. Specifically, the role of (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of partition functions and associated $q$-series are discussed. Examples of these connection in quantum field theory are considered (in particular, within the AdS/CFT correspondence), as the AdS$_{3}$ case where one has Ruelle/Selberg spectral functions, whereas on the CFT side, partition functions and modular forms arise. These objects are here shown to have a common background, expressible in terms of Euler-Poincaré and Macdonald identities, which, in turn, describe homological aspects of (finite or infinite) Lie algebra representations. Finally, some other applications of modular forms and spectral functions (mainly related with the congruence subgroup of $SL(2, {\mathbb Z})$) to partition functions, Hilbert schemes of points, and symmetric products are investigated by means of homological and K-theory methods.

math-ph↗
arXiv2011

BRST-Invariant Deformations of Geometric Structures in Sigma Models

We study a Lie algebra of formal vector fields $W_n$ with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. We show that equivalent classes of deformations are describing by a Hochschild cohomology theory of the DG-algebra ${\mathfrak A} = (A, Q)$, $Q =\bar{\partial}+\partial_{\rm deform}$, which is defined to be the cohomology of $(-1)^n Q +d_{\rm Hoch}$. Here $\bar{\partial}$ is the initial non-deformed BRST operator while $\partial_{\rm deform}$ is the deformed part whose algebra is a Lie algebra of linear vector fields ${\rm gl}_n$. We show that equivalent classes of deformations are described by a Hochschild cohomology of ${\mathfrak A}$, an important geometric invariant of the (anti)holomorphic structure on $X$. We discuss the identification of the harmonic structure $(HT^\bullet(X); HΩ_\bullet(X))$ of affine space $X$ and the group ${\rm Ext}_{X^{2}}^n({\cO}_{\triangle}, {\cO}_{\triangle})$ (the HKR isomorphism), and bulk-boundary deformation pairing.

hep-th↗
arXiv2011

Partition Functions for Quantum Gravity, Black Holes, Elliptic Genera and Lie Algebra Homologies

There is a remarkable connection between quantum generating functions of field theory and formal power series associated with dimensions of chains and homologies of suitable Lie algebras. We discuss the homological aspects of this connection with its applications to partition functions of the minimal three-dimensional gravities in the space-time asymptotic to AdS3, which also describe the three-dimensional Euclidean black holes, the pure N = 1 supergravity, and a sigma model on N-fold generalized symmetric products. We also consider in the same context elliptic genera of some supersymmetric sigma models. These examples can be considered as a straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of SL(2, Z)) to partition functions represented by means of formal power series that encode Lie algebra properties.

hep-th↗
arXiv2011

AdS Solutions in Gauge Supergravities and the Global Anomaly for the Product of Complex Two-Cycles

Cohomological methods are applied for the special set of solutions corresponding to rotating branes in arbitrary dimensions, AdS black holes (which can be embedded in ten or eleven dimensions), and gauge supergravities. A new class of solutions is proposed, the Hilbert modular varieties, which consist of the $2n$-fold product of the two-spaces ${\bf H}^n/Γ$ (where ${\bf H}^n$ denotes the product of $n$ upper half-planes, $H^2$, equipped with the co-compact action of $Γ\subset SL(2, {\mathbb R})^n$) and $({\bf H}^n)^*/Γ$ (where $(H^2)^* = H^2\cup \{{\rm cusp\,\, of}\,\,Γ\}$ and $Γ$ is a congruence subgroup of $SL(2, {\mathbb R})^n$). The cohomology groups of the Hilbert variety, which inherit a Hodge structure (in the sense of Deligne), are analyzed, as well as bifiltered sequences, weight and Hodge filtrations, and it is argued that the torsion part of the cuspidal cohomology is involved in the global anomaly condition. Indeed, in presence of the cuspidal part, all cohomology classes can be mapped to the boundary of the space and the cuspidal contribution can be involved in the global anomaly condition.

math-ph↗
arXiv2009

Global anomaly and a family of structures on fold product of complex two-cycles

We propose a new set of IIB type and eleven-dimensional supergravity solutions which consists of the $n$-fold product of two-spaces ${\bf H}^n/Γ$ (where ${\bf H}^n$ denotes the product of $n$ upper half-planes $H^2$ equipped with the co-compact action of $Γ\subset SL(2, {\mathbb R})^n$) and $({\bf H}^n)^*/Γ$ (where $(H^2)^* = H^2\cup \{{\rm cusp of} Γ\}$ and $Γ$ is a congruence subgroup of $SL(2, {\mathbb R})^n$). The Freed-Witten global anomaly condition have been analyzed. We argue that the torsion part of the cuspidal cohomology involves in the global anomaly condition. Infinitisimal deformations of generalized complex (and Kähler) structures also has been analyzed and stability theorem in the case of a discrete subgroup of $SL(2, {\mathbb R})^n$ with the compact quotient ${\bf H}^n/Γ$ was verified.

hep-th↗