arXiv subjects

A. Gerasimov

Publications and source records attributed to A. Gerasimov.

23 records · Page 2Linked to original sources

arXiv2000

Harmonic Twistor Formalism and Transgression on Hyperkähler manifolds

In this paper we continue our study of the fourth order transgression on hyperähler manifolds introduced in the previous paper. We give a local construction for the fourth-order transgression of the Chern character form of an arbitrary vector bundle supplied with a self-dual connection on a four dimensional hyperkähler manifold. The construction is based on the harmonic twistor formalism. Remarkably, the resulted expression for the fourth order transgression is given in terms of the determinant of the $\bar{\partial}$-operator defined on fibers of the twistor fibration.

math.DG↗

arXiv2000

Bogolubov's Recursion and Integrability of Effective Actions

The Hopf algebra of Feynman diagrams, analyzed by A.Connes and D.Kreimer, is considered from the perspective of the theory of effective actions and generalized $τ$-functions, which describes the action of diffeomorphism and shift groups in the moduli space of coupling constants. These considerations provide additional evidence of the hidden group (integrable) structure behind the standard formalism of quantum field theory.

hep-th↗

arXiv1996

Liouville Type Models in Group Theory Framework. I. Finite-Dimensional Algebras

In the series of papers we represent the ``Whittaker'' wave functional of $d+1$-dimensional Liouville model as a correlator in $d+0$-dimensional theory of the sine-Gordon type (for $d=0$ and $1$). Asypmtotics 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 2- and 3-point correlators in $1+1$ Liouville model, where emergence of peculiar double-periodicity is observed. The Whittaker wave functions of $d+1$-dimensional non-affine ("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 is originally a Gaussian integral over interior of a $d+1$-dimensional disk with given boundary conditions, as a (non-local) quadratic integral over the $d$-dimensional boundary itself. In the present paper we mostly concentrate on the finite-dimensional case. The results for finite-dimensional "Iwasawa" Whittaker functions were known, and we present their survey. We also construct new "Gauss" Whittaker functions.

hep-th↗

arXiv1994

Generalized Hirota Equations and Representation Theory. I. The case of $SL(2)$ and $SL_q(2)$"

This paper begins investigation of the concept of ``generalized $τ$-function'', defined as a generating function of all the matrix elements of a group element $g \in G$ in a given highest-weight representation of a universal enveloping algebra ${\cal G}$. In the generic situation, the time-variables correspond to the elements of maximal nilpotent subalgebras rather than Cartanian elements. Moreover, in the case of quantum groups such $τ$-``functions'' are not $c$-numbers but take their values in non-commutative algebras (of functions on the quantum group $G$). Despite all these differences from the particular case of conventional $τ$-functions of integrable (KP and Toda lattice) hierarchies (which arise when $G$ is a Kac-Moody (1-loop) algebra of level $k=1$), these generic $τ$-functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The most important applications of the formalism should be to $k>1$ Kac-Moody and multi-loop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (0-loop) algebra $SL(2)$ and its quantum counterpart $SL_q(2)$, as well as for the system of fundamental representations of $SL(n)$.

hep-th↗

arXiv1993

Localization in GWZW and Verlinde formula

Gauged Wess-Zumino-Witten theory for compact groups is considered. It is shown that this theory has fermionic BRST-like symmetry and may be exactly solved using localization approach. As an example we calculate functional integral for the case of SU(2) group on the arbitrary Riemann surface. The answer is the particular case of Verlinde formula for the number of conformal blocks.

hep-th↗