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R. L. Mkrtchyan

Publications and source records attributed to R. L. Mkrtchyan.

31 records · Page 2Linked to original sources
arXiv2016

Universal volume of groups and anomaly of Vogel's symmetry

We show that integral representation of universal volume function of compact simple Lie groups gives rise to six analytic functions on $CP^2$, which transform as two triplets under group of permutations of Vogel's projective parameters. This substitutes expected invariance under permutations of universal parameters by more complicated covariance. We provide an analytical continuation of these functions and particularly calculate their change under permutations of parameters. This last relation is universal generalization, for an arbitrary simple Lie group and an arbitrary point in Vogel's plane, of the Kinkelin's reflection relation on Barnes' $G(1+N)$ function. Kinkelin's relation gives asymmetry of the $G(1+N)$ function (which is essentially the volume function for $SU(N)$ groups) under $N\leftrightarrow -N$ transformation (which is equivalent of the permutation of parameters, for $SU(N)$ groups), and coincides with universal relation on permutations at the $SU(N)$ line on Vogel's plane. These results are also applicable to universal partition function of Chern-Simons theory on three-dimensional sphere. This effect is analogous to modular covariance, instead of invariance, of partition functions of appropriate gauge theories under modular transformation of couplings.

hep-th↗
arXiv2014

On a Gopakumar-Vafa form of partition function of Chern-Simons theory on classical and exceptional lines

We show that partition function of Chern-Simons theory on three-sphere with classical and exceptional groups (actually on the whole corresponding lines in Vogel's plane) can be represented as ratio of respectively triple and double sine functions (last function is essentially a modular quantum dilogarithm). The product representation of sine functions gives Gopakumar-Vafa structure form of partition function, which in turn gives a corresponding integer invariants of manifold after geometrical transition. In this way we suggest to extend gauge/string duality to exceptional groups, although one still have to resolve few problems. In both classical and exceptional cases an additional terms, non-perturbative w.r.t. the string coupling constant, appear. The full universal partition function of Chern-Simons theory on three-sphere is shown to be the ratio of quadruple sine functions. We also briefly discuss the matrix model for exceptional line.

hep-th↗
arXiv2013

Nonperturbative universal Chern-Simons theory

Closed simple integral representation through Vogel's universal parameters is found both for perturbative and nonperturbative (which is inverse invariant group volume) parts of free energy of Chern-Simons theory on $S^3$. This proves the universality of that partition function. For classical groups it manifestly satisfy N \rightarrow -N duality, in apparent contradiction with previously used ones. For SU(N) we show that asymptotic of nonperturbative part of our partition function coincides with that of Barnes G-function, recover Chern-Simons/topological string duality in genus expansion and resolve abovementioned contradiction. We discuss few possible directions of development of these results: derivation of representation of free energy through Gopakumar-Vafa invariants, possible appearance of non-perturbative additional terms, 1/N expansion for exceptional groups, duality between string coupling constant and Kähler parameters, etc.

hep-th↗
arXiv2013

Universal Chern-Simons partition functions as quadruple Barnes' gamma-functions

We show that both perturbative and non-perturbative parts of universal partition functions of Chern-Simons theory on 3d sphere are ratios of four over four Barnes' quadruple gamma functions with arguments given by linear combinations of universal parameters. Since nonperturbative part of partition function is essentially a universal compact simple Lie group's volume, latter appears to be expressed through quadruple Barnes' functions, also. For SU(N) values of parameters recurrent relations on Barnes' functions give the proof of level-rank duality of complete partition function, thus extending that duality on non-integer level and rank. We note that integral representation of universal partition function is defined on few disjoint regions in parameters' space, corresponding to different signs of real parts of parameters, and introduce a framework for discussion of analytic continuation of partition functions(s) from these regions. Although initial integral representation is symmetric under all permutations of parameters (which corresponds particularly to $N \rightarrow -N$ duality of gauge theories with classical groups), analytic continuations are not symmetric under transposition of parameters with different signs of their real parts. For the particular case of SU(N) Chern-Simons this asymmetry appears to be the Kinkelin's functional equation (reflection relation) for Barnes' G-function.

hep-th↗
arXiv2013

A universal formula for the volume of compact Lie groups

We provide a closed formula for the volume of a simple compact Lie group in terms of the universal Vogel parameters. For the unitary groups SU_n this reduces to the integral representation of the classical Barnes G-function.

math.GR↗
arXiv2012

On the map of Vogel's plane

We search points in a Vogel plane with regular universal expression for character of adjoint representation. This gives seven patterns of singularities cancellation, each giving a certain Diophantine equation of third order on three variables. Solutions of these equations are classical series of simple Lie algebras (including an "odd symplectic" one), $D_{2,1,λ}$ superalgebra, the straight line of three-dimensional algebras, and a number of isolated solutions, including exceptional simple Lie algebras. One of these Diophantine equations, namely knm=4k+4n+2m+12 contains all simple Lie algebras, except SO(2N+1). Isolated solutions contain, beside exceptional simple Lie algebras, so called $E_{71/2}$ algebra and also two other similar (unknown) objects with positive dimensions. In addition, there are 47 isolated solutions in "unphysical semiplane" with negative dimensions. Isolated solutions mainly lie on a few straight lines in Vogel plane. All solutions give an integers in universal dimension formulae for first three symmetric powers of adjoint representation.

math-ph↗
arXiv2012

Casimir eigenvalues for universal Lie algebra

For two different natural definitions of Casimir operators for simple Lie algebras we show that their eigenvalues in the adjoint representation can be expressed polynomially in the universal Vogel's parameters $α, β, γ$ and give explicit formulae for the generating functions of these eigenvalues.

math.RT↗
arXiv2002

Little Groups and Statistics of Branes

The little groups (i.e. the subgroups of Lorentz group, leaving invariant given configurations of tensorial charges) of unitary irreps of superstring/M-theory superalgebras are considered. It is noted, that in the case of $(n-1)/n$ (maximal supersymmetric) BPS configuration in any dimensions the non-zero supercharge is neutral w.r.t. the algebra of little group, which means that all members of supermultiplet are in the same representation of that algebra and hence of (generalized with tensorial charges) Poincare algebra. This situation is similar to two-dimensional case and shows that usual spin-statistics connection statement is insufficient in the presence of branes, because different little groups can appear. We discuss the rules for definition of statistics for representations of generalized Poincare, and note that a geometric quantization method seems to be most relevant for that purpose.

hep-th↗
arXiv1995

Coaxial Ring Cyclotron as a Perspective Nuclear Power Engineering Machine

The circuit arrangement of the proposed coaxial ring cyclotron (CRC) is described, and its main advantages, such as simple injection technique, several injected beams summation option, high efficiency, are considered. The proposed proton accelerator is a perspective machine for the solution of the main problems of the present day nuclear power engineering as well as for the next-generation nuclear power plants, representing a combination of subcritical reactors and particle accelerators. The possibility of installation of CRCs into ring accelerators with an average diameter from 60 to $100m$, e.g., the Yerevan electron synchrotron, is considered.

acc-phys↗
arXiv1994

Trace Anomalies and Cocycles of Weyl and Diffeomorphisms Groups

The general structure of trace anomaly, suggested recently by Deser and Shwimmer, is argued to be the consequence of the Wess-Zumino consistency condition. The response of partition function on a finite Weyl transformation, which is connected with the cocycles of the Weyl group in $d=2k$ dimensions is considered, and explicit answers for $d=4,6$ are obtained. Particularly, it is shown, that addition of the special combination of the local counterterms leads to the simple form of that cocycle, quadratic over Weyl field $σ$, i.e. the form, similar to the two-dimensional Lioville action. This form also establishes the connection of the cocycles with conformal-invariant operators of order $d$ and zero weight. Beside that, the general rule for transformation of that cocycles into the cocycles of diffeomorphisms group is presented.

hep-th↗
arXiv1994

Algebra of potentials of the volume-preserving vector fields

The algebra of volume-preserving vector fields is considered. The potentials for that fields are introduced, and induced algebra of potentials is considered. It is shown, that this algebra fails to satisfy the Jacoby identity. Analogy with hamiltonian mechanics is developed, as well as 3-cocycle interpretation of corresponding expressions.

hep-th↗
arXiv1994

Area-preserving structure of 2d-gravity

The effective action for 2d-gravity with manifest area-preserving invariance is obtained in the flat and in the general gravitational background. The cocyclic properties of the last action are proved, and generalizations on higher dimensions are discussed.

hep-th↗