R.L. Mkrtchyan

Nonperturbative universal Chern-Simons theory

A bstractClosed 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 S3. This proves the universality of that partition function. For classical groups it manifestly satisfy N → −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.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. This proves the universality of that partition function. For classical groups it manifestly satisfy N → −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.

TRACE ANOMALIES AND COCYCLES OF WEYL AND DIFFEOMORPHISMS GROUPS

The general structure of trace anomaly, suggested recently by Deser and Schwimmer 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. In particular, 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 Liouville action. This form also establishes the connection of the cocycles with conformal-invariant operators of order d and zero weight. We also give the general rule for transformation of that cocycles into the cocycles of diffeomorphisms group.

On universal knot polynomials

A bstractWe present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel’s plane, respectively and give their exceptional group’s counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel’s plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.

Universality in Chern-Simons theory

A bstractWe show that the perturbative part of the partition function in the ChernSimons theory on a 3-sphere as well as the central charge and expectation value of the unknotted Wilson loop in the adjoint representation can be expressed in terms of the universal Vogel’s parameters α, β, γ. The derivation is based on certain generalisations of the Freudenthal-de Vries strange formula.

Trace anomalies and cocycles of the Weyl group

Abstract The response of the partition function of six-dimensional classically Weyl-invariant theories on a finite Weyl transformation is calculated (this is a cocycle of the Weyl group, connected with the trace anomaly), and it is shown that this cocycle can be chosen quadratic over the Weyl field, in exact analogy with the two-dimensional Liouville action: S = ∫ d 6 x g (σΔ 6 σ+σ × Anomaly ) where Δ 6 is the zero weight conformal-invariant operator of order six, derived in the present letter. The structure of the trace anomaly as a direct consequence of the Wess-Zumino consistensy condition is discussed.

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 Vogels parameters

On duality and negative dimensions in the theory of Lie groups and symmetric spaces

\alpha, \beta, \gamma

Radial Reduction and Cubic Interaction for Higher Spins in (A)dS space

and give explicit formulae for the generating functions of these eigenvalues.

Area-preserving structure of 2d-gravity

We give one more interpretation of the symbolic formulas U( − N) = U(N) and Sp( − 2N) = SO(2N) and show that they can be extended to the classical symmetric spaces using Macdonald duality for Jack and Jacobi symmetric functions.

Exact Chern-Simons / Topological String duality

We present a new version of the radial reduction formalism to obtain a cubic interaction of higher spin gauge fields in AdSd+1 space from the corresponding cubic interaction in a flat (d+2)-dimensional background. We modify the radial reduction procedure proposed previously by T. Biswas and W. Siegel in 2002 [54] and applied to the free higher spin Lagrangian by K. Hallowell and A. Waldron in 2005 [55]. This modified radial reduction scheme is applied to interacting massless higher spin fields in Fronsdalʼs formulation, and all results are expressed in a direct AdSd+1 invariant way with AdS covariant derivatives. We present a consistent algorithm and define new procedure to obtain all corrections proportional to powers of the cosmological constant, and apply these to the main term of the cubic self-interaction.