We show that the perturbative part of the partition function in the Chern-Simons 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 $\alpha, \beta, \gamma.$ The derivation is based on certain generalisations of the Freudenthal-de Vries strange formula.Comment: Slightly revised version with minor correction

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 K\"ahler parameters, etc.Comment: 18 pages, Final journal version, references added and refine

Abstract:We invoke universal Chern-Simons theory to analytically calculate the exact free energy of the refined topological string on the resolved conifold. In the unrefined limit we reproduce non-perturbative corrections for the resolved conifold found elsewhere in the literature, thereby providing strong evidence that the Chern-Simons / topological string duality is exact, and in particular holds at arbitrary N . In the refined case, the non-perturbative corrections we find are novel and appear to be non-trivial. We show that non-perturbatively special treatment is needed for rational valued deformation parameter. Above results are also extended to refined Chern-Simons with orthogonal groups.

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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. *

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 $\alpha, \beta, \gamma$ and give explicit formulae for the generating functions of these eigenvalues.Comment: Slightly revised versio

A special embedding of the SU(4) algebra in SU(10), including both spin two and spin three symmetry generators, is constructed. A possible five dimensional action for massless spin two and three fields with cubic interaction is constructed. The connection with the previously investigated higher spin theories in AdS 5 background is discussed. Generalization to the more general case of symmetries, including spins 2, 3, . . . s, is shown. * These algebras should correspond to the representations of su(2, 2) (the latter can serve as defining representations for these algebras) found in [14] and should be discrete cases of the one-parameter family of algebras of [15]. * The identity (A.16) relates two possible expressions for the spin 2 action.