Abstract
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.
Highlights
T r means an arbitrarily normalized invariant bilinear form in simple Lie algebra of compact gauge group
We show that partition function of Chern-Simons theory on three-sphere with classical and exceptional groups can be represented as ratio of respectively triple and double sine functions
Triple sine function has the product representation, immediately giving GV representation, plus terms non-perturbative w.r.t. the string coupling constant, see section 3. We extend this approach to SO/Sp groups in sections 4, 5, 6 and obtain similar representation, which includes double sine function
Summary
In calculation of partition function according to (1.2) the main element is universal character We consider it on lines γ = k(α + β) with natural k. Since for k = 1, 2 X(x/t, k) is divisible on (−1 + ex), the double gamma functions will appear from the X(x/δ) terms, only. The gamma functions corresponding to typical term of X above are (one minus is coming from denominators, another one from connection between free energy and partition function): Γ2(. Terms X and Y are not convergent, under integral sign, at upper limit separately We add to this the l.h.s. of identity below:. All four terms are convergent on upper limit, and we can transform them separately into gamma functions. All uniple gamma functions cancel, only double sine functions remain, up to explicit elementary functions of z, δ, and numerical multipliers.
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