Exact Chern-Simons / Topological String duality

Abstract

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.

Partition function of Chern–Simons theory as renormalized q-dimension

“…This simultaneous regularization and renormalization by deformation of integration contour is in general agreement with regularization of plethystic sums, suggested in e.g. ([7], 5.10), since both lead to answers in terms of multiple Barnes' gamma functions and multiple sine functions, see for Chern-Simons theory [2,8,9,3], for supersymmetric Yang-Mills theory e.g. [7], [10].…”

“…which is O(x 2 ) at x → 0, so integrals below converge. Cancellation in the last lines of (7) and correspondingly representation (8) are first observed in [3].…”

“…for Chern-Simons theory κ becomes an integer coupling k (level). As shown in [3], expression (1) makes sense for values of universal parameters out of Vogel's table, also. Particularly, it contains non-perturbative corrections to the Gopakumar-Vafa partition function of dual topological string.…”

“…We postpone discussion of this observation to Conclusion, and below in Section 1 first consider partition function of Chern-Simons theory on three-dimensional sphere [1]. We present it in the special (universal) form [2,3], then in Section 2 calculate the q-dimension [4] of highest weight representation with highest weight kΛ 0 , k ∈ Z + of affine Kac-Moody algebras, and in Section 3 show that these expressions have coinciding integrands and coincide exactly after natural renormalization of latter in q → 1 limit. In Conclusion we also discuss possible applications.…”

X 2 series of universal quantum dimensions

“…Another application of universal formulae is the derivation of non-perturbative corrections to Gopakumar-Vafa partition function [17,16] by gauge/string duality from the universal partition function of Chern-Simons theory. This shows the relevance of the "analytical continuation" of the universal formulae from the points of Vogel's table (1) to the entire Vogel's plane.…”