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.
Highlights
Include the usual Chern-Simons theories, but, after some extension of range of parameters, as well the refined versions thereof, as shown in [13]
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
The universal Chern-Simons partition function constitutes an integral representation of the partition functions and thereby provides an analytic continuation in the parameters, e.g., simple Lie groups are parametrized by the two-dimensional Vogel’s plane [14]
Summary
Where we defined the effective coupling constant δ := κ + t with κ the usual Chern-. Simons partition functions, performed in [11] and [13], is completely analytical and exact Besides that it simultaneously encodes the usual and refined Chern-Simons theories in an unified way, other benefits of the integral representation (2.5) are that it naturally extends Chern-Simons theory to non-integer and/or negative values of N and the refinement parameter β, to non-integer values of δ (cf., [21]), and more generally to wide ranges of complex values of parameters. Where {x∗+} denotes the set of poles of the integrand, one has to establish that one can deform the integration contour without picking up an extra contribution That this is the case follows from the multiple sine representation we will discuss below. Depending on the particular values the parameters take (e.g., integer multiplies of each other), enhancement to higher order poles may occur for some subsets of poles
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