Abstract
We find that an S-duality in SL(2) Chern-Simons theory for hyperbolic 3-manifolds emerges by the Borel resummation of a semiclassical expansion around a particular flat connection associated to the hyperbolic structure. We demonstrate it numerically with two representative examples of hyperbolic 3-manifolds.
Highlights
Let us briefly summarize our main statement in this paper
We find that the perturbative expansion around a saddle point corresponding to a particular flat connection, A = Aconj defined in (2.4), is Borel summable, and its Borel resummation has the S-duality property, while the resummations around the other saddle points do not
Our analysis provides supporting evidence for the conjecture in [23,24,25] saying that only the flat connection Aconj on M contributes to the Sb3 partition function of the corresponding 3d theory T [M ] in (2.26)
Summary
A is an SL(2) gauge field on a 3-manifold M and S[A; M ] is the Chern-Simons functional. The state-integral can be interpreted as a squashed 3-sphere partition function of a 3d N = 2 gauge theory T [S3\41] associated to the knot complement upon a proper choice of integral contour. In this identification, the formal SL(2) CS level k is related to the squashing parameter b by the relation k = b−2. When p = −5, the 3-manifold (S3\41)p=−5 is called the Thurston manifold, which is known to be the second smallest hyperbolic 3-manifold with vol (S3\41)p=−5 = 0.981369 In this case, there are two saddle points (zconj±, uconj±) corresponding to the flat connection Aconj (zconj+, uconj+) = (−0.929172 + 1.90501i, −0.721568 − 1.15121i) , (zconj−, uconj−) = (1.59632 + 2.79266i, 0.721568 + 1.15121i). R-charge determined by F-maximization [44]
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