Resurgent transseries and the holomorphic anomaly



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PIRSA Number: 
13100123

Abstract

Topological string theory is restricted enough to be solved
completely in the perturbative sector, yet it is able to compute
amplitudes in physical string theory and it also enjoys large N
dualities. These gauge theory duals, sometimes in the form of matrix
models, can be solved past perturbation theory by plugging transseries
ansätze into the so called string equation. Based on the mathematics of
resurgence, developed in the 80's by J. Ecalle, this approach has been
recently applied with tremendous success to matrix models and their
double scaling limits (Painlevé I, etc). A
natural question is if something similar can be done directly in the
topological closed string sector. In this seminar I will show how the
holomorphic anomaly equations of BCOV provide the starting point to
derive a master equation which can be solved with a transseries ansatz. I
will review the perturbative sector of the solutions, its structure,
and how it generalizes for higher instanton nonperturbative sectors.
Resurgence, in the guise of large order behavior of the perturbative
sector, will be used to derive the holomorphicity of the instanton
actions that control the asymptotics of the perturbative sector, and
also to fix the holomorphic ambiguities in some cases. The example of
local CP^2 will be used to illustrate these results.
This work is based on 1308.1695 and on-going research in collaboration with J.D. Edelstein, R. Schiappa and M. Vonk.