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- Resurgent transseries and the holomorphic anomaly

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PIRSA Number:

13100123

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.

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.

©2012 Perimeter Institute for Theoretical Physics