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- Holomorphic Floer quantization, wall-crossing structures and resurgence

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1. The notion of wall-crossing structure (as defined by Maxim Kontsevich and myself in arXiv: 1303.3253)

provides the universal framework for description of different types of wall-crossing formulas (e.g. Cecotti-Vafa in 2d or KSWCF in 4d). It also gives

a language and tools for proving algebraicity and analyticity of arising generating series (e.g. for BPS invariants).

2. Holomorphic Floer theory is the Floer theory of a pair of complex Lagrangian subvarieties of a complex symplectic manifold (maybe infinite-dimensional).

This geometry underlies several important topics, both in mathematics and physics. Those include questions about analytic continuation of exponential integrals

(e.g. Feynman integrals),

deformation quantization of holomorphic symplectic manifolds (and related Riemann-Hilbert correspondences), Geometric Langlands correspondence, etc.

3. It was known for a long time that many a priori divergent series (like e.g. formal WKB series for solutions of equations with a small parameter) become analytic or meromorphic functions after taking their Laplace transform

(Borel resummation). This property was called the resurgence property of the divergent series. The relation of the resurgence phenomenon to simplest wall-crossing formulas was realized in the early 90's in the work of French mathematicians (Ecale, Voros, Pham , Malgrange and others).

I plan to discuss how the combination of 1 and 2 can be applied to 3 in a very general situation.

Main idea goes back to our theory of Donaldson-Thomas invariants (arXiv: 0811.2435).

Namely, analyticity of the formal series follows from existence of a global analytic object which is glued from the local ones by means of the formal series.

This analytification of an a priori formal variety is a by-product of the growth estimates on the data of the underlying wall-crossing structure.

In case of the Holomorphic Floer theory (which underlies e.g. the Stokes phenomenon for the WKB solutions of PDE or difference equations) one needs an estimate on the number of pseudo-holomorphic discs

with the boundary on the union of our Lagrangian subvarieties. In practice it often appears as an estimate on the number of gradient lines between two critical points

of the action or potential.

©2012 Institut Périmètre de Physique Théorique