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## Banach Center Publications

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## An invitation to 2D TQFT and quantization of Hitchin spectral curves

### Volume 114 / 2018

Banach Center Publications 114 (2018), 85-144 MSC: Primary: 14H15, 14N35, 81T45; Secondary: 14F10, 14J26, 33C05, 33C10, 33C15, 34M60, 53D37. DOI: 10.4064/bc114-3

#### Abstract

This article consists of two parts. In Part 1, we present a formulation of two-dimensional topological quantum field theories in terms of a functor from a category of ribbon graphs to the endofunctor category of a monoidal category. The key point is that the category of ribbon graphs produces all Frobenius objects. Necessary backgrounds from Frobenius algebras, topological quantum field theories, and cohomological field theories are reviewed. A result on Frobenius algebra twisted topological recursion is included at the end of Part 1.

In Part 2, we explain a geometric theory of quantum curves. The focus is placed on the process of quantization as a passage from families of Hitchin spectral curves to families of opers. To make the presentation simpler, we unfold the story using $SL_2(\mathbb C)$-opers and rank $2$ Higgs bundles defined on a compact Riemann surface $C$ of genus greater than $1$. In this case, quantum curves, opers, and projective structures in $C$ all become the same notion. Background materials on projective coordinate systems, Higgs bundles, opers, and non-Abelian Hodge correspondence are explained.

#### Authors

• Olivia DumitrescuDepartment of Mathematics
Central Michigan University
Mount Pleasant, MI 48859, U.S.A.
and
Simion Stoilow Institute of Mathematics
21 Calea Grivitei Street
010702 Bucharest, Romania
e-mail
• Motohico MulaseDepartment of Mathematics
University of California
Davis, CA 95616–8633, U.S.A.
and
Kavli Institute for Physics and Mathematics of the Universe
The University of Tokyo
Kashiwa, Japan
e-mail

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