Modular functors, cohomological field theories, and topological recursion

@article{Andersen2018ModularFC,
title={Modular functors, cohomological field
theories, and topological recursion},
author={J{\o}rgen Ellegaard Andersen and Gaetan Borot and Nicolas Orantin},
journal={Proceedings of Symposia in Pure
Mathematics},
year={2018}
}
• Published 4 September 2015
• Mathematics
• Proceedings of Symposia in Pure Mathematics
Given a topological modular functor $\mathcal{V}$ in the sense of Walker \cite{Walker}, we construct vector bundles over $\bar{\mathcal{M}}_{g,n}$, whose Chern classes define semi-simple cohomological field theories. This construction depends on a determination of the logarithm of the eigenvalues of the Dehn twist and central element actions. We show that the intersection of the Chern class with the $\psi$-classes in $\bar{\mathcal{M}}_{g,n}$ is computed by the topological recursion of \cite…
8 Citations

Figures from this paper

The ABCD of topological recursion
• Mathematics
• 2017
Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of math-ph/0702045, seeing it as a quantization of certain quadratic Lagrangians in $T^*V$ for some vector
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles
• Mathematics
• 2017
Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees
Topological recursion, topological quantum field theory and Gromov-Witten invariants of BG
The purpose of this paper is to give a twisted version of the Eynard-Orantin topological recursion by a 2D Topological Quantum Field Theory. We define a kernel for a 2D TQFT and use an algebraic
The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition
In this series of lectures, we (re)view the "geometric method" that reconstructs, from a geometric object: the "spectral curve", an integrable system, and in particular its Tau function,
Lectures on the Topological Recursion for Higgs Bundles and Quantum Curves
• Mathematics
The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles
• 2018
The paper aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the
Primary invariants of Hurwitz Frobenius manifolds
• Mathematics
Proceedings of Symposia in Pure Mathematics
• 2018
Hurwitz spaces parameterizing covers of the Riemann sphere can be equipped with a Frobenius structure. In this review, we recall the construction of such Hurwitz Frobenius manifolds as well as the
The Verlinde formula for Higgs bundles
• Mathematics
• 2016
We propose and prove the Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. This generalizes the equivariant Verlinde
Topological recursion and geometry
This paper aims at explaining some incarnations of the idea of topological recursion: in two-dimensional quantum field theories (2d TQFTs), in cohomological field theories (CohFT), and in the compu...

References

SHOWING 1-10 OF 83 REFERENCES
A Generalized Jacobi Theta Function and Quasimodular Forms
• Mathematics
• 1995
In this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf [1, Theorem 2 and Corollary]. Let \( {\tilde
Congruence Subgroups and Generalized Frobenius-Schur Indicators
• Mathematics
• 2010
We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category $${\mathcal {C}}$$ , an equivariant indicator of an object in $${\mathcal {C}}$$ is defined
Hypergeometric $${\tau}$$τ -Functions, Hurwitz Numbers and Enumeration of Paths
• Mathematics
• 2014
A multiparametric family of 2D Toda $${\tau}$$τ -functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of
CONSTRUCTING TQFTS FROM MODULAR FUNCTORS
We prove in this paper that any 2 dimensional modular functor satisfying that S1,1≠0 induces a family of 2+1 dimensionally topological quantum field theories. We do this for two kinds of modular
Topological gauge theories and group cohomology
• Mathematics
• 1990
We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH4(BG,Z). In a
Blobbed topological recursion: properties and applications
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2016
Abstract We study the set of solutions (ωg,n ) g⩾0,n⩾1 of abstract loop equations. We prove that ω g,n is determined by its purely holomorphic part: this results in a decomposition that we call
Invariants of algebraic curves and topological expansion
• Mathematics
• 2007
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties.
The structure of 2D semi-simple field theories
I classify the cohomological 2D field theories based on a semi-simple complex Frobenius algebra A. They are controlled by a linear combination of κ-classes and by an extension datum to the
Moduli of vector bundles on curves with parabolic structures
• Mathematics
• 1980
Let H be the upper half plane and F a discrete subgroup of AutH. When H mo d F is compact, one knows that the moduli space of unitary representations of F has an algebraic interpretation (cf. [7] and
ABELIAN CONFORMAL FIELD THEORY AND DETERMINANT BUNDLES
• Mathematics
• 2003
Following [10], we study a so-called bc-ghost system of zero conformal dimension from the viewpoint of [14, 16]. We show that the ghost vacua construction results in holomorphic line bundles with