• Corpus ID: 119283378

Lecture notes on topological recursion and geometry

@article{Borot2017LectureNO,
title={Lecture notes on topological recursion and geometry},
author={Gaetan Borot},
journal={arXiv: Mathematical Physics},
year={2017}
}
• G. Borot
• Published 28 May 2017
• Physics
• arXiv: Mathematical Physics
These are lecture notes for a 4h mini-course held in Toulouse, May 9-12th, at the thematic school on "Quantum topology and geometry". The goal of these lectures is to (a) explain some incarnations, in the last ten years, of the idea of topological recursion: in two dimensional quantum field theories, in cohomological field theories, in the computation of Weil-Petersson volumes of the moduli space of curves; (b) relate them more specifically to Eynard-Orantin topological recursion (revisited…

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References

SHOWING 1-10 OF 25 REFERENCES
Gromov-Witten classes, quantum cohomology, and enumerative geometry
• Mathematics
• 1994
The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic
Airy structures and symplectic geometry of topological recursion
• Mathematics
Proceedings of Symposia in Pure Mathematics
• 2018
We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental
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
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
GROMOV - WITTEN INVARIANTS AND QUANTIZATION OF QUADRATIC HAMILTONIANS
We describea formalism based on quantizationof quadratichamil- tonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about
The Virasoro conjecture for Gromov-Witten invariants
The Virasoro conjecture is a conjectured sequence of relations among the descendent Gromov-Witten invariants of a smooth projective variety in all genera; the only varieties for which it is known to
Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure
• Mathematics
• 2014
We identify the Givental formula for the ancestor formal Gromov–Witten potential with a version of the topological recursion procedure for a collection of isolated local germs of the spectral curve.
Two-dimensional Topological Quantum Field Theories and Frobenius Algebras
We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra
Tautological relations and the r-spin Witten conjecture
• Mathematics
• 2010
A geometric interpretation of Y.P. Lee’s algorithm leads to a much simpler proof of the fact that every tautological relation gives rise to a universal relation, and implies that in any semi-simple Gromov–Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautology relations, the formal and the geometric Gronov– Witten potentials coincide.