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Introduction to Topological Recursion theory and moduli spaces of curves

External Lecturer: 
Danilo Lewański
Course Type: 
PhD Course
Master Course
Academic Year: 
20 h

Topological Recursion (TR) can be thought as a universal procedure, or algorithm, to generate solutions of enumerative geometric problems related directly or indirectly to moduli spaces of curves. 
 - the input is the so-called spectral curve (think e.g. an algebraic curve with two particular meromorphic functions on it), 
 - the output is an infinite list of numbers (think e.g. Gromov-Witten invariants of some kind).
In fact the literature around TR boomed after its discovery 15 years ago, as different communities of mathematicians and of physicists were realising how enumerative problems in their field can be proved to be (or instead are supposed to be, via conjectures still open) generated by TR.
TR is now applied to random matrix models, Gromov-Witten theory, Hurwitz theory, to the enumeration of many graphs drawn on surfaces (Grothendieck dessins d’enfants, constellations, combinatorial maps, ..), WKB analysis of classical ordinary differential equations, Painlevé equations, polynomial invariants of knots, moduli spaces of Higgs bundles, BPS states, free probability, and more…

TR is intrinsically related to Frobenius manifolds introduced by Prof. Dubrovin (together with other ideas of Dubrovin, such as the Dubrovin superpotential employed in several examples to construct global spectral curves) and moduli spaces of curves.

The goal of the course is to introduce TR, especially in relation with Frobenius manifolds and moduli spaces of curves (which will also be introduced), and empower the audience in computations involving the intersection theory of the moduli spaces of curves, both by hand and by an ad-hoc Sage package developed for cohomological field theories, towards open problems and, possibly, germs of projects.

More info on TR:
TR was discovered in 2004 and promoted to a mathematical theory in 2007 by Eynard and Orantin. Random matrix models carry a fundamental object called spectral curve, and TR generates the asymptotic expansion of the correlation functions of the model simply from this spectral curve. 
TR has at least three valuable features: 
1. The algorithm only needs the spectral curve to work (the matrix model can be non-existent or extremely hard to find), 
2. The algorithm is universal: it is the same for very different enumerative problems. 
3. A cohomological description (the CohFT) is provided directly from the spectral curve, and the output numbers are its integrals: all numbers produced are intersection numbers, and this statement is constructive.

A few examples of TR enumerative problems:
In the early nineties E. Witten formulated a fundamental conjecture, shortly after proved by M. Kontsevich, establishing a quantum theory for two-dimensional gravity in terms of enumerative algebraic geometry. More precisely, he conjectured that the generating series of certain numbers satisfies an integrable hierarchy of type KdV — an infinite list of partial differential equations in infinitely many variables used in mathematical physics, that determines the numbers uniquely after fixing initial conditions. 
The simplest possible spectral curve (the parabola) produces the simplest possible CohFT: the Witten-Kontsevich case, i.e. the identity. A slightly more complicated curve (the sine) produces the volumes of moduli spaces of hyperbolic structures, computed recursively in the beautiful work of M. Mirzakhani. TR provided a new proof of the two results, using the same unifying universal formula. The cosine has been shown by Norbury, via the recent work of Witten-Stanford in 2019, to produce the super-Weil-Petersson volumes. Hurwitz theory provides a wide fan of enumerative problems (the spectral curve being in some cases a variation of the Lambert curve), which are now all proved to be generated by TR.

Indicative program:

  • Week 1: Introduction to moduli spaces of curves
  • Week 2: Introduction to Topological recursion. Examples in Hurwitz theory.
  • Week 3: Dubrovin-Frobenius manifolds, cohomological field theories and Givental formalism.
  • Week 4: Identification between Topological Recursion and Givental formalism. Examples in Hurwitz theory and ELSV formulae.
  • Week 5: Towards open problems. Computations empowerment: use of the admcycles Sage package.

References for the course:

[1] B.Eynard, “Counting Surfaces”, book (introduction to Topological Recursion)
[2] R.Cavalieri, E.Miles “A First Course in Hurwitz Theory”, book.
[3] B.Eynard, M.Mulase, B.Safnuk “the Laplace transform of the cut-and-join equation and the Bouchard-Mariño conjecture on Hurwitz numbers”, research article. (Proof of the Bouchard-Mariño conjecture)
[4] D.Lewański, “on ELSV-type formulae, Hurwitz numbers and topological recursion”, review article. (For those more interested to the moduli spaces applications of TR)
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