Mathematics and Mechanics Faculty, St.Petersburg State University



Saint Petersburg Colloquium on Topological Recursion

Chebyshev Laboratory, SPbU

October 19-20, 2015
Topological recursion (also known as Eynard-Orantin, or Chekhov-Eynard-Orantin recursion) is a universal recursive formula that is applicable to a variety of enumerative problems in various areas of mathematics and mathematical physics. It is not just ubiquitous, but is also effective in the sense that it allows for an actual computation. 
Topological recursion was initially derived as a recursion for the coefficients of asymptotic expansions in random matrix theory. However, it soon appeared to be useful for counting combinatorial objects of different nature (in particular, for computing Hurwitz numbers, Gromov-Witten invariants, volumes of moduli spaces, Jones polynomials and many more). It is also closely tied with quantum and cohomological field theories. 
The aim of the colloquium is to bring together the specialist, from the renowned researches to postdocs and graduate students, to discuss the modern trends and the latest results in the actively developing field of topological recursion.
Program of Saint Petersburg Colloquium on Topological Recursion

10:30—11:30 G. Borot, The topological content of rational 2d CFTs.

11:30—12:00 Coffee break

12:00—13:00 M. Kazarian, Polynomiality in the enumeration of maps and hypermaps.

13:00—15:00 Lunch

15:00—16:00 P. Norbury, Superpotential and topological recursion I.

16:00—16:30 Coffee break

16:30—17:30 P. Dunin-Barkowski, Superpotential and topological recursion II.

17:30—18:30 Refreshment

19:00—21:00 Dinner

October 20, 2015

10:30—11:30 L. Chekhov, Hypergeometric Hurwitz numbers and topological recursion

11:30—12:00 Coffee break

12:00—13:00 M. Karev, Monotone orbifold Hurwitz numbers.

13:00—15:00 Lunch

G. Borot, The topological content of rational 2d CFTsI will introduce modular functors, which are a way to axiomatise 2d rational CFTs.Out of a given modular functor, we construct a collection of bundles over the Deligne-Mumford moduli space, and show that, when the modular functor is unitary, their Chern character defines a semi-simple cohomological field theory (in the example of Wess-Zumino-Witten theory, this coincides with the bundle of conformal blocks, and the CohFT statement was recently proved by Marian et al.). It follows from earlier work of Dunin-Barkowski et al. that the correlation functions of this CohFT satisfies topological recursion, for a local spectral curve we explicitly describe. As a particular case, we show that Verlinde formula is a consequence of the topological recursion. This is a joint work with J. E. Andersen and N. Orantin.

L. Chekhov, Hypergeometric Hurwitz numbers and topological recursion. I give a review on applications of topological recursion to describing enumeration problems pertaining to Grothendieck's dessins d'enfants and their generalisations to hypergeometric Hurwitz numbers --numbers of coverings of a sphere by a Riemann surface of genus $g$ with fixed number $n$ of branching points. For any $n$ the corresponding generating function is a tau function of the KP hierarchy and we can find its spectral curve encoding all terms of expansion in $g$ in terms of the topological recursion (based on joint papers with J. Ambjorn, NBI,Copenhagen).

P. Dunin-Barkowski, Superpotential and topological recursion II. This talk is a continuation of Paul Norbury's talk "Superpotential and topological recursion I"; it is based on the same joint paper with Norbury, Orantin, Popolitov and Shadrin. In this talk I will outline the proof of the statement that Dubrovin's superpotential under certain conditions provides the spectral curve for the corresponding cohomological field theory, and I will illustrate it with some examples including the case of $A_n$-singularities and the case of hypermaps. 

M. Karev, Monotone orbifold Hurwitz numbersThe talk is based on the joint work with N. Do (Monash University).
Both monotone Hurwitz numbers and orbifold Hurwitz numbers have been studied in the literature and are known to be governed by the topological recursion. In my talk I will introduce the notion of monotone orbifold Hurwitz numbers which are simultaneously variations of the orbifold case and generalisations of the monotone case. I will explain the derivation of the corresponding cut-and-join recursion, and state an explicit conjecture relating monotone orbifold Hurwitz numbers to the topological recursion.

M. Kazarian, Polynomiality in the enumeration of maps and hypermaps. We show that many generating series appearing in the problem of enumeration of maps and hypermaps are algebraic meaning that with a suitable change of independent variable they are govern by finitely many coefficients of an explicitly computed polynomials. As a consequence we provide closed efficient formulas for these functions. This result is a formal consequence of the relations of topological recursion obtained earlier. This is a joint work with P. Zograf. 

P. Norbury, Superpotential and topological recursion I. One construction of Frobenius manifolds, originating in Saito's work on singularities, uses a Landau-Ginzburg superpotential.  This is a family of pairs consisting of a curve equipped with a meromorphic function whose critical values are canonical coordinates on the Frobenius manifold.  The metric on the Frobenius manifold arises from an admissible differential on the curve.  Conversely, Dubrovin associated to any semi-simple conformal Frobenius manifold a Landau-Ginzburg superpotential that retrieves the Frobenius manifold structure.  In joint work with Dunin-Barkowski, Orantin, Popolitov and Shadrin we prove that topological recursion applied to the superpotential equipped with an appropriate bidifferential stores correlators of the CohFT associated to the Frobenius manifold.  I will introduce the four main ingredients in this theorem - Frobenius manifold, cohomological field theory, Landau-Ginzburg superpotential and topological recursion.  This is the first of two talks.  Part II will be given by P. Dunin-Barkowski.



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