# MoSCART VI / abstracts

### 6th Workshop on Combinatorics of Moduli Spaces, Cluster Algebras, and Topological Recursion Moscow / June 4-9 / 2018

#### | OVERVIEW | PRACTICAL DETAILS | PARTICIPANTS | PROGRAM | ABSTRACTS |

[ last update: June 2, 2018 ]

### Titles and abstracts of the talks:

Simonetta Abenda / Bologna Univ. //
KP theory, planar bipartite networks in the disk and rational degenerations of M-curves

• We associate real algebraic-geometric data à la Krichever to any real regular multiline soliton solution of the Kadomtsev-Petviashvili II (KP) equation. These solutions correspond to a certain finite-dimensional reduction of the Sato Grassmannian and their asymptotic behavior is known to be classified in terms of the combinatorial structure of the totally non-negative part of real Grassmannians $Gr^{TNN} (k,n)$. In our construction, to any network representing a given soliton data in $Gr^{TNN} (k,n)$, we uniquely associate a rational degeneration of an M–curve of genus g and a real and regular degree g KP divisor on it. In particular, if we use the Le-network, then g is minimal and is equal to the dimension of the positroid cell to which the soliton datum belongs (joint research project with P.G. Grinevich (LITP, RAS))

Nezhla Aghaei / Bern Univ. //
Super pentagon relation and mapping class group

• Teichmueller space (TS) is a fundamental space that is important in many areas of mathematics and physics.The generalizations of this space on supermanifolds called super-Teichmueller spaces (STS). Super means that the structure sheaf is Z/2Z graded and contains odd or anti-commuting coordinates. The STS arise naturally as higher Teichmueller spaces corresponding to supergroups, which play an important role in mathematical physics. We construct a quantisation of the TS of super Riemann surfaces using coordinates associated to ideal triangulations of super Riemann surfaces. A new feature is the non-trivial dependence on the choice of a spin structure which can be encoded combinatorially in a certain refinement of the ideal triangulation. By constructing a projective unitary representation of the groupoid of changes of refined ideal triangulations we demonstrate that the dependence of the resulting quantum theory on the choice of a triangulation is inessential. Super pentagon relations is the main equation in the super-groupoid relations (arxive:1512.02617).
We also find the super generalisation of the generator of mapping class group acts on the Hilbert space of the once-puncture torus. We will show how these result will help to find CS invariant of mapping torus which may be the limit of the partition function of particular type of 3d, N=2 theory. This is based on the ongoing project with M. Pawelkiewicz and M. Yamazak

Alexander Alexandrov / IBS-CGP, Pohang & ITEP//
Weighted Hurwitz numbers and topological recursion

• The KP and 2D Toda tau-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral spectral curves are derived. The pair correlators are given a finite Christoffel-Darboux representation. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations. This is a joint work with G. Chapuy, B. Eynard, and J. Harnad

Alexei Basalaev / Skoltech //
Givental-type reconstruction at a non-semisimple point

• Out of all cohomological field theories (CohFT for brevity) the best understood are so-called semisimple CohFTs, for which one can compute Givental’s group element whose action on the tensor product of several Gromov-Witten theories of a point is exactly the CohFT given. One says then that this Givental’s group element reconstructs the CohFT. However this is only applicable to semisimple CohFTs.
This talk deals with the Givental-type reconstruction without semisimplicity. We show it for one particular example of the certain orbifold Gromov-Witten theory. To achieve the result we first solve modern mirror symmetry conjecture about CY/LG correspondence. In this sense this talk could be seen as an application of mirror symmetry results to the more classical problems

Alexander Belavin / Landau Inst., Chernogolovka //
Special geometry on Calabi-Yau moduli spaces and Q-invariant Frobenius rings

• I will talk about a new approach to computing the volumes of the compact CY threefolds which may be realized as the hypersurface in the weighted projective space given by the quasihomogenious polynomial $W(x)$. As known the volume of CY manifold defines the Kahler potential of the CY moduli space. The main idea of the approach is to use the one to one correspondence between Hodge structure of the middle Cohomology of the CY manifold and Hodge structure of the Invariant Frobenius ring related with the isolated singularity defined by the same polynomial $W(x)$.
This correspondence is realized by the Oscillatory integral presentation for the periods of the holomorphic Calabi-Yau 3-form which makes it possible to efficiently compute the periods without using the Picard-Fuchs equations. The use of the approach is demonstrated by computing the volume for the full complex structures moduli space of Fermat threefold

Mikhail Bershtein / Skoltech & HSE, Moscow) //
Deautonomization of cluster integrable systems, III

Alexei Bondal / Steklov Inst., Moscow //
Perverse sheaves on Riemann surfaces

• I will discuss a description of topological perverse sheaves on Riemann surfaces and orbifold Riemann surfaces via representations of suitable algebras. Algebras are obtained via the homotope construction from the fundamental groups of punctured Riemann surfaces. They have remarkable properties which allow to understand the perverse sheaves via ribbon graphs

Gaetan Borot / MPIM, Bonn //
Towards new Mirzakhani-McShane identities

• I will present a general recursive machinery which produces continuous functions on the moduli space of bordered Riemann surfaces, from a small amount of initial data. Under mild growth conditions on the initial data, the resulting functions are integrable with respect to the Weil-Petersson metric, and their integration produces functions of boundary lengths which satisfy the topological recursion. Conversely, any initial data for the topological recursion can be refined to initial data for this new machinery, which we call “geometric recursion” (GR). For suitable initial data, GR can produce the constant function 1 (via Mirzakhani-McShane identities), and linear statistics of the hyperbolic length spectra. The latter example can be thought as a new family of Mirzakhani-McShane identities. This is based on joint work with Andersen and Orantin

Vincent Bouchard / Univ. Alberta //
Higher Airy structures and W-algebras

• In this talk I will define the notion of higher Airy structures. I will show that twisted modules for W(A_n)-algebras provide an example of higher Airy structures, and in fact can be used to reconstruct the correlation functions obtained from the generalized topological recursion on spectral curves with higher ramification

Alexander Buryak / Univ. of Leeds //
Extended r-spin theory in all genera and the discrete KdV hierarchy

• In a recent work with E. Clader and R. Tessler we observed that the intersection numbers on the moduli space of disks, constructed by R. Pandharipande, J. Solomon and R. Tessler, are equal to the intersection numbers on the moduli space of genus 0 stable curves with a certain class, which we called the extended 2-spin class. In a joint work with P. Rossi, trying to generalize this result to higher genera, we construct a cohomological field theory type system of classes on the moduli spaces of stable curves in all genera and proved that the intersection numbers with these classes are controlled by the discrete KdV hierarchy. We also conjecture that our intersection numbers should correspond to the Hodge integrals of certain type on the moduli space of Riemann surfaces with boundary

Mattia Cafasso / Univ. Angers //
Calogero-Painlevé systems, their isomonodromic formulation and non-commutative monodromy data

• I present our recent results on the isomonodromic formulation of the so-called Calogero-Painlevé systems. We will also discuss, for the case of Painlevé II, how the associated monodromy data can be interpreted, to some extent, as a sort of quantization of the classical cubic associated to the “classical” Painlevé II equation (joint work with Marco Bertola and Vladimir Roubtsov)

Norman Do / Monach Univ., Melbourne //
Generalisations of the Harer-Zagier recursion for 1-point functions

• In their work on Euler characteristics of moduli spaces of curves, Harer and Zagier proved a recursion to enumerate gluings of a 2d-gon that result in an orientable genus g surface. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: How large is the family of problems for which these so-called 1-point recursions exist?
In joint work with Anupam Chaudhuri, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer-Zagier recursion, but our methodology also applies to the enumeration of dessins d’enfant, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs simple Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions. We conclude with a brief discussion of relations between 1-point recursions and the theory of topological recursion

Boulos El-Hilany / MPIM, Bonn //
Real Hurwitz numbers and simple rational functions

Vladimir Fock / Univ. Strasbourg & ITEP, Moscow //
A-coordinate on cluster integrable systems are special values of a tau-function

• The statement that a-coordinates on cluster integrable system has
something to do with tau-functions since they are given by infinite dimensional determinants was around for several years. We will try to give to it a precise meaning. Namely we will give a “bosonic” definition of a tau-function as a limit of generating function of flat Abelian connections on a Riemann surface and defined on generalized divisors. Then we show that the value of this function on ordinary
divisors supported at infinity are just cluster variables in the corresponding integrable system

Sergey Fomin / Univ. of Michigan //
Morsifications and mutations

• I will discuss a connection between the topology of isolated singularities of plane curves and the mutation equivalence of quivers associated with their morsifications. Joint work with Pavlo Pylyavskyy, Eugenii Shustin, and Dylan Thurston, arXiv:1711.10598

Pavlo Gavrylenko / Skoltech, & HSE, Moscow //
Deautonomization of cluster integrable systems, II

Ezra Getzler / Northwestern Univ. //
The Batalin-Vilkovisky formalism for field theory and general covariance

• AKSZ field theories are a class of field theories generalizing the Chern-Simons model. They exhibit general covariance under diffeomorphisms of spacetime. In this talk, I will recast general covariance as a Maurer-Cartan equation for a certain curved differential graded Lie algebra.
In joint work with Sean Pohorence, I have shown that the superparticle, a toy model of the (Green-Schwarz) superstring, while not an AKSZ model, does exhibit general covariance in the sense that the curved Maurer-Cartan equation mentioned above may be solved. The proof requires the use of the Thom-Whitney normalization of curved differential graded Lie algebras, introduced by Sullivan in his work on rational homotopy theory

Maxim Kazarian / Steklov Inst., HSE, and Skoltech //
ELSV-type formula for the monotone Hurwitz numbers

• Hurwitz numbers enumerate factorization of permutations into the product of a given number of transpositions. Monotone Hurwitz numbers enumerate similar factorizations satisfying certain monotonicity condition. The classical now ELSV formula expresses Hurwitz numbers as intersection numbers on moduli spaces of complex curves. We announce an analogue of this formula in the monotone case. Its derivation is based on the topological recursion for these numbers obtained earlier by Do, Dyer, and Mathews

Piotr Kucharski / Warsaw Univ. //
Physics and geometry of knots-quivers correspondence

• In this talk I will talk about the physical and geometric interpretation of knots-quivers correspondence in terms of 3d N=2 theories and holomorphic disks. I will keep it very basic, but nevertheless I encourage everyone to pay attention to Piotr Sulkowski’s presentation on Thursday as it will provide an excellent introduction. My talk is a report on ongoing work with Tobias Ekholm and Pietro Longhi

Danilo Lewanski / Amsterdam Univ. //
Virasoro contraints for simple maps and free probability

• Symplectic invariance is a known feature of topological recursion at the theoretical level. An explicit instance of such phenomenon has been recently found and proved by Borot and Garcia-Failde: two enumerative geometric problems satisfying topological recursion, whose spectral curves are related by the swap of x and y — The enumeration of usual maps and the enumeration of maps with an extra combinatorial condition, named “simple”.
The transition coefficients between their correlators is given by monotone Hurwitz numbers, another enumerative problem satisfying topological recursion. Since the Fock space operators for this problem is known, we can link the two partition functions involved in the symplectic invariance, and compute the Virasoro algebra of the simple maps from the usual maps one. This has interesting applications in the context of free probability, in particular towards the computation of higher order free cumulants and towards the introduction of the the concept of genus for such cumulants (from a joint work in progress w/ G.Borot and E.Garcia Failde)

Chiu-Chu Melissa Liu / Columbia Univ. //
Holomorphic anomaly equations and Gromov-Witten invariants

• I will describe some conjectures and theorems on (extended) holomorphic anomaly equations which are related to (open) Gromov-Witten invariants under mirror symmetry.

Andrei Marshakov / Skoltech, HSE, ITEP & Lebedev Inst., Moscow //
Deautonomization of cluster integrable systems, I

Paul Norbury / Melbourne Univ. //
A new cohomology class on the moduli space of stable curves

• We define a collection of cohomology classes on the moduli space of curves. We prove that a generating function for the intersection numbers involving these new cohomology classes is a tau function of the KdV hierarchy, analogous to the Kontsevich-Witten theorem. The cohomology classes can be used to define a new type of Gromov-Witten invariant for any target variety

Nicolas Orantin / EPF, Lausanne //
Topological recursion, isomonodromic systems and WKB

• Among other applications, topological recursion is believed to provide a way to compute the coefficients of the WKB expansion of formal solutions of some linear differential systems depending on a small parameter $\hbar$. If the characterization of such systems is a hard problem, isomonodromic systems are conjectured to provide examples of such systems. In this talk, I will report on a work in progress aiming at building such a $\hbar$-deformed system from any isomonodromic system by using the underlying Poisson structure, giving partial results in the $sl_2$ case.
Based on a joint work with Olivier Marchal

Valentin Ovsienko / Univ. Reims //
Partitions of unity in $SL(2,\mathbb Z)$, negative continued fractions and dissections of polygons

• We characterize sequences of positive integers $(a_1, a_2,\dots, a_n)$ for which the $2\times 2$ matrix given by the product of the elementary matrices $\left(\begin{array}{cc}a_j & -1 \1 & 0 \end{array}\right)$is either the identity matrix Id, its negative − Id, or square root of − Id. This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction

Alex Popolitov / Amsterdam Univ., & ITEP, Moscow //
Towards bosonic map for plane partitions

• To a line partition (Young diagram) \lambda one can associate
a monomial in time variables in a simple way: \lambda \rightarrow p_{\lambda_1} \dots p_{\lambda_n}. There are hints that this should generalize to plane partitions and I will discuss the progress in building this map

Chaiho Rim / Sogang Univ. //
Open KdV hierarchy and minimal gravity on disk

• We show that the minimal gravity of Lee-Yang series on disk is a solution to the open KdV hierarchy proposed for the intersection theory on the moduli space of Riemann surfaces with boundary

Vladimir Roubtsov / Univ. Angers //
Decorated quantum character variety

• We introduce the notion of decorated character variety to generalize the Betti moduli space. This decorated character variety is the quotient of the space of representations of the fundamental groupoid of arcs by a product of unipotent Borel sub-groups (one per each bordered cusp). We demonstrate that this representation space is endowed with a Poisson structure induced by the Fock{Rosly-type bracket and show that the quotient by unipotent Borel subgroups giving rise to the decorated character variety is a Poisson reduction. We present a construction of quantum decorated character varieties for SL(k;C) on any Riemann surface ∑g;s;n of genus g, s > 0 holes, and with n > 0 bordered cusps endowed with Borel unipotent radicals, based on elementary “quantum triangle relation” M1(1) M2(2) = M2(2) M1(1) R12(q)

Michael Shapiro / Michigan State Univ. //
Exotic cluster algebras

Vasilisa Shramchenko / Sherbrook Univ., Canada //
Counting ribbon graphs using quantum field theory

• Enumeration of rooted ribbon graphs started with the work by Tutte in the 1960’s. We call such graphs one-rooted graphs and introduce a notion of more general N-rooted ribbon graphs. This definition is motivated by the bijective correspondence we establish between the N-rooted ribbon graphs with e edges and the (e-N+1)-loop Feynman diagrams of a certain quantum field theory. This result is used to obtain explicit expressions and relations for the generating functions of N-rooted maps and for the numbers of N-rooted maps with a given number of edges using the path integral approach applied to the corresponding quantum field theory. This is a joint work with K. Gopala Krishna and Patrick Labelle

Jake Solomon / Hebrew Univ., Jerusalem //

• I will discuss the notion of a graded Riemann surface and how it gives rise to open descendent integrals at arbitrary genus. This is joint work with Ran Tessler

Piotr Sulkowski / Univ. of Warsaw //
Knots, quivers, and combinatorics

• I will present a recently found surprising relation between knot invariants and quiver representation theory, that we refer to as the knots-quivers correspondence. Consequences of this relation include the proof of one of the famous Labastida-Marino-Ooguri-Vafa integrality conjecture of BPS invariants, explicit (and unknown before) formulas for colored HOMFLY-PT polynomials for various knots, new viewpoint on knot homologies, a novel type of categorification, new dualities between quivers, solutions to certain combinatorial problems, and many others

Leon Takhtajan / (Stony Brook Univ. & Euler Inst. Math., St.Petersburg //
Symplectic forms on moduli spaces of orbifolds

• I will review construction of Goldman symplectic space on character varieties of orbifold Riemann surfaces and will discuss the analog of Kawai theorem in the orbifold case, as well as its generalization for the moduli spaces of parabolic vector bundles

Ran Tessler / Hebrew Univ., Jerusalem //
The combinatorial formula for open descendent integrals

• I will explain how to calculate, in terms of sums over Feynman diagrams, the all-genus open intersection numbers defined in Jake Solomon’s talk

Pavel Tumarkin / Durham Univ. //
Quivers with non-integer weights

• Mutations of quivers can be generalized to the case of non-integer weights of arrows. We provide a geometric model for mutation of all rank 3 quivers (here the invariant responsible for the type of the model is the Markov constant), and use this to classify all non-integer quivers of finite mutation type. This is a joint work with Anna Felikson

Alexander Veselov / Loughborough Univ. & Moscow Univ. //
Geometry and dynamics of Markov triples

Siye Wu / National Tsing Hua Univ., Taiwan //
Moduli space from a non-orientable surface and duality

• In this talk, I will recall the geometric and topological structure of the Hitchin moduli space associated to a non-orientable surface and explore its relation to dualities in two and four dimensional physics

Paul Zinn-Justin / Melbourne Univ. //
Some numerology around Schubert calculus

• We’ll discuss some intriguing aspects of my recent work in collaboration with A. Knutson on Schubert calculus on d-step flag varieties (di=4), and possible connections to cluster algebras

Peter Zograf / EIMI, St.Petersburg & Chebyshev Lab, St.Petersburg Univ. //
Enumeration of meanders and volumes of moduli spaces of quadratic differentials

• A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversely. In physics, meanders provide a model of polymer folding, and their enumeration is directly related to the entropy of the associated dynamical systems. We combine recent results on Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials in genus zero with a fact that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated to study the asymptotic enumeration of meanders. As a result, we get an explicit asymptotic formula for the number of closed meanders with fixed number of minimal arcs when the number of crossings goes to infinity (after a joint work with V.Delecroix, E.Goujard and A.Zorich)

Anton Zorich / Skoltech & Inst. Math. Jussieu //
Masur-Veech volumes, Siegel-Veech constants and intersection numbers of moduli spaces

• The cotangent space to the moduli space of complex curves of genus g with n marked points can be identified with the moduli space of pairs (C,q), where C is a complex curve of genus g, and q is a meromorphic quadratic differential on C with n simple poles and no other poles. This cotangent space comes with a natural symplectic form and the associated volume form is called the Masur-Veech volume form. We provide a formula for the volume of the level hypersurface of quadratic differentials of area 1/2. We also provide a formula of similar nature for the Siegel-Veech constant.
As a concrete application, we get a large table of exact numerical values of the volumes and Siegel-Veech constants for all small g and n extending previously known data. This data was obtained by Goujard by completely different approach designed by Eskin and Okounkov.
Both the volume and the Siegel-Veech constant are expressed as polynomials in the intersection numbers of psi-classes supported on the boundary components of the Deligne-Mumford compactification of the moduli space of curves. The formulae are derived from lattice point counting involving the Kontsevich volume polynomials that also appear in Mirzakhani’s topological recursion for the Weil-Petersson volumes of the moduli space of curves. Analogous formula for the Masur-Veech volume of the moduli space of holomorphic quadratic differentials was first obtained by Mirzakhani by different method (after a joint work with V.Delecroix, E.Goujard and P.Zograf)

Contact email of the Organizing Committee
(Leonid Chekhov)
(Sergei Lando)