Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Mathematical physics seminar

Date: Tuesdays 11:15-12:45 Uhr

Venue: 1.023 (BMS Room, Haus 1, ground floor), Rudower Chaussee 25, Adlershof, 12489 Berlin

Organiser: Gaëtan Borot

 

Sommersemester 2025

22. April 2025

Vincent Bouchard (University of Alberta)

W-constraints for the ancestor potential of (r,s)-theta classes
Witten’s conjecture, which was proved by Kontsevich, states that the generating series for intersection numbers on the moduli space of curves is a tau-function for the KdV integrable hierarchy. It can be reformulated as the statement that the ancestor potential of the trivial cohomological field theory is the unique solution to a system of differential constraints that form a representation of the Virasoro algebra. In this talk I will present a new generalization of this celebrated result. The object of study becomes the ancestor potential of an interesting set of cohomology classes on the moduli space of curves, called (r,s) theta classes, which form a non-semisimple cohomological field theory. (Here, r is a positive integer and s is a positive integer between 1 and r-1.) The (r,s) theta classes are constructed as the top Chern classes of the Chiodo classes, and can be understood as a higher generalization of the Norbury classes. We show that the ancestor potential is the solution of a system of differential constraints that form a representation of a W-algebra. However, the differential constraints form an Airy structure (and thus uniquely fix the ancestor potential) only in the cases s=1 and s=r-1. An equivalent way of stating this result is that we determine loop equations satisfied by the correlators for all (r,s), but only when s=1 or s=r-1 are the loop equations uniquely solved by (shifted) topological recursion.
 
This talk is based on work in progress with N. K. Chidambaram, A. Giacchetto and S. Shadrin and previous work by many people such as https://arxiv.org/abs/2412.09120, https://arxiv.org/abs/2408.02608, https://arxiv.org/abs/2205.15621, https://arxiv.org/abs/1812.08738, to name a few.
 

29. April 2025
Thomas Creutzig (Erlangen Universität)

Verlinde's formula in logarithmic conformal field theory 

Verlinde's formula for rational two-dimensional conformal field theory says that the fusion rules can be computed from the modular transformations of characters. Thanks to Yi-Zhi Huang this is a theorem for rational vertex operator algebras. I will give a historical introduction to the subject and then introduce a setting in which this statement also holds for logarithmic conformal field theories.

6. Mai 2025 - No seminar

SPECIAL DATE: Monday 12. Mai 2025, 2-3.30 p.m.
Leonid Chekhov (Michigan State University)

Symplectic groupoid and cluster algebras

The symplectic groupoid is a set of pairs (B,A) with A unipotent upper-triangular matrices and B in GLn being such that the matrix A~ = BABT is itself unipotent upper triangular. It turned out recently that the problem of description of such pairs can be explicitly solved in terms of Fock--Goncharov--Shen cluster variables; moreover, for B satisfying the standard semiclassical Lie--Poisson algebra, the matrices B, A, and A~ satisfy the closed Poisson algebra relations expressible in the r-matrix form. Since works of J.Nelson, T.Regge and B.Dubrovin, it was known that  entries of A can be identified with geodesic functions on Riemann surfaces with holes. In our approach, we are able to construct a complete set of geodesic functions for a closed Riemann surface. We have a complete description for genus two; I'm also about to discuss moduli spaces of higher genera. Based on my joint papers with MIsha Shapiro and our students.

TRI-seminar 16. Mai 2025

followed by the PhD defense of Giacomo Umer at 3pm.

9-10 a.m.

Paolo Rossi (Padova) - Moduli spaces of curves and the classification of integrable systems

I will present several results and conjectures on the classification of different classes of integrable systems of evolutionary PDEs, up to the appropriate transformation groups. These include Hamiltonian systems, tau symmetric systems and systems of conservation laws. I will then explain in what sense we expect that integrable systems arising from intersection theory on the moduli space of stable curves are universal objects with respect to these classifications. In the rank one case I will present strong evidence in support of these claims. This is joint work with A. Buryak.

10-11 a.m.

Danilo Lewanski (Trieste) - On the DR/DZ equivalence 

There are two main recipes to associate to a Cohomological Field Theory (CohFT) an integrable hierarchy of hamiltonian PDEs: the first one was introduced by Dubrovin and Zhang (DZ, 2001), the second by Buryak (DR, 2015). It is interesting to notice that the latter relies on the geometric properties of the Double Ramification cycle — hence the name DR — to work. As soon as the second recipe was introduced, it was conjectured that the two had to be equivalent in some sense, and it was checked in a few examples. In the forthcoming years several papers by Buryak, Dubrovin, Guerè, Rossi and others followed, checking more examples of CohFTs, making the conjecture more precise, proving the conjecture in low genera, and eventually turning the statement of the conjecture in a purely intersection theoretic statement on the moduli spaces of stable curves. Lately, the conjecture was proved in its intersection theoretic form, employing virtual localisation techniques. (j.w.w. Blot, Rossi, Shadrin).

11.30am-12.30 p.m.

Reinier Kramer (Milano Bicocca) - Leaky Hurwitz numbers and topological recursion

Leaky Hurwitz numbers were introduced by Cavalieri-Markwig-Ranganathan by extending the branching morphism from the logarithmic double ramification cycle to its pluricanonical counterpart. These numbers also have a natural interpretation in terms of tropical geometry and yield (non-hypergeometric) KP tau functions.
I will explain how to think about these numbers, and how we can extend the recent works of Alexandrov-Bychkov-Dunin-Barkowski-Kazarian-Shadrin to prove (at least blobbed) topological recursion. Along the way, I will interpret the cut-and-join operator as a hamiltonian whose flow generates the spectral curve. This is joint work in progress with M. A. Hahn.

SPECIAL DATE: Monday 19. Mai 2025, 1.45-2.45 p.m.

Silvia Ragni (Università di Padova)

Teleman’s reconstruction theorem: adaptation for semi-simple F-CohFTs

In his paper "The structure of 2D semi-simple field theories", Teleman showed that the Givental group acts transitively on semi-simple CohFTs, so that any semi-simple flat unit CohFT can be reconstructed from its cohomological 0-degree part by means of its associated Frobenius manifold. 
In "Semisimple flat F-manifolds in higher genus" Arsie, Buryak, Lorenzoni and Rossi found an adaptation of the Givental group and they showed that it can be used to produce a F-CohFT FΩ' from the cohomological 0-degree part of an original flat unit F-CohFT FΩ by means of the FΩ-associated Flat F-Manifold. 
I am going to adapt Teleman’s proof to show that FΩ and FΩ' actually coincide after taking the restriction to a partial compactification of the moduli space of Riemann Surfaces, namely to the union of those strata whose dual graphs are stable trees.

27. Mai 2025

Jan Pulmann (Charles University)

Batalin-Vilkovisky formalism, half-densities and Lagrangian relations

Lagrangian relations model maps and more general correspondences between physical systems. In Batalin-Vilkovisky formalism, it is natural to generalize Lagrangian relations to distributional half-densities, as advocated by Ševera. We give a rigorous definition of linear distributional half-densities and describe their composition, thus constructing a linear version of a quantum odd symplectic category. As an application, we describe the computation of the BV effective action as a composition in this category. Based on [arXiv:2401.06110], joint with B. Jurčo and M. Zika.

3. Juni 2025

Davide Scazzuso (HU Berlin)

Remodelling the Gauged Linear Sigma Model

I will discuss the topological gauged linear sigma model (GLSM) on a disc as a way to encode B-brane physics in the equivariant open-closed A-model topological string theory on toric Calabi-Yau threefolds. I will introduce the disc GLSM partition function via a localization argument and explain its geometric nature as the quantum volume of the target. I will then show how to construct cigar and cylinder partition functions and prove a statement of equivariant mirror symmetry between A-model B-branes and Lagrangian A-branes in the  B-model, given by cycles on the equivariant mirror curve. Finally, I will state the Remodelling theorem which provides a definition of GLSM partition functions on arbitrary worldsheets through spectral curve topological recursion and mirror symmetry. Based on upcoming work. 

10. Juni 2025

Kunal Gupta (Uppsala University Sweden)

From knots to quivers via exponential networks

I will present our proposal for a mirror derivation of the quiver description of open topological strings known as the knots-quivers correspondence, based on enumerative invariants of augmentation curves encoded by exponential networks. Quivers are obtained by studying M2 branes wrapping holomorphic disks with Lagrangian boundary conditions on an M5 brane, through their identification with a distinguished sector of BPS kinky vortices in the 3d-3d dual QFT. Our proposal suggests that holomorphic disks with Lagrangian boundary conditions are mirror to calibrated 1-chains on the associated augmentation curve, whose intersections encode the linking of boundaries. This is based on works arXiv:2407.08445, arXiv:2412.14901 with Pietro Longhi.

17. Juni 2025 - No seminar

24. Juni 2025

Niklas Martensen (HU Berlin)

Geometric quantization via real polarizations

A quantization of a symplectic manifold is a process that transforms classical observables into operators that act on a Hilbert space and satisfy certain commutation relations. However, there is no general theory that allows for such quantization. Geometric quantization attempts to solve this problem for a certain class of symplectic manifolds by a two-step process: First, a pre-quantization that leads to a Hilbert space that is too big, which is then reduced by choosing a (in this talk real) polarization. In this talk, I will try to give a gentle introduction to geometric quantization by focusing on many examples to see what structures are necessary to make the process work. If time permits, I will also talk about possible applications of topological recursion.

SPECIAL DATE: Thursday 3. Juli 2025, 9 a.m.

Gerard Bargalló i Gómez (HU Berlin)

Dynamics and disc filling methods

First, we give a gentle introduction to contact geometry through the exploration of dynamical questions leading to the Weinstein conjecture (now Taubes' theorem). The true goal of the talk is to explain how holomorphic curve theory can help study dynamical questions, in this case the existence of closed orbits. After an introduction to basic features of holomorphic curves (closed, with boundary and with punctures) we use them to explain Hofer's proof of the Weinstein conjecture for 3-manifolds that are overtwisted or have $\pi_2$.

8. Juli 2025

SPECIAL TIME: 2 p.m.

Subrabalan Murugesan (Heriot-Watt University)
A geometric recipe for constructing coordinates on the moduli space of flat connections
The moduli space M(C, G) of flat G-connections on a Riemann surface C is an object that appears in various places in physics, especially in the context of supersymmetric gauge theories. It was shown that one could construct a particularly nice set of coordinates on M(C, G) provided C is equipped with some additional structure known as a spectral network W. This construction uses a technique called W-abelianisation. W-abelianisation, and the inverse map W-nonabelianisation, define a diffeomorphism between the space M(C, G) and a (possibly) simpler space M(\Sigma, GL(1, C)) where \Sigma is a branched cover over C. In this talk, I shall give a pedagogical introduction to the abelianisation and nonabelianisation maps that were introduced in [Gaiotto-Moore-Neitzke] and later expanded upon by [Hollands-Neitzke]. This talk should be accessible to anyone comfortable with differential geometry and complex geometry.

15. Juli 2025

Rob Klabbers (HU Berlin)

How to freeze elliptic spin Ruijsenaars models

The connection between integrable quantum many body systems and spin chains is formed by `freezing', an idea that goes back to Polychronakos. I will discuss a reformulation of freezing in the language of deformation quantisation, that builds on recent work by Mikhailov and Vanhaecke. I will show that there exists an SL(2,Z) famiy of equilibria on which one can freeze, yielding an infinite family of integrable spin chains. Based on work with Jules Lamers.