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, 2-3.30pm
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.

9-10am

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.

10-11am

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-12.30pm

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, 1.45-2.45pm

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

Abstract: TBA

10. Juni 2025

TBA

24. Juni 2025

Niklas Martensen (HU Berlin)

Title: TBA

Abstract: TBA

1. Juli 2025

Gerard Bargalló i Gómez (HU Berlin)

Title: TBA

Abstract: TBA

8. Juli 2025

TBA

15. Juli 2025

Rob Klabbers (HU Berlin)

Title: TBA

Abstract: TBA