Mikhail
Gorskii
Hamburg Universität
Counting in Calabi-Yau categories
Mathematical Physics Seminar 📅
Institute
Head
Gaëtan Borot
Usual time
Tuesdays from 11:15 to 12:45
Usual venue
1.023 (BMS Room, Haus 1, ground floor), Rudower Chaussee 25, Adlershof, 12489 Berlin
Number of talks
33
February 2025
Raffaele
Vitolo
University of Salento
Bi-Hamiltonian geometry of WDVV equations: general results
Abstract.
It is known (work by Ferapontov and Mokhov) that a system of N-dimensional WDVV equations can be written as a pair of N-2 commuting quasilinear systems (first-order WDVV systems). In recent years, particular examples of such systems were shown to possess two compatible Hamiltonian operators, of the first and third order. It was also shown that all $3$-dimensional first-order WDVV systems possess such bi-Hamiltonian formalism. We prove that, for arbitrary N, if one first-order WDVV system has the above bi-Hamiltonian formalism, than all other commuting systems do. The proof needs some interesting results on the structure of the WDVV equations that will be discussed as well. (Joint work with S. Opanasenko).
January 2025
Daniel
Roggenkamp
MiS Leipzig
Olivier
Marchal
Univ. St-Etienne
Isomonodromic deformations, quantization and exact WKB
Abstract.
In this talk, I will review the theory of isomonodromic deformations of meromorphic connections on gl2 and the underlying symplectic structure. In particular, I will explain how to obtain explicit formulas for the Hamiltonian systems and the Lax pairs. Next, I will explain how one can formally reconstruct these results using the quantization of the classical spectral curve using topological recursion. Finally, I will explain the current challenges and results to upgrade this formal reconstruction to an analytic one focusing on the genus zero case where one can use Borel resummation of WKB solutions. The talk is supposed not to require any knowledge in integrable systems, topological recursion of Borel resummation.
Türkü Özlüm
Çelik
MPI of Molecular Cell Biology and Genetics, Dresden
Gaëtan
Borot
HU Berlin
Limits in topological recursion
Abstract.
I will discuss the subtle aspects of the behavior of topological recursion (= seen as the construction of W-algebra modules from branched covers of curves) along families of spectral curves. To do so, I will review basics of singularities of plane curves and discuss (maximal) equigeneric families.
Türkü Özlüm
Çelik
MPI of Molecular Cell Biology and Genetics, Dresden
Gaëtan
Borot
HU Berlin
Limits in topological recursion
Abstract.
I will discuss the subtle aspects of the behavior of topological recursion (= seen as the construction of W-algebra modules from branched covers of curves) along families of spectral curves. To do so, I will review basics of singularities of plane curves and discuss (maximal) equigeneric families.
December 2024
Silvia
Ragni
University of Padua/HU Berlin
Teleman reconstruction of CohFTs
Abstract.
Given a CohFT, it is possible to define the action of the so called Givental group. Teleman in 2010 managed to prove that, given certain conditions, it is possible to reconstruct a whole CohFT from its 0-degree part. In this talk we are going to deal with the main ideas behind Teleman's reconstruction proof.
Jan
Pulmann
HU Berlin
Moduli spaces of flat connections as Kähler spaces, part III
Abstract.
Building on earlier sessions, we will propose the existence of an approach to obtain Kähler structures on the moduli spaces analogous to the quasi-Hamiltonian reduction.
Niklas
Martensen
HU Berlin
and
Jan
Pulmann
HU Berlin
Moduli spaces of flat connections as Kähler spaces, part III
Abstract.
Building on earlier sessions, we will propose the existence of an approach to obtain Kähler structures on the moduli spaces analogous to the quasi-Hamiltonian reduction.
Gaëtan
Borot
HU Berlin
Limits in topological recursion
Abstract.
I will discuss the subtle aspects of the behavior of topological recursion (= seen as the construction of W-algebra modules from branched covers of curves) along families of spectral curves. To do so, I will review basics of singularities of plane curves
Niklas
Martensen
HU Berlin
and
Jan
Pulmann
HU Berlin
Moduli spaces of flat connections as Kähler spaces, part II
Abstract.
Building on the introduction of the various moduli spaces in part I, we will sketch the proof of the non-abelian Hodge correspondence for flat bundles on compact Kähler manifolds, which provides a homeomorphism between the moduli space of (semisimple) flat connections and of (certain) Higgs bundles. Moreover, we will propose the existence of an approach to obtain Kähler structures on the moduli spaces analogous to the quasi-Hamiltonian reduction.
Niklas
Martensen
HU Berlin
and
Jan
Pulmann
HU Berlin
Moduli spaces of flat connections as Kähler spaces, part I
Abstract.
Moduli spaces of flat connections on surfaces appear in physics and geometry. Consequently, they themselves carry various geometric structures. We will recall the constructions of Poisson and complex structures on these moduli spaces, as well as the non-abelian Hodge correspondence. This talk will be mainly a literature review; our goal for the second part is to discuss the problem of inventing 'quasi' generalizations of Kähler structures.
November 2024
Rob
Klabbers
HU Berlin
Elliptic long-range quantum integrable systems
Abstract.
There are at least two seemingly distinct realms of quantum integrability. The first domain is formed by the (short-range) Heisenberg spin chains, connected to the quantum inverse scattering method, which play a role in many different contexts both in physics and mathematics. The second domain is formed by the Calogero-Sutherland models and their deformations, which are families of differential or difference operators associated to root systems, with close ties to harmonic analysis, orthogonal Jack and Macdonald polynomials, and Knizhnik-Zamolodchikov equations. Their integrability follows from a connection to affine Hecke algebras. Understanding how these two realms are connected goes through the elliptic CS models and their generalisations, which are also interesting in their own right. I will discuss in what way this bridge between worlds is formed and how far we are in building it. Along the way I will try to point out connections to different research areas.
Guillaume
Baverez
Aix-Marseille Université
Uniqueness of Malliavin-Kontsevich-Suhov measures
Abstract.
About 20 years ago, Kontsevich & Suhov conjectured the existence and uniqueness of a family of measures on the set of Jordan curves, characterised by conformal invariance and another property called 'conformal restriction'. This conjecture was motivated by (seemingly unrelated) works of Schramm, Lawler & Werner on stochastic Loewner evolutions (SLE), and Malliavin, Airault & Thalmaier on 'unitarising measures'. The existence of this family was settled by works of Werner-Kemppainen and Zhan, using a loop version of SLE. The uniqueness was recently obtained in a joint work with Jego. I will start by reviewing the different notions involved before giving some ideas of our proof of uniqueness: in a nutshell, we construct a family of 'orthogonal polynomials' which completely characterise the measure. In the remaining time, I will discuss the broader context in which our construction fits, namely the conformal field theory associated with SLE.
Gernot
Akemann
Bielefeld Universität
Three universality classes in non-Hermitian random matrices
Abstract.
Non-Hermitian random matrices with complex eigenvalues have important applications, for example in open quantum systems in their chaotic regime. It has been conjectured that amongst all 38 symmetry classes of non-Hermitian random matrices only 3 different local bulk statistics exist. This conjecture has been based on numerically generated nearest-neighbour spacing distributions between complex eigenvalues so far. In this talk I will present first analytic evidence for this conjecture. It is based on expectation values of characteristic polynomials in the three simplest representatives for these statistics: the well-known Ginibre ensemble of complex normal matrices, complex symmetric and complex self-dual random matrices. After giving a basic introduction into the complex eigenvalue statistics of the Ginibre ensemble, I will present results for all three ensembles for finite matrix size N as well as in various large-N limits. These are expected to be universal, that is valid beyond ensembles with Gaussian distribution of matrix elements. This paper is based on joint work with Noah Aygün, Mario Kieburg and Patricia Päßler in arXiv/2410.21032
Guilherme
Feitosa de Almeida
Hannover Universität
1D Landau-Ginzburg superpotential of big quantum cohomology of CP2
Abstract.
Using the inverse period map of the Gauss-Manin connection associated with QH∗(CP2) and the Dubrovin construction of Landau-Ginzburg superpotential for Dubrovin Frobenius manifolds, we construct a one-dimensional Landau-Ginzburg superpotential for the quantum cohomology of CP2. In the case of small quantum cohomology, the Landau-Ginzburg superpotential is expressed in terms of the cubic root of the j-invariant function. For big quantum cohomology, the one-dimensional Landau-Ginzburg superpotential is given by Taylor series expansions whose coefficients are expressed in terms of quasi-modular forms. Furthermore, we express the Landau-Ginzburg superpotential for both small and big quantum cohomology of QH∗(CP2) in closed form as the composition of the Weierstrass ℘-function and the universal coverings of C \ (Z ⊕ jZ) and C \ (Z ⊕ zZ) respectively. This seminar is based on the results of arXiv/2402.09574.
October 2024
Bernadette
Lessel
Bonn Universität
On the early history of quantum gravity
Abstract.
Quantum gravity, in the sense of a formal quantization of general relativity, had a first beginning in the year 1930. That year, the Belgian physicist Léon Rosenfeld published a seminal paper called "Zur Quantelung der Wellenfelder" in which he developed Heisenberg and Pauli's recently constructed method to quantize the electromagnetic field in order to apply it to the tetrad formulation of general relativity. In my talk, I aim to shed light on a perhaps surprising crucial historical influence that made this piece of intellectual work possible: that of unified field theory. A purely classical program, most prominently pursued by Albert Einstein and Hermann Weyl, to formally reduce the (classical) electromagnetic field to the gravitational field as described by general relativity.
February 2024
Anne
Spiering
HU Berlin
Elliptic Feynman integrals from a symbol bootstrap
Abstract.
A Feynman integral is a multi-dimensional integral that encodes the probability amplitude for particle interactions within the framework of quantum field theory. While Feynman integrals play a crucial role in connecting theoretical models with experimental data, their evaluation can pose significant challenges. The “symbol bootstrap” has proven to be a powerful tool for calculating specific (polylogarithmic) Feynman integrals that bypasses a direct integration. I will discuss a generalisation of this method to the elliptic case, mainly focusing on the so-called double-box integral where elliptic structures appear in the integration.
January 2024
Hugo
Parlier
Universite du Luxembourg
Crossing the line: from graphs to curves
Abstract.
The crossing lemma for simple graphs gives a lower bound on the necessary number of crossings of any planar drawing of a graph in terms of its number of edges and vertices. Viewed through the lens of topology, this leads to other questions about arcs and curves on surfaces. Here is one: how many crossings do a collection of m homotopically distinct curves on a surface of genus g induce? The talk will be about joint work with Alfredo Hubard where we explore some of these, using tools from the hyperbolic geometry of surfaces in the process.
Pietro
Longhi
Uppsala University
Open topological strings and symplectic cuts
Abstract.
The study of A-branes as boundary conditions for open topological strings has extensive ramifications across physics and mathematics. Yet, from a mathematical perspective a generally valid definition of open Gromov-Witten invariants is still lacking, while on the physics side computations rely heavily on the use of large N dualities and mirror symmetry. In this talk I will present a novel approach to the computation of genus-zero open topological string amplitudes on toric branes based on a worldsheet description. We consider an equivariant gauged linear sigma model whose target is a certain modification of the Calabi-Yau threefold, known as symplectic cut and determined by the toric brane data. This leads to equivariant generating functions of open and closed genus-zero string amplitudes that extend smoothly across the entire moduli space, and which provide a unifying description of standard Gromov-Witten potentials.
Kento
Osuga
Tokyo University
Recent progress in refined topological recursion
Abstract.
I will first present recent progress in the formulation of refined topological recursion with a brief overview of previous attempts. I will then show its interesting properties such as refined quantum curves, the refined variational formula, and refined BPS structures. I will also discuss an intriguing relation between refined topological recursion, W-algebras, and b-Hurwitz numbers. Finally, I will conclude with open questions and future directions. This talk is partly based on joint work with Kidwai, and also partly joint work in progress with Chidambaram and Dolega.
December 2023
Xavier
Coulter
University of Auckland
A one-parameter deformation of the monotone Hurwitz numbers
Abstract.
The monotone Hurwitz numbers are involved in a wide array of mathematical connections, linking topics such as integration on unitary groups, representation theory of the symmetric group, and topological recursion. In recent work, we introduce a one-parameter deformation of the monotone Hurwitz numbers and show that the resulting family of polynomials admits a similarly broad network of connections. We will discuss these results and some non-trivial conjectures on the roots of these polynomials.
Thomas
Buc-d'Alché
ENS Lyon
Fay-like identities for hyperelliptic curves
Abstract.
Fay's identity is a determinantal formula between Riemann theta functions associated to the period matrix of a Riemann surface. In random matrix theory, the theta function appears in the asymptotic expansion of the partition function of the β-model. Using Pfaffian formulae for averages of characteristic polynomials when β = 1 or β =4, we derive Pfaffian identities involving the theta function associated to half or twice the period matrix of a hyperelliptic curve. This is joint work with Gaëtan Borot.
November 2023
Yannik
Schüler
University of Sheffield
Gromov-Witten theory from the 5-fold perspective
Abstract.
The observation that the Gromov-Witten theory of a Calabi-Yau threefold X may be viewed as a mathematical realisation of the A-model topological string on this target is the corner stone of some of the most exciting developments in Enumerative Geometry in the last decades. Despite this, the so called refined topological string so far lacked a mathematical description. In this talk I will make a proposal for a rigorous formulation in terms of equivariant Gromov-Witten theory on the fivefold X x C^2. To convince you of our construction I will mention several precision checks our proposal passes. Most of these results were expected by physics but some are new.
Jarkko
Kari
University of Turku
From Wang Tiles to the Domino Problem: A Tale of Aperiodicity
Abstract.
This presentation delves into the remarkable history of aperiodic tilings and the domino problem. Aperiodic tile sets refer to collections of tiles that can only tile the plane in a non-repeating, or non-periodic, manner. Such sets were not believed to exist until 1966 when R. Berger introduced the first aperiodic set consisting of an astonishing 20,426 Wang tiles. Over the years, ongoing research led to significant advancements, culminating in 2015 with the discovery of a mere 11 Wang tiles by E. Jeandel and M. Rao, alongside a computer-assisted proof of their minimality. Simultaneously, researchers found even smaller aperiodic sets composed of polygon-shaped tiles. Notably, Penrose's kite and dart tiles emerged as early examples, and most recently, a groundbreaking discovery was made - a solitary aperiodic tile known as the "hat" that can tile the plane exclusively in a non-periodic manner. Aperiodic tile sets are intimately connected with the domino problem that asserts how certain tile sets can tile the plane without us ever being able to establish their tiling nature with absolute certainty. Moreover, aperiodic tilings hold a distinct visual aesthetic allure. In today's musical presentation, their artistic appeal transcends the visual domain and extends into the realm of music.
Murad
Alim
Hamburg Universität
Resurgence, BPS structures and topological string S-duality
Abstract.
The partition function of topological string theory is an asymptotic series in the topological string coupling and provides in a certain limit a generating function of Gromov-Witten (GW) invariants of a Calabi-Yau threefold. I will discuss how the resurgence analysis of the partition function allows one to extract Donaldson-Thomas (DT) or BPS invariants of the same underlying geometry. I will further discuss how the analytic functions in the topological string coupling obtained by Borel summation admit a dual expansion in the inverse of the topological string coupling leading to another asymptotic series at strong coupling and to the notion of topological string S-duality. This S-duality leads to a new modular structure in the topological string coupling. I will also discuss relations to difference equations and the exact WKB analysis of the mirror geometry. This is based on various joint works with Lotte Hollands, Arpan Saha, Iván Tulli and Jörg Teschner as well as on work in progress.
Paolo
Gregori
IPhT - CEA Saclay
New results in non-perturbative topological recursion
Abstract.
I will present recent techniques which combine topological recursion with ideas from the theory of resurgence. In this framework, one can compute non-perturbative contributions to the formal power series one usually obtains from topological recursion, upgrading them to resurgent 'transseries'. The computation of such contributions serves two main purposes: on the one hand, it allows for an in-depth study of instanton effects in 2d gravitational theories such as Jackiw-Teitelboim gravity. On the other hand, it leads to new formulas for the large genus asymptotics of a large class of enumerative invariants, such as Weil-Petersson volumes and intersection numbers.
Martin
Markl
Czech Academy of Sciences
Transfers of strongly homotopy structures as Grothendieck bifibrations
Abstract.
It is well-known that strongly homotopy structures can be transferred over chain homotopy equivalences. Using the uniqueness results of Markl & Rogers we show that the transfers could be organized into a discrete Grothendieck bifibration. An immediate application is e.g. functoriality up to isotopy.
October 2023
Davide
Scazzuso
HU Berlin
Topological gravity, volumes and matrices
Abstract.
Jackiw-Teitelboim (JT) gravity is a simple model of two-dimensional quantum gravity that describes the low-energy dynamics of any near-extremal black hole and provides an example of AdS_2/CFT_1. In 2016 Saad, Shenker and Stanford showed that the path integral of JT gravity is computed by a Hermitian matrix model, by reinterpreting Mirzakhani's results on the volumes of moduli spaces of Riemann surfaces through the lenses of Eynard and Orantin's topological recursion. Thus, a beautiful threefold story connecting quantum gravity in two dimensions, random matrices and intersection theory emerged. In this talk I will review such connection from the point of view of physics and touch upon its generalization to N=1 JT supergravity and super Riemann surfaces.
Emanuel
Malek
HU Berlin
Exceptional generalised geometry and Kaluza-Klein spectra of string theory compactifications
Abstract.
Most interesting solutions of string theory are of the form M x C, where M is some D-dimensional non-compact space (e.g. Minkowski or Anti-de Sitter), and C is some (10-D)- or (11-D)-dimensional compact space, known as a compactification. Many interesting questions about string theory then reduce about understanding the properties of the 'Kaluza-Klein spectra' of certain differential operators on C. Because these operators often involve a complicated interplay between the p-forms arising in string theory and the metric on C, few general results are known. Generalised geometry is the study of structures on TM + T*M and similar extensions of TM, and naturally 'geometrises' the interaction between p-forms and metric in string theory. I will review generalised geometry and show how it allows us to study the Kaluza-Klein spectra for a large class of string theory compactifications.
Pedro
Tamaroff
HU Berlin
Differential operators of higher order and their homotopy trivializations