Archive of Mathsphys seminar
Sommersemester 2023
18. Apr. 2023
Niklas Martensen (HU)
Nonperturbative partition functions and variations of spectral curves
The topological recursion associates to any compact spectral curve a set of numbers Fg called free energies. These free energies are encoded in the perturbative partition function Z which is a formal Laurent series in a parameter N. However, from the viewpoint of string theory and matrix models, the perturbative partition function does not have good transformation properties under a change of the symplectic basis of the spectral curve. To restore modularity, one introduces the nonperturbative partition function which contains correction terms to the perturbative partition function. In this talk, I will recall the basic theory to define the non-perturbative partition function and discuss its properties with an emphasis on its behavior under the variation of the spectral curve.
Gaëtan Borot (HU Berlin)
Special Kähler geometry in topological recursion
I will review special Kähler geometry and its relation to complex integrable systems. In particular, tThe base ofHitchin integrable system carries a special Kähler structure, and I will explain how variations of the special Kähler metric can be computed from the genus 0 sector of topological recursion. I'll put this in context of the properties of topological recursion under deformations of spectral curves.
On maps with tight boundaries and slices
Maps are discrete surfaces obtained by gluing polygons, and form an important model of 2D random geometry. Among the many approaches developed to study them, the bijective method has been instrumental in understanding their metric properties and their scaling limits. Originally the method consisted in finding bijections between planar maps and certain labeled/decorated trees, called blossom trees or mobiles. It was more recently realized that the recursive structure of trees could be directly implemented at the level of maps, via the so-called "slice decomposition". I start by presenting the main ideas of this method. Then, I will present some recent work done in collaboration with Emmanuel Guitter and Grégory Miermont. Our long-term goal is to extend the slice decomposition to arbitrary topologies, and I will report on two first steps in that direction. In arXiv:2104.10084, we obtain a very simple formula for the generating function of maps with the topology of a pair of pants. Our derivation is bijective and is reminiscent of hyperbolic geometry, hinting that our approach may be universal in a sense to be determined. In arXiv:2203.14796, we consider quasi-polynomials counting so-called tight maps (which I will precisely define during the talk). Such quasi-polynomials were previously encountered by Norbury in the context of the enumeration of lattice points in the moduli space of curves. We give a fully explicit expression for these quasi-polynomials in the genus 0 case.
No seminar. Instead: Spring school 3 facets of Gravity
The study of N=2 supersymmetric gauge theories has proven to be a fruitful field to explore new ideas in both physics and mathematics. These theories always have non-chiral matter representations and thus have no hope of directly describing the real world. However, the existence of two sets of supersymmetries lets us study the dynamics of such systems in great detail, leading to a host of surprising results. In this talk I will review the Lagrangian construction of such theories and describe the Seiberg-Witten (SW) solution for the case of pure N=2 supersymmetric SU(2) Yang-Mills theory. I will then describe the SW and UV curves and explore their physical meaning through the twisted compactitication of the 6d N=(2,0) theory. Finally, if time allows, I will illustrate some ideas behind the AGT correspondence and 2d/4d dualities.
Agostino Patella (HU Berlin)
Large-L expansion of electromagnetic current 2-pt function in QCD and applications to the muon g-2
The quantity of interest of this talk is the electromagnetic current 2-point function, calculated in Quantum Chromodynamics (QCD) on the Euclidean spacetime R x S_1^3. I will sketch the derivation of an asymptotic expansion of the 2pt function valid for L -> infinity, where L is the length of each compact dimension. This formula can not be obtained in the context of perturbative QCD, since the long-distance behaviour of QCD is governed by non-perturbative physics. In order to circumvent this difficulty, we use the conceptual framework of Effective Field Theory (EFT). The desired asymptotic expansion is obtained using a fully general EFT of hadrons, and is shown to be valid at any order in the perturbative expansion of the EFT independently of the microscopic details of the interactions.
This work is motivated by the effort to determine the hadronic contributions to the anomalous magnetic moment (g-2) of the muon by means of Lattice QCD simulations. Beyond one loop, the muon g-2 gets contributions from quarks and gluons. The leading QCD contribution to g-2 (also known as HVP, hadron vacuum polarization) is calculated from the 2-point function of the electromagnetic current in QCD at energy scales around 1GeV. Due to the non-perturbative nature of QCD at these energy scales, these contributions can not be calculated with perturbative techniques and one must rely on Lattice QCD simulations. When dealing with Lattice QCD, one usually considers QCD in a Euclidean compact spacetime (e.g. a 4-torus). Observables are calculated in this setup, and the infinite-volume limit is taken by extrapolation. Asymptotic formulae, like the one presented in this talk, are needed in order to control the extrapolation and to obtain a reliable estimate of the associated systematic errors.
The study of N=2 supersymmetric gauge theories has proven to be a fruitful field to explore new ideas in both physics and mathematics. These theories always have non-chiral matter representations and thus have no hope of directly describing the real world. However, the existence of two sets of supersymmetries lets us study the dynamics of such systems in great detail, leading to a host of surprising results. In this talk I will review the Lagrangian construction of such theories and describe the Seiberg-Witten (SW) solution for the case of pure N=2 supersymmetric SU(2) Yang-Mills theory. I will then describe the SW and UV curves and explore their physical meaning through the twisted compactitication of the 6d N=(2,0) theory. Finally, if time allows, I will illustrate some ideas behind the AGT correspondence and 2d/4d dualities.
Wintersemester 2023/24
17. Okt. 2023
Pedro Tamaroff (HU Berlin)
Differential operators of higher order and their homotopy trivializations.
In the classical Batalin–Vilkovisky formalism, the BV operator is a differential operator of order two with respect to a commutative product; in the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a genus zero level cohomological field theory induced on homology. In this talk, we will explore generalisations of non-commutative Batalin-Vilkovisky algebras for differential operators of arbitrary order, showing that homotopically trivial operators of higher order also lead to interesting algebraic structures on the homology. This is joint work with V. Dotsenko and S. Shadrin.
24. Okt. 2023
Emanuel Malek (HU Berlin)
Exceptional generalised geometry and Kaluza-Klein spectra of string theory compactifications
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.
31. Okt. 2023
Davide Scazzuso (HU Berlin)
Topological gravity, volumes and matrices
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.
7. Nov. 2023
Martin Markl (Czech Academy of Sciences)
Transfers of strongly homotopy structures as Grothendieck bifibrations
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 aplication is e.g. functoriality up to isotopy.
14. Nov. 2023
Paolo Gregori (IPhT - CEA Saclay)
New results in non-perturbative topological recursion
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.
21. Nov 2023 (last minute)
Murad Alim (Hamburg Universität)
Resurgence, BPS structures and topological string S-duality
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.
Wednesday 22. Nov 2023, 19.30Uhr
Public lecture + Concert
"A history of the domino problem"
Fritz-Reuter-Saal (HU@Hegelplatz), Dorotheenstr. 24
Jarkko Kari (University of Turku)
From Wang Tiles to the Domino Problem: A Tale of Aperiodicity
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.
Following the talk, the Kali Ensemble will play a series of pieces by Michael Winter.
Jarkko Kari will also give the Math+ Fridays colloquium (for mathematicians) on 17. Nov. 2023: Low-complexity colorings of the two-dimensional grid.
28. Nov. 2023
Yannik Schüler (University of Sheffield)
Gromov-Witten theory from the 5-fold perspective
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.
5. Dez. 2023
Thomas Buc-d'Alché (ENS Lyon)
Fay-like identities for hyperelliptic curves
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.
12. Dez. 2023
Xavier Coulter (University of Auckland) - hybrid
A one-parameter deformation of the monotone Hurwitz numbers
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.
19. Dez. 2023
Cancelled
9. Jan. 2024
Kento Osuga (Tokyo University) - hybrid
Recent progress in refined topological recursion
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.
16. Jan. 2024
Pietro Longhi (Uppsala University)
Open topological strings and symplectic cuts
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.
Pietro Longhi will also speak at the QFT colloquium on 19. Jan. 2024.
23. Jan. 2024
Hugo Parlier (Universite du Luxembourg)
Crossing the line: from graphs to curves
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.
30. Jan. 2024
Cancelled
6. Feb. 2024
Anne Spiering (HU Berlin)
Elliptic Feynman integrals from a symbol bootstrap
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.
13. Feb. 2024
Cancelled.