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

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.

Jérémie Bouttier (IPhT - CEA Saclay)
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.

9. Mai 2023
No seminar. Instead: Spring school 3 facets of Gravity

Guillaume Baverez (HU Berlin)

In this first talk, I will introduce the semigroup of annuli and construct a (projective) representation of this semigroup on the Liouville Hilbert space, which is the L2 space of the Gaussian free field on the circle. The infinitesimal generator of this semigroup is a family of unbounded operators on the Hilbert space, and they represent the Virasoro algebra in a suitable sense.

 

Guillaume Baverez (HU Berlin)

In this second talk, I will build on this construction in order to define the conformal blocks of Liouville CFT, and I will show how to recover the algebra-geometic formulation (Ward identities). This allows us to give geometric and analytic content to an otherwise formal framework. Along the way, I will also point out to some purely geometric questions related to the semigroup of annuli (central extensions, Kähler structures, Weil-Petersson metric etc.).

 

Davide Scazzuso (HU Berlin)

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.

 

Giacomo Umer (HU Berlin)

Cohomological field theories (CohFTs) were introduced by Kontsevich and Manin in 1994 to grasp the formal properties of Gromov-Witten classes and show their natural recursive structure. I will present their classification due to Givental-Teleman in the semisimple case, provide some examples and illustrate the connection to Frobenius structures. I will then express the CohFT’s partition function by means of topological recursion, exploiting the notion of Airy structure. If time permits, I will briefly mention a generalisation of this picture starting from the notion of F–manifolds, defined by Manin in 1998, eventually introducing the less–constrained F–CohFTs and a version of topological recursion with non–symmetric outputs.

 

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.

 

Omid Amini (Ecole Polytechnique)

I will discuss a mathematical approach to realize the idea suggested by physicists that quantum filed theory amplitudes should arise as a limit of string theory amplitudes.

 

Christoph Chiaffrino (HU Berlin)

I give an introduction to the BV (Batalin-Vilkovisky) formalism and presents its relation to integration theory in finite dimensions. I then show how it can be used to compute quantum expectation values, which can be thought of as performing a Feynman path integral over an infinite dimensional space of functions.

 

Davide Scazzuso (HU Berlin)