# Interactions between Poisson Geometry and Quantisation

20 - 24 March 2023, Göttingen

## Graduate School in Mathematical Physics

The aim of the School is to bring together experts in the fields of Poisson Geometry, Quantum Field Theory and Mathematical Physics for one week, from Monday 20 to Friday 24 of March 2023. In a lively and stimulating atmosphere, we will have the exciting possibility to learn directly from them and discuss in depth some of the most advanced and recent developments concerning the role of Poisson Structures in Quantisation

## VERY IMPORTANT!

We were recently informed that on the weekend of Saturday 25 and Sunday 26 of March, an ordnance will be cleared in Göttingen. An area of approximately 1 km around the train station will have to be evacuated by 6 am of Saturday 25 of March, until the end of the operations. Some of the hotels suggested for your staying in Göttingen are included in this area.This also means that on Saturday 25 of March it will be impossible to travel by train from Göttingen (although some alternative bus lines to the closest train stations might be set up). Please take into careful account this information when preparing your travel to Göttingen.

For more information you can visit here the webpage of the city of Göttingen (Liveblog). You can also find a map of the area that will have to be evacuated here.

We will make sure to keep you constantly informed with the most recent updates on the situation.

## SPEAKERS

Alejandro Cabrera
Universidade Federal do Rio de Janeiro

Jean-Marie Lescure
Université Paris-Est Créteil

Catherine Meusburger
Friedrich-Alexander-Universität Erlangen-Nürnberg

Alexander Schenkel
University of Nottingham

## Mini-Courses

Alejandro Cabrera - About quantization of Poisson brackets and local symplectic groupoids

Abstract: In this minicourse, the idea is to review general facts relating star products to the Lie theory of the underlying Poisson brackets and the corresponding (local) symplectic groupoids. We aim at presenting 3 recent results by the author in that context. As an introduction, we first review the relevant definitions, concepts and basic examples. We then go into the first group of results concerning existence and characterization of generating functions for local symplectic groupoids. The second result involves asymptotic expansions of these generating functions and a direct comparison to an extract of Kontsevich's quantization formula in a coordinate space. The third result involves a functional (or "field-theoretic") presentation of the previous structures in which we consider a system of PDE's associated with the "Poisson Sigma Model" on a disk. Time permitting, we shall sketch a work in progress (together with R. Fernandes) relating quantization to the "integrability" of the underlying Poisson structure.

Jean-Marie Lescure - Fourier integral operators on Lie groupoids

Abstract: The purpose of these lectures is to present a calculus for Fourier Integral (invariant) Operators (FIO) on Lie groupoids and to show that the fundamental solution of evolution equations $\partial_tu +iPu=0$ belongs to this class of operators. We will begin by recalling basic useful facts about Lie groupoids and their convolution algebras of $C^\infty$ functions, before explaining how to extend the convolution product to distributions. We will give an answer using wave front sets and this is where we will encounter the cotangent symplectic groupoid of Coste-Dazord-Weinstein. Next, we will briefly recall the notion of Lagrangian distributions and explain under which conditions they are stable for convolution. This will lead to the suitable notion of FIOs on a Lie groupoid, generalizing the well known pseudodifferential calculus. After reviewing its properties, we will turn to its main application (so far): the construction of an approximate solution of evolution equations on Lie groupoids.

Catherine Meusburger - Topological quantum field theories with defects

Abstract: The lectures give an introduction to oriented topological quantum field theories with and without defects in dimension d=2 and d=3. In the first part of the course we introduce the general formalism for oriented topological quantum field theories (TQFTs) in terms of the cobordism category Cob_{d,d-1}. We give their classification in dimensions d=1and d=2 and describe state sum contructions of TQFTs in dimensions d=2 and d=3, the Fukuma-Hosano-Kawai model and Turaev-Viro-Barrett-Westbury model, respectively. In the second part we explain the algebraic data for defects in these models and show how to formulate TQFTs with defects in dimensions d=2 and d=3. If time permits, we comment on the relation to defects in models from condensed matter physics such as Kitaev's quantum double model and Levin-Wen models.

Alexander Schenkel - Higher structures and quantization

Abstract: These lectures give an introduction to derived algebraic geometry, shifted symplectic and Poisson structures, and their deformation quantizations. After a non-technical exposition of the theory of derived stacks and their relevance for geometry and also mathematical physics, we will focus mostly on simple examples and explicit constructions, based on derived affine schemes and derived quotient stacks. As particular applications, we will discuss the shifted symplectic geometry of derived critical loci (this is a mathematical version of the BV formalism from physics) and the quantization of derived cotangent stacks over quotient stacks along their canonical symplectic structure.

## Schedule

Pre-School

The pre-School takes place at the Mathematical Institute in Göttingen, on the week before the School, from Monday 13 to Wednesday 15 of March. Attendance is open to anyone, but in particular the event is meant for locals, so no financial support will be offered. The aim is to make sure that all the participants can start with the same background knowledge.
We will give one introductory lecture for each of the minicourses of the School plus, thanks to the collaboration of Prof. E. Schrohe (Leibniz University Hannover), a more extensive review on Pseudo-differential Operators and Fourier Integral Operators.

In order to grant access to all the participants, the lectures will be streamed and recorded.

(Links for the streaming and the recordings will appear soon).

Programme of the pre-School

The Programme of the pre-School can also be downloaded here

Programme of the School

The Programme of the School can also be downloaded here

→ The on-site registration procedure will take place in the main hall (Hilbertraum) of the Mathematical Institute, Monday 20.03, 8:30-9:30.

Lectures and other activities will be held in the Sitzungszimmer.

## Poster Session

We look forward to get to know more about each other's work and interests! That's why we strongly encourage all the participants to contribute to the Poster Session.

## Arriving in Göttingen

Closest airports: Frankfurt Main International, Hannover

Göttingen is well connected via fast trains to most major cities in Germany. The travel time from Frankfurt is about 2h and from Hannover less than 1h.

Walking from the train station to the Mathematical Institut takes no more than 20 minutes passing through the town centre. Most hotels are in walking distance from the main building, however there is also a chance of using the buses.

## Hotels

Here is a list of recommended Hotels in Göttingen:

Leine Hotel
Eden Hotel
Hotel Central
G Hotel
Novostar