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TQFTs via Cut and Paste Techniques

English title TQFTs via Cut and Paste Techniques
Applicant Cattaneo Alberto Sergio
Number 192080
Funding scheme Project funding (Div. I-III)
Research institution Institut für Mathematik Universität Zürich
Institution of higher education University of Zurich - ZH
Main discipline Mathematics
Start/End 01.04.2020 - 31.03.2023
Approved amount 590'720.00
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All Disciplines (2)

Discipline
Mathematics
Theoretical Physics

Keywords (11)

BV algebras; coisotropic submanifolds; extended symplectic category; Poisson manifolds; symplectic reduction; symplectic groupoids; configuration spaces; BFV formalism; BV quantization; perturbative quantum field theories; deformation quantization

Lay Summary (Italian)

Lead
Questo è un progetto di matematica pura che trae ispirazione da concetti di base della fisica moderna e che si propone di sviscerarne le principali conseguenze.
Lay summary

Un’idea fondamentale di tutta la fisica moderna è il concetto di località, cioè il fatto che ci siano solo influenze nelle immediate vicinanze e che tutti gli effetti a distanza siano in realtà mediati da effetti appunto locali.

 

Nella descrizione matematica di cui facciamo uso la nozione di località è espressa come la possibilità di descrivere la fisica di una regione come una combinazione della fisica di regioni più piccole in cui la decomponiamo. Ciascuna di queste regioni più piccole comunica con le regioni circostanti attraverso interfacce comuni e un problema matematico interessante consiste nel descrivere queste interfacce e le informazioni che queste ereditano dalle regioni che racchiudono.

 

Un secondo aspetto fondamentale della fisica moderna è la traducibilità tra diverse descrizioni. P.es. possiamo guardare allo stesso fenomeno da punti d’osservazione diversi, ma possiamo anche decidere di misurare le grandezze in gioco con unità diverse. Il dizionario tra le diverse descrizioni ci permette di comunicare con altri osservatori. Questa traducibilità dei punti di vista diventa particolarmente interessante quando è combinata con la località: quando cerchiamo di incollare regioni diverse lungo un’interfaccia comune dobbiamo tener conto che le singole regioni possono essere presentate in linguaggi diversi. 

 

Lo scopo generale di questo progetto consiste appunto nello sviluppare le idee matematiche che nascono da questi due concetti concetti di località e di traducibilità. Da una parte questo potrebbe darci strumenti più potenti per capire la realtà fisica e dall’altra ci potrebbe essere d’aiuto a sviluppare dei nuovi formalismi matematici, con potenziali applicazioni anche in altre aree. In particolare ci si concentrerà su alcune teorie fisiche ben note, come quelle che descrivono le particelle elementari o la gravitazione, ma anche su teorie di maggior interesse matematico, come le cosiddette teorie topologiche che permettono di caratterizzare strutture geometriche diverse.

Direct link to Lay Summary Last update: 27.03.2020

Responsible applicant and co-applicants

Employees

Publications

Publication
Convolution algebras for relational groupoids and reduction
Contreras Ivan, Moshayedi Nima, Wernli Konstantin (2021), Convolution algebras for relational groupoids and reduction, in Pacific Journal of Mathematics, 313(1), 75-102.
Gravitational Constraints on a Lightlike Boundary
Canepa G., Cattaneo A. S., Tecchiolli M. (2021), Gravitational Constraints on a Lightlike Boundary, in Annales Henri Poincaré, 22(9), 3149-3198.
Symplectic microgeometry, IV: Quantization
Cattaneo Alberto S., Dherin Benoit, Weinstein Alan (2021), Symplectic microgeometry, IV: Quantization, in Pacific Journal of Mathematics, 312(2), 355-399.
General Relativity and the AKSZ Construction
Canepa G., Cattaneo A. S., Schiavina M. (2021), General Relativity and the AKSZ Construction, in Communications in Mathematical Physics, 385(3), 1571-1614.
On quantum obstruction spaces and higher codimension gauge theories
Moshayedi Nima (2021), On quantum obstruction spaces and higher codimension gauge theories, in Physics Letters B, 815, 136155-136155.
Formal Global AKSZ Gauge Observables and Generalized Wilson Surfaces
MoshayediNima (2020), Formal Global AKSZ Gauge Observables and Generalized Wilson Surfaces, in Annales Henri Poincaré, 21, 2951 -2995.

Collaboration

Group / person Country
Types of collaboration
ETHZ Switzerland (Europe)
- Publication
University of Notre Dame United States of America (North America)
- Publication
Aix-Marseille University France (Europe)
- Publication
UC Berkeley United States of America (North America)
- Publication
Google Ireland (Europe)
- Publication
Stanford University United States of America (North America)
- Publication
Queen Mary University of London Great Britain and Northern Ireland (Europe)
- Publication
Amherst College United States of America (North America)
- Publication

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Differentiable Stacks, Poisson Geometry and related geometric structures Talk given at a conference Poisson structures from corners of field theories 07.02.2022 SwissMAP Research Station (SRS), Les Diablerets, Switzerland Cattaneo Alberto Sergio;
San Diego Focus Meeting Talk given at a conference On BV-BFV 10.01.2022 La Jolla, United States of America Cattaneo Alberto Sergio;
Corfu2021: Workshop on Quantum Geometry, Field Theory and Gravity Talk given at a conference The BFV formalism for Palatini-Cartan gravity and corner structure 25.09.2021 Corfu, Greece Cattaneo Alberto Sergio;
A gauge summer with BV Talk given at a conference The reduced phase space of general relativity 06.09.2021 Scalea, Italy Canepa Giovanni;
Geometry for Higher Spin Gravity: Conformal Structures, PDEs, and Q-manifolds Talk given at a conference An introduction to the BV-BFV formalism (three one-hour lectures) 26.08.2021 ESI, Vienna, Austria Cattaneo Alberto Sergio;
ICMP2021 Poster Palatini–Cartan formulation of general relativity in the BV-BFV formalism 02.08.2021 Geneva, Switzerland Canepa Giovanni;
Higher Structures in QFT and String Theory Individual talk AKSZ construction for general relativity 14.07.2021 Zurich (online), Switzerland Canepa Giovanni;
Prague Mathematical Physics Seminar Individual talk AKSZ Construction for General Relativity 26.11.2020 Prague (online), Czech Republic Cattaneo Alberto Sergio;
Global Poisson Webinar Individual talk Hamilton-Jacobi and Quantum Chern-Simons on Cylinders 29.10.2020 Geneva (online), Switzerland Cattaneo Alberto Sergio;
Joint seminar series on theoretical and mathematical physics Individual talk Hamilton-Jacobi and Quantum Chern-Simons on Cylinders 29.06.2020 Arnold Sommerfeld Center, Munich, Germany Cattaneo Alberto Sergio;
Séminaire "Groupes de Lie et espaces des modules" Individual talk The BV-BFV Formalism 21.04.2020 University of Geneva, Switzerland Cattaneo Alberto Sergio;


Self-organised

Title Date Place
A gauge summer with BV 06.09.2021 Scalea, Italy
Quantum Field Theory Thematic Session at ICMP2021 02.08.2021 Geneva, Switzerland
Higher Structures in QFT and String Theory 12.07.2021 Zurich (online), Switzerland
A Gauge Summer with BV: Online 24.06.2020 Online, Switzerland

Associated projects

Number Title Start Funding scheme
172498 TQFTs via Cut and Paste Techniques 01.04.2017 Project funding (Div. I-III)

Abstract

The main theme underlying this project is the mathematical understanding of classical and quantum field theories on manifolds with boundaries and corners, of the novel mathematical structures that arise in this study, and of the effective methods for pasting theories together both at the classical and at the quantum level. Based on an already successfully started program, this project aims at developing new ideas and techniques for a vast range of theories, encompassing not only topological ones.Among the goals of this project, we aim at constructing the Segal-Bargmann transform from one representation to another in the context of perturbative field theories, at understanding the structures that arise on space-time corners of higher codimensions, at developing the methods for computing quantum field theories by gluing manifolds with corners (and not only with boundaries) together. We also plan to keep applying these ideas to classical general relativity and to the quantization of Poisson manifolds, to extend them to quantum field theories involving fermions, and to develop the general formalism for discretized versions.The main methods to be used are mathematical tools we developed over the years stemming, among others, from the now classical approaches of Batalin-Vilkovisky and Batalin-Fradkin-Vilkovisky for the quantization and for the homological description of reduced phase spaces of field theories, from the approach of Kijowski and Tulczyjew for the geometrical determination of the reduced phase space of a theory, from formal geometry `a la Bott-Gelfand-Kazhdan, from ideas in deformation quantization, especially after Kontsevich, and from techniques on configuration spaces as developed by Axelrod and Singer, after Fulton and McPherson.This research fits in the renovated interest in the mathematical study of quantum field theories based, on the one hand, on ideas by Batalin and Vilkovisky for a “homotopical approach” to them and, on the other hand, on Segal’s and Atiyah’s ideas on cutting and pasting them in the case of manifolds with boundaries (and corners after Baez-Dolan and Lurie, although only in the topological case). Among the main players in different areas related to this line of research, I may cite (in a clearly not comprehensive list) Calaque, Costello, Felder, Fredenhagen, Getzler, Gwilliam, Kazhdan, Losev, Pantev, Schwarz, Rejzner, Stolz, Teichner, Toën, Vaquier, Vezzosi, in addition of course to my collaborators Mnëv and Reshetikhin.
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