Important: Due to Covid-19 the first three weeks of the program have been shifted to online events.
The programme is devoted to the fruitful interaction between the theory of higher structures and mathematical approaches to field theory. This interaction goes in both directions.
On the one hand, field theory constantly inspires new developments in almost all fields of mathematics. And higher stuctures including the modern homotopy theory and supergeometry are among the first recepients of these ideas. There is also a renewed interest in mathematical approaches to field theory including algebraic perturbative quantum field theory, factorization homology etc.
On the other hand, many tools of the higher stucture theory naturally enter the language of quantum field theory. Among the examples are A-infinity and L-infinity stuctures, Batalin-Vilkovisky structures (first discovered in physics, they made their home in mathematics, and are now back on the physics arena) and many others. An impressive recent development is the use of higher structures in the classification of states of matter.
The programme will bring together specialists in higher structures and classical and quantum field theory to create more interaction on these exciting topics. Among the highlights of the programme are a focus week on the use of higher structures for a classification of states of, what is called, topological matter (during week 1), a workshop on higher structures (week 3), a focus week on the interaction between quantum field theory and higher structures (week 5), and a focus week on supergeometry and gauge theory (week 7). Among the programme events there will be mini-courses on perturbative algebraic quantum field theory, BV-methods, factorization algebras, and supergeometry (see below for a detailed description).
Overview:
Week 1 (August 3 - 7, 2020) Focus week on Topological Matter - online, have a look at "Associated Events"
Week 2 (August 10 - 14, 2020) Mini-course on The Poisson sigma model and integrable systems (N. Reshetikhin) - online, have a look at "Associated Events"
Week 3 (August 17 - 21, 2020) Workshop on Higher Structures - online, have a look at "Associated Events"
Week 4 (August 24 - 28, 2020) Mini-course on Advances in Algebraic Quantum Field Theory (K. Fredenhagen)
Week 5 (August 31 - September 4, 2020) Focus week on Higher Structures in Quantum field theory
Week 5 (August 31 - September 4, 2020) Mini-course on Higher structures in algebraic quantum field theory (A. Schenkel)
Week 6 (September 7 - 11, 2020) Mini-course on Courant algebroids, generalized Ricci flow, and T-duality (P. Severa)
Week 7 (September 14 - 18, 2020) Focus week on Supergeometry and Gauge Theory
Week 7 (September 14 - 18, 2020) Mini-course on Geometry of Q-manifolds and Gauge Theories (A. Kotov)
Details of the courses:
Week 2: N. Reshetikhin: The Poisson sigma model and integrable systems
The goal of the mini-course is to explain the relation between the semiclassical quantization of the Poisson sigma model and semiclassical amplitudes in quantum integrable systems.
Lecture 1: We recall the notion of quantum integrable systems and review the semiclassical asymptotics of joint eigenfunctions.
Lecture 2: For two integrable systems, we rewrite the semiclassical asymptotics of the scalar product of joint eigenfunctions of quantum integrals as a path integral of the Poisson sigma model with special boundary conditions.
Lecture 3: We will survey the BV-BFV quantization of the Poisson sigma model and its relation to the Kontsevich star product. This will then be used to find the semiclassical expansion to all orders of eigenfunctions of quantum integrable systems.
Week 4: K. Fredenhagen: Advances in Algebraic Quantum Field Theory
After an introduction into the formalism of algebraic quantum field theory recent developments will be discussed, in particular the relation to perturbative quantum field theory, to quantum gravity and to conceptual questions as entanglement and black hole evaporation.
Lecture 1: Framework of AQFT
Tuesday, August 25, 9:00 a.m.
Lecture 2: Perturbative approach towards AQFT
Thursday, August 27, 9:00 a.m.
Lecture 3: Perspectives and open problems
Friday, August 28, 9:00 a.m.
Week 5: A. Schenkel: Higher structures in algebraic quantum field theory
Algebraic quantum field theory (AQFT) is a well-established framework to axiomatize and study quantum field theories on Lorentzian manifolds, i.e. spacetimes in the sense of Einstein’s theory of general relativity. In this mini course, I will give a brief introduction to AQFT and its recent higher categorical generalizations. This will cover the following topics:
Lecture 1: Operads and universal constructions in AQFT;
Monday, August 31, 10:00 a.m.
Lecture 2: Local-to-global properties of quantum gauge theories;
Tuesday, September 1, 9:00 a.m.
Lecture 3: Construction of simple examples by homological techniques. Throughout the course, I will introduce the relevant mathematical techniques from operad theory and homological algebra.
Thursday, September 3, 9:00 a.m.
Week 6: P. Severa: Courant algebroids, generalized Ricci flow, and T-duality
The aim of these lectures is to give an introduction to Courant algebroids (AKA generalized geometry), how they can be used to formulate T-duality known from String theory, and how their geometry implies that T-duality is compatible with the Ricci flow and with the string background equations. The main objects we shall study are Courant algebroids, their reductions and pullbacks, Dirac generating operators, and generalized metrics and the corresponding Ricci flows and Laplacians.
Lecture 1: Courant algebroids (CAs), classification of exact CAs, generalized metrics, pullbacks and reductions, Poisson-Lie T-duality
Monday, September 7, 10:00 a.m.
Lecture 2: divergences, generalized Ricci flow, generalized Laplacian, T-duality is compatible with the Ricci flow and the (bosonic or type I / heterotic) string background equations
Tuesday, September 8, 9:00 a.m.
Lecture 3: CAs in terms of symplectic dg manifolds and generating Dirac operators, T-duality and type II SUGRA
Thursday, September 10, 9:00 a.m.
Week 7: A. Kotov: Geometry of Q-manifolds and Gauge Theories
The focus of this mini-course lies on applications of differential graded (DG) geometry to gauge theories. The underlying grading of the dg manifolds (sometimes also called Q-manifolds) can be valued in Z,N, or Z_2, yielding different notions of super manifolds. Such a manifold is equipped with a degree one, and thus odd, vector field Q which squares to zero.
The gauge theories describe scalar fields coupled to a tower of differential form gauge fields. They are of a generalized Yang-Mills type (in the physically interesting cases) or of topological nature (like in the AKSZ model).
Lecture 1. Basics about the category of dg-manifolds with various important examples such as Lie algebras, Lie algebroids, and Lie infinity algebroids.
Monday, September 14, 10:00 a.m.
Lecture 2. We will introduce the notion of (compatible) G-structures on Q-manifolds. Furthermore, we will discuss characteristic classes in bundles in the category of Q-manifolds and their relation to the topological AKSZ sigma model, which generalizes important examples such as the Chern-Simons gauge theory or the Poisson sigma model to arbitrary dimensions.
Tuesday, September 15, 9:00 a.m.
Lecture 3. We show how to apply these techniques to the construction of Yang-Mills type higher gauge theories, with and without additional scalar fields. Particular attention will be paid to curved Yang-Mills-Higgs gauge theories, which constitute a non-trivial deformation of the gauge theoretical setting used in the current standard model of particle physics. While the standard theory of scalar fields coupled to 1-form gauge fields is governed by Lie groups or algebras and their representations, the more general theory is goverend by quadratic Lie algebroids, the geometry of which will be highligted in this course as well.
Thursday, September 17, 9:00 a.m.