A mathematical guide to fully local topological field theories and relatives

by Prof. Claudia Scheimbauer (Technische Universität München)

Tuesday 28 Nov 2023, 17:00 → 18:00 Europe/Berlin

H4 (Geom)

In the past 15 years the celebrated Cobordism Hypothesis has given us new tools to obtain fully local topological field theories, albeit in a non-constructive way, and doesn’t give a recipe on how to extend bordism invariants or partition functions to general bordisms. On the other hand, an explicit construction often is more desirable for computations. I will give a tour on the mathematical description of fully local topological field theories using higher categories (in an informal and black box way) and give a glimpse on the challenges in using, or not using, the Cobordism Hypothesis. At the end I will give an outlook on relative (field theorie)s, which are central in a modern approach to describe symmetries.

15:00 - 15:45 - Pre-talk für interessierte MSc / PhD Studenten und Postdocs:

A guide to the guide — finite homotopy theories
 

In this talk I will focus on a family of fully local topological field theories which is often used as a symmetry TFT: finite homotopy theories. They generalize Dijkgraaf-Witten theory (using BG) and can be obtained via a finite path integral procedure. Mathematically, this uses centrally the notion of (higher) semiadditivity of (higher) categories (Hopkins-Lurie, Harpaz). Semiadditivity recently has had prominent applications in chromatic homotopy theory, leading to the answer to two of the main conjectures in the field (Carmeli-Schlank-Yanowski, with Barthel, Ben-Moshe). I will focus on this example to illustrate the ideas which will appear in the Kolloquium. This is joint work with Tashi Walde.