TQFTs and Gauge theory seminar

Jan 2025

talk notes seminar

Summary:

Learning seminar on extended TQFTS and how to gauge them, Building up to Teleman’s 2014 ICM address

Presented at:

UC Berkeley, spring 2025

Gauging TQFTs reading seminar

Spacetime coordinates: Tuesdays 4-5 pm, western time, Evans 891
Zoom Information: https://berkeley.zoom.us/j/94584838842

Schedule

Tentatively:

Date	Speaker	Topic	Notes
Week 1	Elliot , Zechen	Hyroglyphics of TQFTS + organization	Link
Week 2	Chris	Extended TQFTs 1
Week 3		Extended TQFTs 2
Week 4	Zechen	Extended TQFTs 3
Week 5	Elliot	Topological group actions on categories 1
Week 6		Topological group actions on categories 2
Week 7	Jacob	Gauging topological theories
3/28		Spring break!
Week 8)		Abelian gauge theories 1
Week 9		Abelian gauge theories 2 (Examples)
4/19		Nonabelian case: Teleman’s conjecture
4/26		Relationships with Coloumb branches

Description

We are trying to understand Teleman’s 2014 ICM address. This provides a conjectural picture organizing categorical representations of groups using the framework of 2 and 3 dimensional mirror symmetry. It’s notorious for being confusing, but we’ll try our best. We’ll start with understanding TQFTs, in their fully extended sense, defined using higher categories. (The seminar would still be a success if we understood this!) Then, we’ll move on to gauging, understanding this by group actions on categories. Finally, we’ll try to understand Teleman’s hiroglyphics:

This is a picture of the Toda space of a group, a holomorphic symplectic manifold built from the group. The KRS two-category should encode and organize topological actions of this group. Each of these objects are bundles of categories over lagrangians in this space.

If we understand this, then we have a geometric picture of categorical representations!

Resources

Extended TQFTs:

Topological group actions on categories”

Nadler and Zaslow, Constructible Sheaves and the Fukaya Category
Seidel, pi 1 of symplectic automorphism groups and invertibles in quantum homology rings

Gauging topological theories:

Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups

KRS 2-category:

Abelian Gauging:

Teleman, Gauge theory and mirror symmetry
Seidel, pi 1 of symplectic automorphism groups and invertibles in quantum homology rings. This doesn’t use the same language as Teleman, but aparently Teleman’s perspective can be easily extracted.

Full Nonabelian theory:

Teleman, Gauge theory and mirror symmetry
- ICM address, lays out the story
Teleman, The rˆole of Coulomb branches in 2D gauge theory
- Extends the ICM picture to include gauge theories with matter, also known as coloumb branchs. Ties it into more usual 3D mirror symmetry
Teleman, Coulomb branches for quaternionic representations
- Extends the last paper from matter of the form $T^* V$ to general quaternionic vector spaces
Teleman, Pomerleano, Quantization commutes with reduction again: the quantum GIT conjecture I
- Extracts and proves a conjecture from the ICM picture, relating quantum cohomology of symplectic reductions and equivariant quantum cohomology. Proved entirely in Floer theory. **

Talk details

Week.1

Speaker: Elliot Kienzle, Zechen Bian
Topic: Hieroglyphics for extended TQFTs
Notes: Link to PDF
Summary: Elliot gave a schematic introduction to TQFTs and how they give rise to higher categories. Fairly physically minded, I start with 1 dimensional TQFTs ( Quantum mechanics), and build up the pictures for 2 and 3 dimensional TQFTs. This follows section 2.1-2.2 of Topological Field Theory, Higher Categories, and Their Applications by Kapustin. After this, Zechen gave an plan for the rest of the semester.