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:

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:

• Jacob Lurie, On the classification of topological field theories
  • Sketches proof of the cobordism hypothesis
• Maxim Kontsevich, Yan Soibelman, Notes on A-infinity algebras, A-infinity categories and non-commutative geometry
• Kevin Costello, Topological conformal field theories and Calabi-Yau categories
• Witten, Topological quantum field theories
• Kapustin, Topological Field Theory, Higher Categories, and Their Application
  • Provides a physical perspective on extendted 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.

Week.2

Speaker: Chris Li
Topic: Extended TQFTs
Notes: none :(
Summary: Discussed the Atiyah-style definition of a TQFT. Gave examples in 1D and 2D TQFTs. Introudced the Lurie-style notion of a TQFT

Week.3

Speaker: Langwen Hui
Topic: Cobordism hypothesis
Notes: slides , annotated
Summary: Discussed the framed bordism category, and tangental structures / G-actions on the bordism category.

Week.4

Speaker: Zechen Bian
Topic: Dijkgraaf-Witten theory
Notes:
Summary: Introduced quantum field theory as a path integral. Gave Chern-Simons theory as an example of a physics TQFT. Discussed the analogue of Chern-Simons theory with a finite gauge group, known as Dijkgraaf-Witten theory. Described the TQFT structure in terms of counts of $G$ bundles for $G$ a finite group, and discussed the twisted case.

Week.5

Speaker: Zechen Bian
Topic: Dijkgraaf-Witten theory part 2
Notes:
Summary: Finished up the twisted digraff-witten theory computation from last time. Discussed Topological sigma models with arbitrary targets. Follows Freed Teleman Lurie Hopkins, TOPOLOGICAL QUANTUM FIELD THEORIES FROM COMPACT LIE GROUPS

Week.6

Speaker: Elliot Kienzle
Topic: A-model TQFT and the Seidel representation
Notes: Link to PDF
Summary: I define the TQFT associated to a symplectic manifold, coming from the A-twisted topological sigma model. On surfaces, these produce invariants called gromov-witten invariants. On a circle, this gives quantum cohomology. I describe an action of loops of hamiltonian diffeomorphisms on quantum cohomology, known as the Seidel representation

Week.7

Speaker: Jacob Erlikhman Topic: group actions on Fukaya categories, part 1
Notes:
Summary: Discussed Fukaya categories and wrapped Fukaya categories. Describe the algebra of chains $C^(G)$, and the algebra of cochains $C_(\Omega G)$, and how they are Kouzul dual to one another.

Week.9

Speaker:
Topic:
Notes:
Summary:

Week.10

Speaker:
Topic:
Notes:
Summary: