Dmitri Pavlov's Lectures on Functorial Field Theory

This is an online Zoom class on functorial field theory. Meeting ID: 963 0871 5590. For password, email me or ask one of the participants.

The initial goal is to understand the statement of the geometric cobordism hypothesis, including the relevant background material in homotopy theory. Additional goals include constructing elementary examples of functorial field theories using GCH. Further goals include looking at more advanced examples, including geometric factorization homology.

Time: Sunday, 15:00–17:00 UTC. (Remember to convert to your time zone.)

Handwritten lecture notes.

Lecture 1: Sunday, October 15. Introduction. Video.

Lecture 2: Sunday, October 22. Functor of points: smooth sets. Video.

Lecture 3: Sunday, November 5. Functor of points: smooth sets, C^∞-rings. Video.

Lecture 4: Sunday, November 12. Functor of points: C^∞-rings. Video.

Lecture 5: Sunday, November 19. Sheaves. Video.

Lecture 6: Sunday, November 26. Stacks. Video.

Lecture 7: Sunday, December 3. Čech nerves and cocycle data.

References:

• [1]: Daniel Grady, Dmitri Pavlov: Extended field theories are local and have classifying spaces.
• [2]: Daniel Grady, Dmitri Pavlov: The geometric cobordism hypothesis.

Preliminary syllabus

Homotopy theory

• Simplicial homotopy theory, including simplicial sets, simplicial maps, simplicial weak equivalences, Quillen's Theorem~A, Kan's Ex^∞ functor, and the simplicial Whitehead theorem. References: My notes, Goerss–Jardine, Dugger–Isaksen.
• Simplicial categories, Dwyer–Kan weak equivalences, the homotopy coherent nerve functor and its left adjoint. References: Bergner [Sections 4 and 7].
• Model categories, including model structures, left Quillen functors, injective and projective model structures on presheaves with values in a model category, Reedy model structures, combinatorial model categories, left Bousfield localizations. References: Hovey, Hirschhorn, Barwick's paper, as well as Lurie [HTT, Appendix A] and the survey of Balchin.
• Homotopy limits and colimits. References: Bousfield–Kan, Hirschhorn, Shulman, Riehl.
• Segal's Γ-objects. References: Segal, Bousfield–Friedlander, Schwede.
• Rezk's complete Segal spaces and their generalization to n-fold Segal spaces by Barwick. References: Rezk, Barwick–Schommer-Pries, Bergner–Rezk.
• Simplicial presheaves and descent. References: Dugger–Hollander–Isaksen and Jardine.
• Diffeological spaces and smooth sets. References: Iglesias-Zemmour, [1].
• Elementary Morse theory. References: Milnor's Morse Theory.

Higher category theory

• Multiple and globular (∞,d)-categories.
• Smooth multiple and globular (∞,d)-categories.
• Smooth symmetric monoidal (∞,d)-categories.
• Smooth symmetric monoidal (∞,d)-categories with duals.
• Functor categories.
• Categories with isotopies.

Smooth bordism categories

• A survey of the existing bordism categories and their deficiencies.
• Geometric structures.
• Cuts, cut tuples, and cut grids.
• Categories of bordisms.
• Categories of bordisms with geometric structure.
• Categories of bordisms with isotopies and geometric structure.
• Axioms for bordism categories.

Locality and codescent

The geometric cobordism hypothesis

Examples

Additional sources

Stephan Stolz, Peter Teichner. Supersymmetric field theories and generalized cohomology.

Stephan Stolz, Peter Teichner. What is an elliptic object?

Henning Hohnhold, Matthias Kreck, Stephan Stolz, Peter Teichner. Differential forms and 0-dimensional super symmetric field theories.

Henning Hohnhold, Stephan Stolz, Peter Teichner. From minimal geodesics to supersymmetric field theories.

Daniel Berwick-Evans.

• The families Clifford index and differential KO-theory
• How do field theories detect the torsion in topological modular forms?
• The families analytic index for 1|1-dimensional Euclidean field theories

Elke Market. Field theory configuration spaces for connective ko-theory.

Corbett Redden. String structures and canonical 3-forms.

Florin Dumitrescu.

• Superconnections and Parallel Transport.
• Addendum to "Superconnections and Parallel Transport".
• Connections and Parallel Transport.
• 1|1 Parallel Transport and Connections.
• A new look at connections.
• A geometric view of the Chern character.

Peter Teichner. Elliptic cohomology via conformal field theory. (Lecture notes for a 2007 class at Berkeley.)

Fei Han. Supersymmetric QFT, Super Loop Spaces and Bismut-Chern Character.

Stephan Stolz, Peter Teichner. Traces in monoidal categories.