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 UTC. (Remember to convert to your time zone, taking into account daylight saving time.)

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Homework 1. (Lectures 1–8.)

Handwritten lecture notes.

Video recordings of all lectures.

Lecture 1: Sunday, October 15. Introduction.

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

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

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

Lecture 5: Sunday, November 19. Sheaves.

Lecture 6: Sunday, November 26. Stacks.

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

Lecture 8: Sunday, December 10. Homotopy categories and categories of fractions. Sheaves of sets as a category of fractions.

Lecture 9: Sunday, December 17. Homotopy theory. Simplicial sets. Nerves of categories, classifying spaces of monoids.

Lecture 10: Sunday, January 7. Model categories. Transferred model structures. Model structures on simplicial presheaves.

Lecture 11: Sunday, January 14. Left Bousfield localizations.

Lecture 12: Sunday, January 21. Local model structures on simplicial presheaves.

Lecture 13: Sunday, January 28. Segal objects in cartesian model categories.

Lecture 14: Sunday, February 4. Examples of Segal objects. Categories with finite products. Monoidal categories.

Lecture 15: Sunday, February 11. Examples of Segal objects. Categories of spans. Complete Segal spaces. The Rezk nerve functor.

Lecture 16: Sunday, February 18. The bordism category as a Segal Δ-object.

Lecture 17: Sunday, February 25. Segal Γ-objects.

Lecture 18: Sunday, March 3. Complete n-fold Segal Δ-objects and higher categories.

Lecture 19: Sunday, March 10. Examples of higher categories.

No lecture on Sunday, March 17.

Lecture 20: Sunday, March 24. Higher categories of bordisms. Smooth categories and smooth category theory.

Lecture 21: Sunday, March 31. Smooth higher categories of bordisms. Bordisms with geometric structures.

No lecture on Sunday, April 7.

Lecture 22: Sunday, April 14. Bordisms with isotopies.

Lecture 23: Sunday, April 21. All bordisms are dualizable. All cylinders are invertible. Examples of dualizable and fully dualizable objects.

Lecture 24: Sunday, April 28. Statements of the locality theorem and geometric cobordism hypothesis.

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.