This is the page for the Winter 2016 Semester working seminar on topological field theory. In this seminar we will discuss 1,2,3, and 4-dimensional topological field theories, their classification, and the emerging tool of factorization homology.
Unless otherwise specified, talks take place in JCMB 4312.
Calendar of talks:
References:
Lecture notes for the working seminar, compiled by students: lecture_notes_tft_seminar.pdf
Before cobordism hypothesis:
- B. Bartlett. Master's thesis. http://arxiv.org/pdf/math/0512103v1.pdf This gives a nice survey of much of the material to be covered in the first half of the seminar.
- J. Abrams. Two dimensional topological field theories and Frobenius algebras. http://home.gwu.edu/~labrams/docs/tqft.ps This proves rigorously the folklore result that fully extended 2D TFT's are in (functorial) bijection with finite-dimensional semi-simple Frobenius algebras. It involves only 2-categories, not "higher" categories in the more technical sense.
- C. Schommer-Pries's PhD thesis http://arxiv.org/abs/1112.1000 This proves the classification of 2-dimensional TFT's directly, in the language of 2-categories, and without invoking the cobordism hypothesis. It gives a "presentation" by "generators and relations" of the fully extended 2-dimensional cobordism category. It presages an alternative approach to Lurie's, in proving the cobordism hypothesis in low dimensions (which are anyways the most interesting).
- Bakalov-Kirrilov, Lectures on Tensor categories and Modular functors. http://www.math.stonybrook.edu/~kirillov/tensor/tensor.html This is a great reference of the state of the art for when it is written (2000). It has many of the algebraic constructions such as braided tensor categories, modular tensor categories, modular functors, and an explicit technique to compute Witten-Reshetikhin-Turaev invariants of 3-manifolds using a Haegard splitting.
- K. Walker. On Witten's 3-manifold invariants http://canyon23.net/math/1991TQFTNotes.pdf and TQFT's http://canyon23.net/math/tc.pdf. While informal and never completed for publication, these notes not only summarize the state of the art as of the 90's of TFT's, they presage many of the ideas that later went into factorization homology.
- E. Witten. Quantum Field Theory and the Jones polynomial https://projecteuclid.org/euclid.cmp/1104178138 This paper gives Witten's answer to Atiyah's challenge: Construct the Jones polynomial in a manifestly 3-dimensional way, using ideas from quantum field theory. The paper is a fascinating read, even today, and helps to build a dictionary between QFT and TFT.
Quantum groups
- C. Kassel. Quantum Groups. (Not online, that I know of.) This is a really clear and well-written textbook, which explains the graphical calculus for working with ribbon categories, the construction of quantum SL(2) in detail. This also explains the connection between quantum group constructions and the Jones polynomial, which is a central motivator.
- P. Tingly. A minus sign that used to annoy me but now I know why it is there.http://arxiv.org/abs/1002.0555 This paper explains the ribbon structure on the quantum group and how one can recover the Kaufman bracket and Jones polynomial from its category of representations.
- Chari and Pressley. A Guide to Quantum Groups. Chapter 11 gives a reasonably clear description of the representation theory of quantum groups at roots of unity.
Cobordism hypothesis, higher categories, and factorization homology:
- J. Lurie. On the Classification of Topological Field Theories. http://www.math.harvard.edu/~lurie/papers/cobordism.pdf More or less the standard reference for the cobordism hypothesis. Covers a lot of material, including a sketch proof of the cobordism hypothesis.
- D. Calaque and C. Scheimbauer. A note on the (infinity,n)-category of cobordisms. http://arxiv.org/pdf/1509.08906.pdf Covers Segal spaces and the definition of the categories Bord_n in detail.
- C. Scheimbauer. PhD thesis. http://guests.mpim-bonn.mpg.de/scheimbauer/ScheimbauerThesisJan23.pdf Overlaps with the Calaque and Scheimbauer note on Segal spaces and the bordism categories. Also contains material on factorisation homology.
- ...
Lecture notes:
- David Jordan. Introduction. TeXed notes: TQFT_lecture_1 Scanned notes: Lec1-scan.pdf
- Juliet Cooke. TFT's in dimensions 1 and 2. TQFTlect2
- Jenny August. Quantum Groups and Knot Invariants. TFTLectures34
- Noah White. Invariants of 3-manifolds, 3d TFTs and Quantum Groups. TFT lecture 5 - Noah
- Matt Booth. Higher Categories, Complete Segal Spaces, and the Cobordism Hypothesis. TFTs_lectures67.pdf
- Tim Weelinck. Introduction to Factorization Homology. TFT_Lecture_8.pdf
Schedule of talks:
- Week 1: Introduction and overview. Speaker: David Jordan. Note-taker: Juliet Cooke
- Week 2: Classifications in dimensions one and two. Speaker: Juliet Cooke. Note-taker: Juliet Cooke
- Week 3: Introduction to quantum groups, especially SL2. Speaker: Jenny August. Note-taker: Jenny August.
- Week 4: ....