Topological Quantum Field Theory

Lecturer: Minhyong Kim

Times: Fridays from 11:00 to 13:00 for eight weeks starting on 22 January, 2021

This course is hosted by the Taught Course Centre and is available to graduate students at the Universities of Bath, Bristol, Oxford, Warwick and Imperial College London. Instructions for registering are on the TCC webpage.

Synopsis

Topological quantum field theories (TQFTs) are quantum field theories in which the evolution of the quantum states of fields on space from one time to another depends only on the topology of the intervening spacetime. For physicists, topological theories arise mostly as 'topological sectors' associated to full quantum field theories that carry operators of square zero with respect to which one can take the cohomology of the space of states. There are also some theories, such as Chern-Simons theories or BF theories, in which the action functional on fields is itself topological.

In this course, we will not consider almost at all the physical origins other than as inspiration. In fact, we will not even be very rigorous about the topological part of the theory. Rather, we will examine the formalism of TQFT from the point of view of an algebraist with eventual applications to arithmetic geometry at the back of our minds. In particular, no prerequisite from physics is necessary. However, some familiarity with manifolds, basic algebraic structures, and the language of category theory will be convenient.

The main technical goal of this course will be to obtain a concrete intuition for TQFTs by focusing on known facts about theories in dimensions 1, 2 and 3. The course will give a reasonably complete outline of the classification of TQFTs in dimensions 1 and 2 (including so-called extended TQFTs) before moving on to the construction of 3 dimensional TQFTs associated to modular tensor categories, on which the bulk of time will be spent.

For an algebraist, these developments indicate the way in which geometry can underlie algebra. In the 1960s, the scheme-theoretic foundations of algebraic geometry made it plausible that algebra is somehow more fundamental than geometry, as it became possible to progressively abstract the notion of a space. The developments in TQFT, on the other other hand, seem to indicate that various types of algebraic operations are themselves reliant on an underlying physical geometry. We will try to understand this interaction between geometry and algebra by way of examples.

Lecture Notes

(Updated 20 February, 2021)


Preliminary version Will be periodically expanded, corrected, and hopefully improved.

Preliminary attempt at figures

Assessment

Projects

References

Atiyah, Michael Topological Quantum Field Theories. Inst. Hautes Etudes Sci. Publ. Math. No. 68 (1988), 175--186.

Bakalov, Bojko. Kirillov, Alexander Jr. Lectures on tensor categories and modular functors. University Lecture Series, 21. American Mathematical Society, Providence, RI, 2001. x+221 pp.

Freed, Daniel, S. Lectures on Field Theory and Topology. Published for the Conference Board of the Mathematical Sciences. CBMS Regional Conference Series in Mathematics, 133. American Mathematical Society, Providence, RI, 2019. xi+186 pp.

Kock, Joachim Frobenius algebras and 2D topological quantum field theories. London Mathematical Society Student Texts, 59. Cambridge University Press, Cambridge, 2004.

Lurie, Jacob On the classification of topological field theories. Current developments in mathematics, 2008, 129-280, Int. Press, Somerville, MA, 2009.

Witten, Edward Quantum field theory and the Jones polynomial Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399.

Aspinwall, Paul S. et. al. Dirichlet branes and mirror symmetry. Clay Mathematics Monographs, 4. American Mathematical Society, Providence, RI, 2009. x+681 pp.