A comic with four panels, each depicting a pair of scientists in front of a blackboard scribbled with equations and diagrams. The first panel depicts a pair of young theoretical physicists, in 1968, joyously discussing physics in front of a board with Feynman diagrams. The second panel depicts a similar scene in 1968 with a pair of young mathematicians, in front of a board with advanced mathematics. The next two panels depict the same people and same boards, but in 1998. Both pairs of people are now old, and the mathematicians stand in front of the board with physics, perplexed, as perplexed as the physicists in front of the board with mathematics.

The Quantization of Gauge Theories

following Kevin Costello

as taught by

John Huerta

Practical information

Course description

Our goal in this course is to explain the combination of effective field theory and the Batalin–Vilkovisky formalism developed by Kevin Costello. This produces a mathematically rigorous, perturbative quantization of gauge theories. Our running example of a gauge theory will be Yang–Mills theory, and we hope to prove the following theorem by the end of the course:

Theorem: Yang–Mills theory is pertubatively renormalizable in 4-dimensions.

This is an advanced course in the theoretical and mathematical physics MSc program, run jointly between LMU and TUM, taught in the Summer Semester 2026.

Where and when?

Lectures: Mondays, 10:00-12:00, 13.04.2026-13.07.2026, in room A450.
Exercises: Wednesdays, 10:30(sharp!)-12:00, 22.04.2026-15.07.2026, in room A348.

Registration

Here is the LSF page.

This week's exercise

23.04.2026

Read Costello, Ch. 2, Sec. 4: Sharp and smooth cut-offs.
What is the heat equation on R^n? What about on a Riemannian manifold? What is the heat kernel?

You can use any reference for these questions; one reference is the book by Berline, Getzler and Vergne.

Lecture notes

Coming soon! In the meantime, you can see a draft version of notes from a previous version of this course:

Notes for the BV track by Rui Peixoto and Björn Gohla, from talks by John Huerta unless noted otherwise.

Schedule

~~13 April: Overview of the course; history of the BV formalism; Wick's lemma.~~
~~15 April: Cancelled.~~
~~20 April: Finite-dimensional Feynman diagrams.~~
23 April: Read Costello, Ch. 2, Sec. 4: Sharp and smooth cut-offs. Answer the following questions: What is the heat equation on R^n? What about on a Riemannian manifold? What is the heat kernel in each case?

References

Kevin Costello, Renormalisation and the Batalin–Vilkovisky formalism, arXiv:0706.1533.
Kevin Costello, Renormalization and effective field theory, American Mathematical Society, Providence, 2011. Draft version.
Kevin Costello and Owen Gwilliam, Factorization algebras and quantum field theory, vol 2, Cambridge University Press, Cambridge, 2021. Draft version.
Chris Elliott, Brian Williams, Philsang Yoo, Asymptotic freedom in the BV formalism, J. Geom. Phys. 123 (2018), 246–283. arXiv:1702.05973.
Owen Gwilliam, Factorization algebras and free field theories, PhD thesis, Department of Mathematics, Northwestern University, 2012.
Pavel Mnev, Quantum field theory: Batalin-Vilkovisky formalism and its applications, American Mathematical Society, Providence, 2019. (Based on lecture notes available as arXiv:1707.08096.)