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

EXAM INFO! EXAM INFO! EXAM INFO!

Exam time and location: 2 PM - 5 PM, Wednesday, 29 July, A449. (Room change!)

Format: The exam will be an oral exam. It will last about 20 minutes. You should prepare a few remarks on something you learned in the course, which can be anything (a definition, an idea, a theorem, a computation), and our discussion will go from there.

I will email you on the morning of the exam with your precise timeslot in the interval 2 PM - 5 PM.

If you haven't already done so, please email me to let me know you want to take the exam: john.huerta@lmu.de

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.

Lecture notes

Here are the notes for our final lectures, including the Zoom lectures:

Notes for 30 June and 1 July.
Notes for 8, 10, 13, and 15 July.

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.

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.)