This is the main page for **Math 392C (Applications of Quantum Field Theory to Geometry)**, unique number **56135**.

I am Andy Neitzke; my office is RLM 9.134. My office hours are Friday 3-4pm.

This course meets MWF from 11am-12pm, in RLM 10.176.

Quantum field theory has found numerous applications to mathematics and particularly to geometry over the last few decades. A particularly significant example is the relationship between Donaldson and Seiberg-Witten invariants, which revolutionized 4-manifold topology in the mid-1990's. In this course I will attempt to give an account of what this relationship is and the physical picture underlying it. This will require us to develop a fair amount of intuition about (four-dimensional, supersymmetric) quantum field theory, and in particular about the notion of "effective" field theory, which in one way or another is underlying many of the deepest applications of quantum field theory to mathematics.

Many elements of the physical picture have not been made into rigorous mathematics yet. It follows that the ratio of theorems to ideas in this course will be relatively low (though I will try to make it as high as practicable). I hope to make the presentation accessible to those without previous exposure to quantum field theory (but some independent reading may be required at points). Some familiarity with quantum mechanics would help to make the learning curve shallower. On the geometric side, basic differential topology and differential geometry will be helpful.

We will start with a general overview and then try to fill in as many of the details as practicable. The first step will be to study quantum field theories in zero dimensions and in one dimension. Already here we will be able to see many of the basic phenomena of interest. Then depending on how things are going, we may go to two dimensions or we may jump directly to four.

- 01: Exercises (last update: 17 May 2021)

Below are my notes from the lectures so far and perhaps slightly into the future. The mapping between files and lectures is not necessarily 1-1. The notes will be updated to correct errors/omissions where they are pointed out or where I notice them later. In practice, so far this policy seems to require a lot of updating. I apologize for whatever errors still remain.

- 01.0: Introduction (last update: 17 May 2021)
- 02.0: QFT in zero dimensions (last update: 17 May 2021)
- 03.0: Fermions (last update: 17 May 2021)
- 04.0: Duistermaat-Heckman formula (last update: 17 May 2021)
- 05.0: Quantum mechanics (last update: 17 May 2021)
- 05.5: Canonical quantization (last update: 17 May 2021)
- 06.0: Harmonic oscillator (last update: 17 May 2021)
- 06.3: Effective quantum mechanics (last update: 17 May 2021)
- 06.7: Clifford algebras (last update: 17 May 2021)
- 07.0: Supersymmetric quantum mechanics (last update: 17 May 2021)
- 08.0: The free boson in two dimensions (last update: 17 May 2021)
- 08.7: Scalar field theory in four dimensions (last update: 17 May 2021)
- 09.0: Abelian gauge theory in four dimensions (last update: 17 May 2021)
- 10.0: Nonabelian gauge theory in four dimensions (last update: 17 May 2021)
- 10.5: Higgs mechanism (last update: 17 May 2021)
- 10.8: Coupling to fermions (last update: 17 May 2021)
- 10.9: Very brief comments on Standard Model (last update: 17 May 2021)
- 11.0: Supersymmetric gauge theory (last update: 17 May 2021)
- 12.0: Twisting (last update: 17 May 2021)
- 12.5: Anomalies (last update: 17 May 2021)
- 13.0: Computing observables (last update: 17 May 2021)
- 14.0: The Seiberg-Witten solution (last update: 17 May 2021)
- 15.0: Mass and the BPS bound (last update: 17 May 2021)
- 16.0: BPS monopoles (last update: 17 May 2021)
- 17.0: Seiberg-Witten equations and Witten's magic formula (last update: 17 May 2021)

Some likely useful references (this list will probably grow as the semester goes on):

- Mirror Symmetry.
Vafa and Zaslow, editors.
Don't be put off by the title: this book contains a lot of stuff that is useful even if our aim is not to understand mirror symmetry. Especially, it contains some nice introductory material on quantum field theory and some of the special features that arise when the theory is supersymmetric. See particularly Chapters 8 and 9 which discuss field theories in zero and one dimension; I expect we will follow the presentation here fairly closely in the beginning of the course.

- Five Lectures on Supersymmetry.
Dan Freed.
This is a nice and geometric introduction to many of the aspects of supergeometry that we will need, paying particular attention to systematic ways of thinking about some of the tricky sign issues that we may gloss over.

- Construction of Wiener measure (PDF).
Found on the web page of Michael E. Taylor; it's a nice no-fuss account of the story.

- Quantum field theory. Folland.
This is an account of the standard textbook material on quantum field theory, written in language that is meant to be congenial to a mathematical reader. (It thus differs from books with titles like "Quantum field theory for mathematicians" which try to rigorize the theory but usually do not get very far.) I do not expect to follow this book closely but it may be useful for orientation and context.

- Renormalization and effective Lagrangians. Polchinski.
This paper which sets out carefully the picture I tried to explain in class, of how the irrelevant couplings are damped out by the renormalization group flow. It treats specifically the example of scalar field theory in 4 dimensions, but the general philosophy is supposed to be much broader.

- Beta functions and the exact renormalization group.Hughes and Liu.
This is the first paper to calculate the beta function of the quartic coupling in scalar field theory in 4 dimensions within the formalism of the "exact renormalization group" (which I would just call the "renormalization group" -- anyway, it is the approach I have sketched in the lectures).

- Topological quantum field theory and four manifolds.
Labastida and Marino.
This book is intended to give a self-contained account of the physics of the relation between Donaldson and Seiberg-Witten theory, in sufficient detail that one can actually determine the precise formula relating the two, while at the same time being self-contained and readable for mathematicians or physicists, and also staying reasonably concise. I think it does a very good job considering the significant tension between these three constraints. In particular it treats the whole subject with a single unified set of conventions, which is valuable. I expect to use it as the main reference for detailed formulas once we get into four-dimensional gauge theory.

- The geometry of four-manifolds. Donaldson and Kronheimer.
This is a fundamental reference on the definition and basic properties of the Donaldson invariants. I intend to learn a lot from it over the course of the semester, and urge you to do the same.

- The Seiberg-Witten equations and applications to the topology of smooth four-manifolds. John Morgan.
This is a basic mathematical reference on the Seiberg-Witten invariants, written very shortly after they first appeared.

- Scaling and renormalization in statistical physics. Cardy.
This is a more advanced text that explains the ubiquitous phenomena of "scaling" and "universality" in many-body systems near critical points. The intuition that one develops here is very useful also in quantum field theory. This book is probably better suited to readers who have some QFT background already; I don't think we will get to use it too directly, although I will sneak something in from it if I possibly can.

I will assign a few exercises; I strongly encourage you to do them. Some exercises will be mentioned during lecture, but I also will post slightly more organized exercise sheets. (In my graduate course last year I aimed to post one sheet per week but wound up with only 8; I would expect something similar this time.)

If you need a grade for the course, you should either turn in (at least) 1 problem from each exercise sheet, or write a short essay about some topic related to the course that interests you. I can suggest topics on request. The due date for all assignments is May 9.

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.