This is the main page for **Math 392C (Applications of Quantum Field Theory to Geometry)**, unique number **54620**.

I am Andy Neitzke; my office is **RLM 9.134**. My office hours are **2-3pm on Monday**, or by appointment.

This course meets TuTh from 9:30am-10:45am, 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, though this situation is improving (even since the last time I taught this course, in fall 2012). 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.

Lecture notes, references and exercises will be compiled into a document here: qft-geometry (last update 28 Nov 2021).

The source is hosted at the Github repository neitzke/qft-geometry .

My hope is that this document can be to some extent collectively authored: I would welcome corrections, contributions, solutions to exercises, etc. The smoothest way of managing contributions would be to use the mechanisms provided by Git and Github. (If it works for the Stacks Project it can work for us!) But I will be happy to take contributions in any form.

I strongly recommend that you do the exercises. It will be difficult to follow the course without doing them. Moreover, some of the computations which I assign as exercises will actually be needed for the following lectures (thus I will be very grateful if at least a few people submit LaTeX solutions, either by email or via Github.)

Notes from a previous iteration of the course are available. The material which we will cover this time should be broadly similar to the content of these notes, but hopefully a bit more precise and with more exercises.

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.

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