Math 392C (Applications of Quantum Field Theory to Geometry), Spring 2012

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.


Lecture notes

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.


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


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.