Math 392C (Applications of Quantum Field Theory to Geometry), Fall 2017

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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.

Lectures, references, exercises

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.

Additional references

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


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