Math 131.8

Quantum Field Theory Seminar

Fall 2004 TIME:
TALKS :

1. QUANTUM FIELD THEORY IN DIMENSION ZERO, Ivan Mirkovic,
2. RIGOROUS TREATMENT OF FEYNMAN DIAGRAMS, Pavel Etingof (colloquium talk),
3. FEYNMAN CALCULUS and APPLICATION, Paul Gunnells,
4. A NOT-SO-SHORT PROOF of FEYNMAN"S PERTURBATION EXPANSION, Ivan Mirkovic,
5. An APPLICATION of FEYNMAN CALCULUS II: MATRIX INTEGRALS, Paul Gunnells,
6. SUPER MATHEMATICS, Ivan Mirkovic,
7. INTEGRALS ON ODD VECTOR SPACES, Ivan Mirkovic,
8. SUSY (SUPERSYMMETRY) of INTEGRALS IN DIMENSION ZERO, Ivan Mirkovic,

SEMINAR NOTES (evolving):

1. Overview and Integrals in dimension 0 (Ivan)
2. Feynman calculus (Pavel Etingof) The notes contain a detailed proof of Feynman's perturbation expansion of the partition function (and its logarithm), for an infinitesimal perturbation of the free (i.e., quadratic) action.
3. Application of matrix integrals to the moduli of curves (Paul Gunnels) -- To appear.
4. Super math notes (old) Sections 1-4 largely follow Deligne-Morgan notes on Bernstein lectures. The rest are various remarks and speculations. The ``super math'' is the background for the ``super'' part of supersymmetry (the remaining ingredient are spinors).
5. Lecture notes on String Theory, (Ivan Mirkovic) The purpose of this introductory course was to get used to physics ideas. So, these (half-revised) notes start at zero and do not go very far, but most of it should be user friendly. The contents:\ ideas of Classical Mechanics, passage to Quantum Mechanics, a bit of Super-Symmetric quantum mechanics (no Quantum Field Theory!). Free strings: classical solutions and quantization by Polyakov path integral and via a Hilbert space. A representation theory view on the Hilbert space quantization. No exciting contemporary String Theory.

GOALS:

This seminar is geared towards mathematical explanations and applications of QFT (in particular the String Theory): learn what QFT can do for mathematicians.
The intention is to have an educational seminar with much discussion. It should be accessible to wide audience and there should be no prerequisites. The initial stages of QFT are elementary (if deep). Say, one starts with the QFT in dimension 0 which is the just the theory of (very) ordinary integrals. In later stages various kinds of mathematics are likely to pop up. Since various segments of the audience will have different background, all ingredients will have to be explained. I expect not all details to be checked, but the ideas should be completely clear.
We would like to learn QFT ideas in a way that leads, as quickly as possible, to ability to do research (such as thesis work) in related topics.

INTEREST in QFT:

In recent past QFT, particularly the String Theory, has generated huge excitements in mathematical community. The first wave concentrated on mathematical verification of mathematical predictions made by physicists -- most famously the Mirror Symmetry predictions in algebraic geometry. It seems to me this is changing and QFT ideas are becoming a part of standard mathematical education.

LITERATURE:

Hori, Kentaro; Katz, Sheldon; Klemm, Albrecht; Pandharipande, Rahul; Thomas, Richard; Vafa, Cumrun(1-HRV); Vakil, Ravi; Zaslow, Eric, a book by Many Mathematicians (and Some Physicists)

Mirror symmetry.

MathSci Review.

Pavel Etingof , lecture notes on his 2002 course

Mathematical ideas and notions of quantum field theory.

R. E. Borcherds, A. Barnard , lecture notes on Borcherds 2001 course

Lectures on Quantum Field Theory

A. Zee , a physicist's introduction book to QFT

Quantum Field Theory in a Nutshell

SPECIAL TOPICS:
We should also have parallel talks on related more advanced topics of current interest. These should be kept separate from the main line of the seminar and if possible they should be distributed through this and other seminars. One of such topics is

Stability :
This is a related but more special topic. In various areas of mathematics (vector bundles, connections, representations of Galois groups) one uses the idea of ``slope'' in a -- seemingly -- similar way. A part of this has been formalized into a notion of ``stability condition on an abelian category'' (Rudakov), and recently also as a ``stability condition on a triangulated category'' (Bridgeland). The Bridgeland version seems to be a new mathematical tool, its novel features come from QFT arising as an attempt to mathematicize a piece of String Theory -- in algebraic geometry the Bridgeland stability conditions parameterize some Super Conformal Field Theories (via the sigma model). I would like to put together the experiences with the stability notion in various areas.