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 NOTSOSHORT 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 14 largely follow
DeligneMorgan 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 (halfrevised) 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
SuperSymmetric 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(1HRV); Vakil, Ravi; Zaslow, Eric,
a book by Many Mathematicians (and Some Physicists)
With a preface by Vafa. Clay Mathematics Monographs, 1.
American Mathematical Society.
This may be the basic source. It
seems to be the first text which
presents the relevant information in an accessible way.
The study of Mirror Symmetry
is the area where most thinking on the
interaction of QFT and mathematics has been done.
Also, the mathematical
applications are to classical mathematics such as Morse theory,
so they are easy to motivate.
So, even if one's interest is not strictly the Mirror Symmetry itself,
some familiarity with Mirror Symmetry seems to offer the best
entrance into mathematics of String Theory.
See also
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
An essentially selfcontained introduction to some of the ideas and terminology of QFT.

A. Zee
,
a physicist's introduction book to QFT
Quantum Field Theory in a Nutshell
It tries to say what is happening,
then explain it by computations with attempts at humor.
Recommended by Aroldo Kaplan.
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.