

* Geometry and String Theory, I &
II are taught every other year and Anomalies in Quantum
Field Theory and Strings, I & II are taught the alternate
years.

GEOMETRY
and STRING THEORY I 
FALL 
Physics 654 / Math 694 

Geometry
and String Theory I is the first term in a two semester interdisciplinary
course taught jointly by Burt Ovrut of the Physics Department and
Ron Donagi of the Mathematics Department. The goal of the course is
to introduce students, postdocs and faculty to the mathematics and
physics associated with recent advances in field theory and superstring
theory. This is not a survey course. In this first term we will introduce,
and use, relatively sophisticated mathematical techniques, such as
homology/cohomology theory, fiber bundles and characteristic classes.
These will be applied to important physics topics such as YangMills
theory, instantons, monopoles and topological defects. 

Syllabus: 


 Fiber bundles and connections
 Groups, Lie algebras and representations
 Characteristic classes
 Sheaf and bundle cohomology
 YangMills theory
 Instantons, monopoles and topological defect


Texts: 


Geometry and String Theory I will not follow any one text. However,
the following books will be very useful for different topics discussed
in the syllabus. They are:
 Geometry, Topology and Physics, M. Nakahara, Adam Hilger
 Topology and Geometry for Physicists, C. Nash and S.
Sen, Academic Press
 Differential Topology and Quantum Field Theory, C. Nash,
Academic Press




GEOMETRY
and STRING THEORY II 
SPRING 
Physics 655 / Math 695 

Geometry and String Theory II is the second term
in a two semester interdisciplinary course taught jointly by Burt
Ovrut of the Physics Department and Ron Donagi of the Mathematics
Department. In this second term we will introduce relatively sophisticated
mathematical techniques, such as elliptic operators, index theory
and topics in algebraic geometry. These will be applied to important
physics topics such SeibergWitten theory, Ftheory and Ftheory/Mtheory
duality in superstrings.
More specifically, the syllabus will be the following, although
topics may vary somewhat according to new developments in string
theory and the interests of the class and the instructors.


Syllabus: 


 Elliptic operators
 Complexes and their index
 Topics in algebraic geometry
 Elliptic fibrations
 SeibergWitten theory
 Ftheory, Mtheory and Ftheory/Mtheory duality


Texts: 


Geometry
and String Theory II will not follow any one text. However,
the following books will be very useful for different topics discussed
in the syllabus. They are:
 Geometry, Topology and Physics, M. Nakahara, Adam Hilger
 Topology and Geometry for Physicists, C. Nash and S.
Sen, Academic Press
 Differential Topology and Quantum Field Theory, C. Nash,
Academic Press


Anomalies
in Quantum Field Theory and Strings I 
FALL 
Physics 656/ Math 696 

Anomalies
in Quantum Field Theory and Strings I is the first term in
a two semester interdisciplinary course taught jointly by Burt Ovrut
of the Physics Department and Ron Donagi of the Mathematics Department.
The goal of the course is to develop the mathematical and physical
theory of anomalies from several equivalent, but formally very different,
points of view. In this first term, we will survey anomalies in YangMills
theories and gravitational anomalies from the point of view of Feynman
diagrams. Both the Abelian and nonAbelian anomalies will then be
derived using Fujikawa's method. The theory of elliptic operators
and complexes will be reviewed and the index evaluated using the AtiyahSinger
index theorem. The relation of this index to anomalies will be discussed.
More specifically, the syllabus will be the following, although
topics may vary somewhat according to new developments in string
theory and the interests of the class and the instructors.


Syllabus: 


 Review of elliptic operators, complexes and the index
 AtiyahSinger index theorem
 Proof of the index theorem using heat kernal methods
 Feynman graphs and anomalies
 Fujikawa's method
 Computation of the Abelian and nonAbelian anomalies
 Relation of anomalies to the index theorem


Texts: 


Anomalies in Quantum Field Theory and Strings I will not
follow any one text. However, the following books will be very useful
for different topics discussed in the syllabus. They are:
 Geometry, Topology and Physics, M. Nakahara, Adam Hilger
 Differential Topology and Quantum Field Theory, C. Nash,
Academic Press
 Anomalies in Quantum Field Theory, R. Bertlmann, Oxford
University Press




Anomalies
in Quantum Field Theory and Strings II 
SPRING 
Math 697 / Physics 657 

Anomalies in Quantum Field Theory and Strings II
is the second term in a two semester interdisciplinary course taught
jointly by Burt Ovrut of the Physics Department and Ron Donagi of
the Mathematics Department. In this second term, we start by discussing
anomalies from the point of view of the families index theorem,
equivariant cohomology and loop spaces. NonAbelian gauge quantum
field theory will be reviewed using the partition function. The
BRST invariance of the gauge fixed/ghost action will be derived
and we will discuss, in detail, the WessZumino consistency condition
and its solution in terms of ``descent equations''. Using these
techniques, the nonAbelian and gravitational anomalies will be
recomputed and the cohomological algebra underlying the descent
equations studied.
More specifically, the syllabus will be the following, although
topics may vary somewhat according to new developments in string
theory and the interests of the class and the instructors.


Syllabus: 


 Family index theorem
 Equivariant cohomology and loop spaces
 Cohomological algebra
 Review of nonAbelian gauge theory
 BRST invariance and the WessZumino consistency condition
 Descent equations and anomalies
 Worldsheet anomalies and string theory


Texts: 


Anomalies
in Quantum Field Theory and Strings II will not follow any
one text. However, the following books will be very useful for different
topics discussed in the syllabus. They are:
 Geometry, Topology and Physics, M. Nakahara, Adam Hilger
 Differential Topology and Quantum Field Theory, C. Nash,
Academic Press
 Anomalies in Quantum Field Theory, R. Bertlmann, Oxford
University Press


MATHEMATICAL FOUNDATION OF THEORETICAL PHYSICS/ TOPICS IN ALGEBRAIC GEOMETRY 
FALL 
Math 694/Math 729 

We will discuss algebraic varieties which are of interest to the physics of string theory. We will start with del Pezzo, K3, elliptic and abelian surfaces and then the higher dimensional generalizations of these surfaces (including Fano and CalabiYau varieties). We will also study singularities (ADE and beyond) on these varieties and their interpretations in physics (gauge theory, anomalies and realistic models).
We will also discuss birational transformations (flips, flops...). 


TOPICS IN MATH/PHYSICS 
SPRING 
Math 724 

Topics
in Math/Physics is a one semester course taught by Antonella
Grassi of the Mathematics Department. This course is inspired by recent
work on the interface of string theory and geometry. On the string
theory side, the focus is on the "dualities" between closed
and open strings and ChernSimons theory. The geometric realization
of these dualities are certain birational contractions of CalabiYau
threefolds followed by complex deformations. The local topology of
the transformations is a surgery between symplectic 6manifolds, and
the ChernSimons theory is considered on Lagrangian submanifolds.
The resulting geometry is very rich and the physics dualities imply
relations between geometric invariants: knot invariants in the Lagrangian
submanifolds and open and closed GromovWitten invariants of the symplectic
manifolds. We will also discuss discuss work which analyzes these
dualities as transformations (Mtheory dualities) between G2 holonomy
spaces.
Prerequisites:
Some familiarity with either algebraic geometry or differential/symplectic
geometry might be helpful, but will not be assumed. We will focus
on the mathematical aspects of the problems, and the different branches
of mathematics which come to interplay in this stringtheoretic
setup. However, every attempt will be made to highlight these results
in terms of their physical content and to make the course accessable
to physicists familiar with string theory.


