          |
 |
* 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, post-docs 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 Yang-Mills
theory, instantons, monopoles and topological defects. |
| |
Syllabus: |
|
| |
- Fiber bundles and connections
- Groups, Lie algebras and representations
- Characteristic classes
- Sheaf and bundle cohomology
- Yang-Mills 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 Seiberg-Witten theory, F-theory and F-theory/M-theory
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
- Seiberg-Witten theory
- F-theory, M-theory and F-theory/M-theory 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 Yang-Mills
theories and gravitational anomalies from the point of view of Feynman
diagrams. Both the Abelian and non-Abelian 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 Atiyah-Singer
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
- Atiyah-Singer index theorem
- Proof of the index theorem using heat kernal methods
- Feynman graphs and anomalies
- Fujikawa's method
- Computation of the Abelian and non-Abelian 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. Non-Abelian 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 Wess-Zumino consistency condition
and its solution in terms of ``descent equations''. Using these
techniques, the non-Abelian and gravitational anomalies will be
re-computed 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 non-Abelian gauge theory
- BRST invariance and the Wess-Zumino 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 Calabi-Yau varieties). We will also study singularities (A-D-E 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 Chern-Simons theory. The geometric realization
of these dualities are certain birational contractions of Calabi-Yau
threefolds followed by complex deformations. The local topology of
the transformations is a surgery between symplectic 6-manifolds, and
the Chern-Simons 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 Gromov-Witten invariants of the symplectic
manifolds. We will also discuss discuss work which analyzes these
dualities as transformations (M-theory 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 string-theoretic
set-up. 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.
|
|
|
