INSTITUTE FOR ADVANCED STUDY
School of Mathematics
Princeton, NJ 08540
QUANTUM FIELD THEORY SEMINARS
SPEAKER: Joseph Bernstein, Tel Aviv University
TIME: 2:00 P.M. TO 4:00 P.M.
LOCATION: M-101
TENTATIVE AGENDA
Tuesday, September 17, 1996
Lecture 1. What physicists have in mind when they talk about SUSY
(supersymmetry).
1. Bosons and fermions - Z_2 grading on the Hilbert space of states.
2. General idea about groups of symmetries. Symmetries of
Lagrangian and symmetries of Hilbert space.
Groups of symmetries containing Poincare group P.
3. Coleman Mandula theorem. Possibility to discover supersymmetries.
4.Supergroups containing P.
5.N-extended supersymmetry algebra.
Thursday, September 19, 1996
Lecture 2. Basics of super mathematics.
1.Super linear algebra. ( Example: Lie superalgebras).
2. Definition of superscheme and supermanifold
3. Differential geometry on supermanifolds:
Tangent bundle.
DeRham complex
4. Frobenius pairing and Frobenius theorem
5. Construction of supermanifolds. Families
Example: V(F).
6. Variety M~. Two constructions. Connection with DeRham theory.
7. Supergroups and super Lie algebras. Lie theory.
Tuesday, September 24, 1996
Lecture 3. Integration theory and \sigma-models
I. Integration theory.
1. Definition of volume forms
2. Definition of integral
3. How to compute super integral?
II. \sigma-models
1. Usual 2-dimensional \sigma model
2. Supersymmetric 2-dimensional \sigma model
3. Reduction to classical integrals - component analisys and
elimination of auxiliery fields.
4. Off-shell and on-shell SUSY
September 26, 1996
Thursday, Lecture 4. Wess-Zumino model on flat space ( d = 4, N = 1).
1. Construction of the flat SUSY space. Structures.
2. Chiral functions.
3. Chiral fields and Young-Mills Lagrangian.
4. General Lagrangian.
5. Reduction to the classical integral
6. Of shell and on-shell SUSY.
7. Generaliztion to \sigma model in Kahler manifolds
8. Dimensional reduction and extended N = 2 off shell theory for d = 2.
Tuesday, October 1, 1996
Lecture 5. Simple supergravity ( d = 4, N = 1).
1. Conformal structure
2. Connection between volume bundles
3. Supergravity
4. Partial connection. Decomposition into direct sum
5. Definition in terms of connection and constrains.
Thursday, October 3, 1996
Lecture 6. Higher dimensional theories.
1. Nahm's theorem
2. 10-dimensional Young-Mills.
3. Dimensional reduction - N = 4 Young-Mills theory in dimension 4
4. 11-dimensional supergravity. reduction to extended
(d=4,N=8)-gravity.
.