Geometric Methods for Quantum Field Theory

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DIRAC OPERATORS AND GEOMETRIC QUANTIZATION
SEIBERG-WIITEN THEORY AND DUALITY IN FIELD THEORY
DIRAC OPERATORS AND GEOMETRIC QUANTIZATION

Course 1: O. HIJAZI (France)
Spectral Properties of Dirac Operators and Geometric Structures: Basic concepts of spin Riemannian Geometry, Dirac Operators and twistors, Weitzenbock formulas, Gauss formula for spinors, Theorem of Lichnerowicz, spectral hole, holonomy group.
Course 2: B. BOOSS-BAVNBECK (Denmark), S. SCOTT (U.K.), K. WOJCIECHOWSKI (U.S.A.)
The Determinant of the Dirac Operator: Determinant bundles, Dirac Operators with boundary conditions and Grassmannians of boundary conditions, Grassmannians and chiral anomaly, eta invariant, additivity formulas for determinants and determinant bundles.
Course 3: T. WURZBACHER (France)
Geometric Quantization and two-dimensional Quantum Field Theory: Geometric Quantization and group actions, second fermionic quantization, Geometric Quantization approach to a fermionic field theory in dimension 2, other examples of Geometric Quantization in infinite dimension.
Course 4: E. LANGMANN (Sweden)
 Representation theory of infinite dimensional groups and algebras in quantum freld theory: Models in two space-time dimensions, relation to anomalies, extensions to higher dimensional quantum field theory models, presentation of recent results.

SEIBERG-WIITEN THEORY AND DUALITY IN FIELD THEORY

Course 1: T. SHEUNG TSUN (U.K.)
 Concepts in Gauge Theory Ieading to Electric-Magnetic Duality: Basic concepts of gauge theory, non-abelian generalization of electromagnetic duality, some consequences of this duality in field theory.
Course 2: R. FLUME (Germany)
Some aspects of N=2 super-symmetric Yang-Mills field theories: Perturbative renormalization of supersymmetric gauge theories and non-renormalization theorems, theorems on non-perturbative non-renormalization, equivariant cohomology, equivariant localization and evaluation of the contribution of an Instanton of the Seiberg-Witten potential.
Course 3: M. MARCOLLI (U.S.A)
Dimensional reduction of the Seiberg-Witten theory, a 3-manifold invariant: Properties of the Seiberg-Witten invariant and its relation with the Casson invariant and Milnor torsion, Seiberg-Witten Floer theory. Seiberg-Witten invariants on contact 3-manifolds.
Course 4: F. QUEVEDO (Mexico)
Duality, Strings and Physics: Duality and global symmetries, duality and string theory, M-theory and its possible consequences.
Course 5: H. OCAMPO (Colombia)
Duality to Seiberg-Witten Curves: Elementary introduction to Seiberg-Witten theory, recent developments generalizing certain models in order to study N=2 supersymmetric gauge theory via M-theory.