ANOMALIES
(topics with * will be reviewed to make the course selfcontained)
*Clifford Algebra in general dimension.Chirality, C,P,T
*Analyticity .Locality
1)Perturbative Chiral anomaly in d=2 and d=4.UV and IR aspects.
2)Algebraic approach to anomalies.Kac-Moody and Virasoro algebra central extensios.WZW model ,Schwarz derivative
3)The Fujikawa approach.Atiyah-Singer Theorem
4)Chiral anomalies in general even dimension.Descent Equations.Pontryaguin Class.Anomaly Inflow.Gravitatinal Anomalies:Generalized Pontryaguin Classes.
5)Conformal Anomalies.Holographic realizations.Restriction on Renormalization Group Flows.
* Spontaneous Symmetry Breaking.Goldstone Theorem
*Instantons
6)Applications of the anomalies in the Standard Model.Gauge Anomaly cancellation,’ Hooft anomaly matching.The axial U(1) problem
7)Applications of the anomalies to String Theory.Critical dimension, Green-Schwarz mechanism.
* Stieffel-Whitney classification :bundles on T_n, S_2xS_2x.. for continuous and discrete gauge groups
8)Global Anomalies: QM ,SU(2) d=4 ,
9)Discrete anomalies.Higher form anomalies. Coupling anomalies.Duality Anomalies Applications: QCD , Topological order in CM.
10)Parity anomaly.Dai-Freed Theorem.