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Advanced Topics in Field Theory
 
The course on "Advanced Topics in Field Theory" consists of three modules, on BRST quantisation, anomalies and instantons.
Introduction BRST Quantisation,      Pietro A. Grassi - Università di Alessandria
 
1) Gauge Symmetries, Lagrangians, S-matrix and Observables. 2) Gauge Fixing, Ghost Fields and BRST Symmetry, Unitarity of the S-matrix and Gauge Independence. 3) Renormalization, Ward Identities and Field-Antifield Formulation (BV). 4) BRST Cohomology, Anomalies.
 
Lecture notes
 
Recommended bibliography includes:
 
    C. Becchi, "Introduction to gauge theories", arXiv:hep-ph/9705211
    C. Becchi, "Introduction to BRS symmetry", arXiv:hep-th/9607181
    G. Barnich, F. Brandt, M. Henneaux, "Local BRST cohomology in gauge theories" Phys. Rept. 338 (2000) 439-569 [arXiv:hep-th/0002245]
    J. Gomis, J. Paris and S. Samuel, "Antibracket, antifields and gauge theory quantization", Phys. Rept. 259 (1995) 1, [arXiv:hep-th/9412228]
    O. Piguet and S.P. Sorella, "Algebraic renormalization: Perturbative renormalization, symmetries and anomalies", Lect. Notes Phys. M28 (1995) 1
 
Anomalies in Field Theory      Luca Griguolo - Università di Parma
 
1) Gauge anomalies. Abelian gauge anomaly from the transformation of the fermionic measure and relation with the index of Dirac operator. Nonabelian anomalies. Current non-conservation and anomalous Ward identities. 2) Gauge anomalies from Feynman diagrams: properties of locality and finiteness. Anomaly cancellation and fermion representations. Anomaly cancellation in the standard model. 3) Anomalies from the Wess-Zumino consistency condition. BRST cohomology. Descent equations. Relation between anomalies and index theorems. 4) Gravitational and mixed gauge-gravitational anomalies. Anomaly cancellation in higher dimensional models relevant for string theory.
 
Lecture notes
 
Recommended bibliography includes:
 
    A. Bilal, "Lectures on Anomalies", arXiv:0802.0634
    R.A. Bertlmann, "Anomalies in Quantum Field Theory" (Oxford University Press)
 
Solitons and Instantons,      Alessandro Tanzini - SISSA Trieste
 
1) Solitons in gauge theories: vortices, monopoles, istantons. 2) Contribution of solitons to the path integral: semiclassical approximation, bosonic and fermionic zero modes, determinants, selection rules for the amplitudes. 3) Instanton calculus in supersymmetric gauge theories: observables, chiral ring. Application to models with extended supersymmetry. Survey on ADHM construction. 4) Duistermaat-Heckman's localisation formula. Topological twist of supersymmetric theories. Survey on multi-instanton amplitudes and check of Seiberg-Witten solution.
 
Lecture notes
 
Recommended bibliography includes:
 
    S. Coleman, "Aspects of symmetry" (Cambridge University Press)
    N.Dorey, T.Hollowood, V.V. Khoze and M.P. Mattis, "The Calculus of many instantons" Phys. Rept. 371 (2002) 231-459 [arXiv:hep-th/0206063]
    N. Berline, E. Getzler and M. Vergne, "Heat kernels and Dirac operators" (Springer-Verlag)
    E. Witten, "Topological Quantum Field Theory" Commun. Math. Phys. 117 (1988) 353