|QUANTUM FIELDS AND TOPOLOGY||FIS/02||LEZIONI||36|
By the end of the course, the students will have acquired knowledge on
general methods of quantization of gauge theories and of the so-called topological field theories, Chern-Simons and BF theories;
basic notions of low dimensional topology: manifolds, knots and links, homotopy equivalence, ambient isotopy equivalence;
polynomial invariants of links, framed links, use of the skein relation, Alexander-Conway, Jones and HOMFLY polynomials, linking number and related Gauss integral, fundamental group of a manifold and of the complement of a link, Seifert surface, surgery on three manifolds and the Lickorish fundamental theorem;
solution of the abelian Chern-Simons theory in a generic 3-manifold.
Introduction to quantum field theories of topological type in 3 dimensions, in which the lagrangian does not depend on the metric. Perturbative properties of the Chern-Simons model and the so-called BF theory. Gauge fixing and BRST symmetry. Connections, holonomies and Wilson lines. Knot and link observables, basic notions of low dimensional topology. Generlized Jones polynomial for oriented, framed links. Surgery rules of Reshethikin-Turaev. Gauge theories in 3-manifolds with nontrivial topology, fundamental group and homology group. Path-integral solution of the abelian Chern-Simons theory in a generic 3-manifold.
Presentation of a specific argument by the student,
final oral examination
Ability to give a talk on a chosen argument, oral examination.
perturbative computations in field theory, determination of the link polynomials, computations of the fundamental group
Elaborate on the relationships between different topics in topological quantum field theories.
the student provides a presentation of a specific argument discussed in the course
Oral examination and talk on a chosen argument.
basic notions of quantum field theory
Basic notions of quantum field theory.
Delivery: face to face
General aspects of perturbative quantum field theory (QFT). Action of Chern-Simons for abelian and non-abelian theories. General covariance, BRST gauge fixing, derivation of the Feynman propagator, Gauss linking number, gauge invariance, Wess-Zumino term. Basic definitions of knot theory, link poynomials, framed links. Gauge variables, holonomy, Wilson line operators. CS expectation values of holonomies associated with links. Skein relation, abelian and non-abelian link invariants obtained in the CS theory. Surgery on 3-manifolds. Functional integration for abelian Chern-Simons theory in a generic 3-manifold.
- E. Witten, Commun. Math. Phys. 121 (1989) 351.
- E. Guadagnini, M. Martellini, M. Mintchev, Phys. Lett. B 227 (1989) 111.
- E. Guadagnini, The link invariants of the Chern–Simons theory, in: O.H. Kegel, V.P. Maslov, W.D. Neumann, R.O. Wells (Eds.), de Gruyter Expositions in Mathematics, vol. 10, de Gruyter, Berlin, 1993.
- E. Guadagnini and F. Thuillier, Path-integral invariants in abelian Chern-Simons theory, Nucl. Phys. B 882 (2014) 450-484.
- E. Guadagnini and F. Rottoli, Perturbative BF theory, Nucl. Phys. B 954 (2020) 114987.