## Thesis etd-09302008-142703 |

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Thesis type

Tesi di laurea specialistica

Author

MANCARELLA, FRANCESCO

URN

etd-09302008-142703

Thesis title

Some computations in Chern-Simons quantum field theory.

Department

SCIENZE MATEMATICHE, FISICHE E NATURALI

Course of study

SCIENZE FISICHE

Supervisors

**Relatore**Prof. Guadagnini, Enore

Keywords

- abelian Chern-Simons
- Chern-Simons
- gauge theory
- Gauss sum.
- homology
- homology spheres
- knot theory
- knots
- non-Abelian Chern-Simons
- Reshetikhin
- surgery
- surgery invariant
- surgery rules
- three-manifolds
- topological field theory
- Turaev
- Wilson loop

Graduation session start date

17/10/2008

Availability

Withheld

Release date

17/10/2048

Summary

RIASSUNTO DELLA TESI DI LAUREA SPECIALISTICA

Titolo: Some computations in Chern-Simons quantum field theory.

In this work, the main properties of the quantum Chern-Simons theory- which is a topological quantum field theory in three dimensions- are exposed, and the computation of a certain class of observables is produced.

In the first part of the thesis, several concepts of three-dimensional topology and of knot theory are introduced. Thus, the Abelian Chern-Simons action functional is considered, and the symmetry properties of the theory are discussed. The gauge-fixing procedure together with the construction of the perturbative expansion are reported; moreover, a definition of the observables associated with oriented framed and coloured links is given. The solution of the theory in R^3 is produced and the rules for the computation of the observables in a generic closed oriented 3-manifold are derived. These rules are applied to compute a set of link observables which are defined on lens spaces.

We give a brief description of the homology groups for closed manifolds; then we concentrate on 3-manifolds which are homology spheres and we give a convenient surgery presentation of these manifolds. We define a combinatorial invariant for closed 3-manifolds which corresponds to the normalized partition function of the Abelian Chern-Simons theory. This invariant is computed for various examples of 3-manifolds and it is shown that it is trivial for homology spheres. It is shown that such an invariant does not depend only on the homology of the manifold.

Titolo: Some computations in Chern-Simons quantum field theory.

In this work, the main properties of the quantum Chern-Simons theory- which is a topological quantum field theory in three dimensions- are exposed, and the computation of a certain class of observables is produced.

In the first part of the thesis, several concepts of three-dimensional topology and of knot theory are introduced. Thus, the Abelian Chern-Simons action functional is considered, and the symmetry properties of the theory are discussed. The gauge-fixing procedure together with the construction of the perturbative expansion are reported; moreover, a definition of the observables associated with oriented framed and coloured links is given. The solution of the theory in R^3 is produced and the rules for the computation of the observables in a generic closed oriented 3-manifold are derived. These rules are applied to compute a set of link observables which are defined on lens spaces.

We give a brief description of the homology groups for closed manifolds; then we concentrate on 3-manifolds which are homology spheres and we give a convenient surgery presentation of these manifolds. We define a combinatorial invariant for closed 3-manifolds which corresponds to the normalized partition function of the Abelian Chern-Simons theory. This invariant is computed for various examples of 3-manifolds and it is shown that it is trivial for homology spheres. It is shown that such an invariant does not depend only on the homology of the manifold.

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