## Thesis etd-09252019-160451 |

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Thesis type

Tesi di laurea magistrale

Author

BEVILACQUA, ANDREA

URN

etd-09252019-160451

Thesis title

On Chern-Simons invariant of 3-manifolds

Department

FISICA

Course of study

FISICA

Supervisors

**relatore**Prof. Benedetti, Riccardo

**relatore**Prof. Guadagnini, Enore

**commissario**Prof. Mannella, Riccardo

**commissario**Prof. Fidecaro, Francesco

**commissario**Prof. Forti, Francesco

**commissario**Prof. Leporini, Dino

**commissario**Prof. Roddaro, Stefano

**commissario**Prof. Shore, Steven Neil

Keywords

- Chern Simons
- hyperbolic geometry
- low dimensional topology
- topology

Graduation session start date

16/10/2019

Availability

Full

Summary

This master thesis is centered on the study of the Chern-Simons (CS) theory and on the calculation of invariants of 3-manifolds. There are several reasons why this is an interesting topic from both a physical and a mathematical point of view.

From a mathematical point of view, there are still many unanswered questions regarding the quantization of the CS theory. More pragmatically, the study of this topic is also an introduction to several technical tools of great importance in contemporary topology and mathematical physics.

From a more physical point of view, one of the main reasons of interest in Chern-Simons theory lies on the fact that it is a non-trivial quantum field theory of topological type, meaning that it is a quantum field theory that exhibits general covariance. Contrary to canonical Quantum Field Theory (QFT), which is formulated in the context of a fixed flat Minkowski space-time background, it can give some insight on the structure of a theory which unifies concepts from General Relativity (GR) and QFT. On more practical side, CS theory is intimately linked to the calculation of manifolds and knots invariants. In this context, it can be linked to the Reshetikhin-Turaev construction for the calculation of invariants [RT91], which believed to constitute an abstraction of the CS procedure; further details on this topic can be found in [Wit89] and especially in [GMT17]. In particular, the formalism used in [RT91]to solve the Yang-Baxter equation allowed in recent times to mathematically formulate the so called Yangian symmetry, which is at the basis of a renewed interest in the reformulation of scattering amplitudes in a fundamentally new way [AHT14].In this thesis we will start with a mathematical introduction to R n and its features, defining calculus in R n . This will be the starting point for the definition of manifolds, fiber bundles and their properties. In particular, we will define tangent bundles and principal bundles, which will play a fundamental role in the rest of the thesis. After this, we will also introduce important constructions on manifolds and fiber bundles, such as forms, vector fields and connections. In particular, we will discuss the properties of connections, and we will motivate the fact that we will usually describe them as 1-forms.

Starting from this background, we will introduce the Chern Simons Lagrangian, both in the abelian case and in the non-abelian one. We will study the properties of the CS action under gauge transformation in both cases, and we will discuss its properties. Furthermore, we will investigate the definition of the CS invariant for manifolds with and without boundary. In this context, we will show two different methods for actually calculating the CS invariant.In particular, in the case of the structural group SU (N ), we will present the approach based on the so called Heegaard splitting developed in [GMT17], and in the case of structural group SL(2, C) the simplicial approach revisited in [Mar10]. In the two cases, we will introduce the necessary additional mathematical notions needed in order to fully understand them.

After the examples of calculation of invariants of 3-manifolds, we will investigate the issue of quantization in the context of the Chern-Simons theory. In particular, in this context, we will clarify the (physical) assumptions made related with the skein relations in [Wit89]. In this discussion, we will also use [Gua94]. In this way, the thesis will lead toward a deeper comprehension of this theory and its features. We will conclude with some remarks about some open questions related to the CS theory.

From a mathematical point of view, there are still many unanswered questions regarding the quantization of the CS theory. More pragmatically, the study of this topic is also an introduction to several technical tools of great importance in contemporary topology and mathematical physics.

From a more physical point of view, one of the main reasons of interest in Chern-Simons theory lies on the fact that it is a non-trivial quantum field theory of topological type, meaning that it is a quantum field theory that exhibits general covariance. Contrary to canonical Quantum Field Theory (QFT), which is formulated in the context of a fixed flat Minkowski space-time background, it can give some insight on the structure of a theory which unifies concepts from General Relativity (GR) and QFT. On more practical side, CS theory is intimately linked to the calculation of manifolds and knots invariants. In this context, it can be linked to the Reshetikhin-Turaev construction for the calculation of invariants [RT91], which believed to constitute an abstraction of the CS procedure; further details on this topic can be found in [Wit89] and especially in [GMT17]. In particular, the formalism used in [RT91]to solve the Yang-Baxter equation allowed in recent times to mathematically formulate the so called Yangian symmetry, which is at the basis of a renewed interest in the reformulation of scattering amplitudes in a fundamentally new way [AHT14].In this thesis we will start with a mathematical introduction to R n and its features, defining calculus in R n . This will be the starting point for the definition of manifolds, fiber bundles and their properties. In particular, we will define tangent bundles and principal bundles, which will play a fundamental role in the rest of the thesis. After this, we will also introduce important constructions on manifolds and fiber bundles, such as forms, vector fields and connections. In particular, we will discuss the properties of connections, and we will motivate the fact that we will usually describe them as 1-forms.

Starting from this background, we will introduce the Chern Simons Lagrangian, both in the abelian case and in the non-abelian one. We will study the properties of the CS action under gauge transformation in both cases, and we will discuss its properties. Furthermore, we will investigate the definition of the CS invariant for manifolds with and without boundary. In this context, we will show two different methods for actually calculating the CS invariant.In particular, in the case of the structural group SU (N ), we will present the approach based on the so called Heegaard splitting developed in [GMT17], and in the case of structural group SL(2, C) the simplicial approach revisited in [Mar10]. In the two cases, we will introduce the necessary additional mathematical notions needed in order to fully understand them.

After the examples of calculation of invariants of 3-manifolds, we will investigate the issue of quantization in the context of the Chern-Simons theory. In particular, in this context, we will clarify the (physical) assumptions made related with the skein relations in [Wit89]. In this discussion, we will also use [Gua94]. In this way, the thesis will lead toward a deeper comprehension of this theory and its features. We will conclude with some remarks about some open questions related to the CS theory.

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