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170569.pdf (684.96 KB)

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Bachelor thesis

Publication date

2020-07-02

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cc-by-nc-nd (c) Jaume Baixas Estradé, 2020
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/170569

Topological quantum field theories

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Abstract

[en] Topological quantum field theories (TQFTs) are functors from the category of bordisms to the category of vector spaces that preserve their monoidal structure. Such functors arose in Physics but have proven to be useful in various fields of Mathematics. TQFTs give topological and geometric invariants of manifolds, and thus may help in understanding and classifying them. In this work, however, we perform the reverse process: the completely known classification of 1- and 2-dimensional manifolds will serve as the ground that permits us comprehend TQFTs in these dimensions and determine their underlying structure. In particular, we give a detailed description of 1- and 2-dimensional TQFTs in terms of finite-dimensional vector spaces and commutative Frobenius algebras, respectively. We conclude by trying to elucidate the relation between TQFTs and Physics. We discuss the common structural properties shared by Hilbert spaces and spacetimes, which motivate the connection of quantum theory with general relativity via TQFTs.

Description

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Joana Cirici

Subject

Espais vectorials topològics, Treballs de fi de grau, Àlgebres de Frobenius, Topologia diferencial, Categories (Matemàtica), K-teoria

Subject (English)

Linear topological spaces, Bachelor's theses, Frobenius algebras, Differential topology, Categories (Mathematics), K-theory

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Treballs Finals de Grau (TFG) - Matemàtiques

Citation

BAIXAS ESTRADÉ, Jaume. Topological quantum field theories. [consulted: 8 of June of 2026]. Available at: https://hdl.handle.net/2445/170569

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