# Introductory lectures on topological quantum field theory

@article{Carqueville2017IntroductoryLO, title={Introductory lectures on topological quantum field theory}, author={Nils Carqueville and Ingo Runkel}, journal={arXiv: Quantum Algebra}, year={2017} }

These notes offer a lightening introduction to topological quantum field theory in its functorial axiomatisation, assuming no or little prior exposure. We lay some emphasis on the connection between the path integral motivation and the definition in terms symmetric monoidal categories, and we highlight the algebraic formulation emerging from a formal generators-and-relations description. This allows one to understand (oriented, closed) 1- and 2-dimensional TQFTs in terms of a finite amount of…

## 17 Citations

Introduction to 2-dimensional Topological Quantum Field Theory

- Mathematics
- 2022

Mostly self-contained script on functorial topological quantum field theories. These notes give a slow introduction to the basic notions of category theory which serve a closer investigation of…

Cylinder topological quantum field theory: A categorical presentation of classical field theory and its symmetries.

- Mathematics
- 2018

We use geometric ideas coming from certain classic algebraic constructions to associate, to every classical field theory, a symmetric monoidal double functor from the double category of cobordisms…

Area-Dependent Quantum Field Theory

- Mathematics
- 2020

Area-dependent quantum field theory is a modification of two-dimensional topological quantum field theory, where one equips each connected component of a bordism with a positive real…

Circuit Complexity in Topological Quantum Field Theory

- Physics
- 2021

Quantum circuit complexity has played a central role in recent advances in holography and many-body physics. Within quantum field theory, it has typically been studied in a Lorentzian (real-time)…

Reflection Positivity: A Quantum Field Theory Connection

- Mathematics, Physics
- 2019

At the heart of constructive quantum field theory lies reflection positivity. Through its use one may extend results for a Euclidean field theory to a relativistic theory. In this dissertation we…

GRAU DE MATEMÀTIQUES Treball final de grau

- Mathematics
- 2020

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…

2d TQFTs and baby universes

- PhysicsJournal of High Energy Physics
- 2021

Abstract
In this work, we extend the 2d topological gravity model of [1] to have as its bulk action any open/closed TQFT obeying Atiyah’s axioms. The holographic duals of these topological gravity…

Equational theories of endomorphism monoids of categories with a topological flavor

- Mathematics
- 2020

It is shown that the endomorphism monoids of the category $2\mathfrak{Cob}$ of all $2$-cobordisms do not have finitely axiomatizable equational theories. The same holds for the {topological annular…

Orbifolds of Reshetikhin-Turaev TQFTs

- Mathematics
- 2018

We construct three classes of generalised orbifolds of Reshetikhin-Turaev theory for a modular tensor category $\mathcal{C}$, using the language of defect TQFT from [arXiv:1705.06085]: (i) spherical…

Phase Space on a Surface with Boundary via Symplectic Reduction

- Mathematics, Physics
- 2021

We describe the symplectic reduction construction for the physical phase space in gauge theory and apply it for the BF theory. Symplectic reduction theorem allows us to rewrite the same phase space…

## References

SHOWING 1-10 OF 25 REFERENCES

Lecture notes on 2-dimensional defect TQFT

- Mathematics
- 2016

These notes offer an introduction to the functorial and algebraic description of 2-dimensional topological quantum field theories `with defects', assuming only superficial familiarity with closed…

Frobenius Algebras and 2-D Topological Quantum Field Theories

- Mathematics
- 2004

This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise…

Integrating quantum groups over surfaces: quantum character varieties and topological field theory

- Mathematics
- 2015

Braided tensor categories give rise to (partially defined) extended 4-dimensional topological field theories, introduced in the modular case by Crane-Yetter-Kauffman. Starting from modules for the…

Quantum field theory and the Jones polynomial

- Mathematics
- 1989

It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones…

Lattice topological field theory in two dimensions

- Mathematics
- 1994

The lattice definition of a two-dimensional topological field theory (TFT) is given generically, and the exact solution is obtained explicitly. In particular, the set of all lattice topological field…

STATE SUM CONSTRUCTION OF TWO-DIMENSIONAL OPEN-CLOSED TOPOLOGICAL QUANTUM FIELD THEORIES

- Mathematics
- 2007

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from…

The Classification of Two-Dimensional Extended Topological Field Theories

- Mathematics
- 2009

We provide a complete generators and relations presentation of the 2-dimensional extended unoriented and oriented bordism bicategories as symmetric monoidal bicategories. Thereby we classify these…

Topological sigma models

- Physics
- 1988

A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surfaceΣ to an arbitrary almost complex manifoldM. It possesses a fermionic BRST-like symmetry,…

Localization and traces in open-closed topological Landau-Ginzburg models

- Mathematics
- 2005

We reconsider the issue of localization in open-closed B-twisted Lan- dau-Ginzburg models with arbitrary Calabi-Yau target. Through careful analsysis of zero-mode reduction, we show that the closed…