# Mathematical Foundations of Topological Quantum Field Theories

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

The field of Topological Quantum Field Theories (TQFT) has always been a source of rich interaction between Mathematics and Physics. This interplay highly relies on the ability to 'translate' the ideas fromPhysics into a rigorous mathematical setting allowing the application of powerful theorems such as the Baez-Dolan Cobordism Hypothesis, which were discovered as purely mathematical statements. The most important tools for studying TQFT in mathematics are higher

category theory and, in particular, the fully-extended bordism category. Realising new 'physical' ideas as mathematical structures in these areas is not only a necessary step in the process of understanding them, but it also is an ample source of examples and motivation of the study of higher categorytheory as a subject, which is already interesting by itself.

In the topic of theoretical condensed matter physics TQFTs arise as lowtemperature limits. They have been of particular interest for the purpose of using topological protection of quantum states in quantum computing.Such TQFTs have been grouped together into different classes, so-called topological phases. Which topological phases can occur in a specific physical situation depends on the parameters of that situation, mainly the dimensiond and the symmetry group G.

There has since been a great interest in classifying topological phasesgiven the parameters (G, d) as such a classification would predict underwhich requirements particular phases, such as topological insulators canbe expected.

Reflection positivity (rp) is a phenomenon that is observed in most physical TQFTs, but so far had no counter-part in the mathematical world. Recently Freed and Hopkins [2] proposed a definition of reflection positivity

for invertible topological phases and argued why one should expect it to be implemented for all TQFTs relevant in Physics. This improvement from the standard definition of a TQFT to a rpTQFT seems to close the gap between the results obtained from the mathematical model and those appearing in the theoretical physics literature. Using tools from stable equivariant homotopy theory they classified invertible rpTQFTs in low dimensions for most interesting symmetry groups reproducing the results known from the theoretical physics literature and generating new results in many interesting cases. From a mathematical perspective it is quite unsatisfying that the FreedHopkins definition of rp only works out for invertible TQFTs. It is therefore a natural question to ask for this definition to be extended to not necessarily invertible TQFTs and to generalize their classification results to this case.

This project mainly falls into the research area of Geometry and Topology, but it also has strong links to several other areas in mathematics and physics. The novelty of the research method lies in combining the physical aspects of TQFT with the theory of equivariant higher categories, and in the invertible case with stable equivariant homotopy theory. This interconnectedness will be useful in either direction.

category theory and, in particular, the fully-extended bordism category. Realising new 'physical' ideas as mathematical structures in these areas is not only a necessary step in the process of understanding them, but it also is an ample source of examples and motivation of the study of higher categorytheory as a subject, which is already interesting by itself.

In the topic of theoretical condensed matter physics TQFTs arise as lowtemperature limits. They have been of particular interest for the purpose of using topological protection of quantum states in quantum computing.Such TQFTs have been grouped together into different classes, so-called topological phases. Which topological phases can occur in a specific physical situation depends on the parameters of that situation, mainly the dimensiond and the symmetry group G.

There has since been a great interest in classifying topological phasesgiven the parameters (G, d) as such a classification would predict underwhich requirements particular phases, such as topological insulators canbe expected.

Reflection positivity (rp) is a phenomenon that is observed in most physical TQFTs, but so far had no counter-part in the mathematical world. Recently Freed and Hopkins [2] proposed a definition of reflection positivity

for invertible topological phases and argued why one should expect it to be implemented for all TQFTs relevant in Physics. This improvement from the standard definition of a TQFT to a rpTQFT seems to close the gap between the results obtained from the mathematical model and those appearing in the theoretical physics literature. Using tools from stable equivariant homotopy theory they classified invertible rpTQFTs in low dimensions for most interesting symmetry groups reproducing the results known from the theoretical physics literature and generating new results in many interesting cases. From a mathematical perspective it is quite unsatisfying that the FreedHopkins definition of rp only works out for invertible TQFTs. It is therefore a natural question to ask for this definition to be extended to not necessarily invertible TQFTs and to generalize their classification results to this case.

This project mainly falls into the research area of Geometry and Topology, but it also has strong links to several other areas in mathematics and physics. The novelty of the research method lies in combining the physical aspects of TQFT with the theory of equivariant higher categories, and in the invertible case with stable equivariant homotopy theory. This interconnectedness will be useful in either direction.

## People |
## ORCID iD |

U Tillmann (Primary Supervisor) | |

Jan Steinebrunner (Student) |

### Publications

*The classifying space of the one-dimensional bordism category and a cobordism model for TC of spaces*in Journal of Topology

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509711/1 | 30/09/2016 | 29/09/2021 | |||

1941474 | Studentship | EP/N509711/1 | 30/09/2017 | 29/09/2020 | Jan Steinebrunner |

Description | The mathematical formulation of topological quantum field theory crucially relies on so-called cobordism categories. The cobordism category is an algebraic object recording how n-dimensional spaces and 'evolve in time' and glue together. It was first introduced to mathematics by Milnor in the 60s to study high-dimensional topology and was then connected to quantum field theory by Segal and Atiyah in the 80s. Recently much research has been conducted about algebraic invariants (for example the cohomology) of certain 'infinity-categories' of cobordisms. The key insight of our research is that the original cobordism category also has interesting cohomology and is in fact more complicated than the infinity category in this respect. We illustrate this new complexity in upcoming work by detailed calculations in the n=1 case, observing unexpected relations to the seemingly distant field of algebraic K-theory. Our research raises the question of what the cohomlogy of this cobordism category is for n>1 and we develop various tools for attacking this type of question. |

Exploitation Route | Our work points out unforeseen relations between the cobordism category and other areas of mathematics such as algebraic K-theory, and possibly tropical geometry. In doing so, we provide new 'categorical models' for objects of interest in these areas. These could be used by researches in those fields to obtain new information about the objects in question. |

Sectors | Other |