The Arf-Brown TQFT of Pin$^-$ Surfaces

@article{Debray2018TheAT,
  title={The Arf-Brown TQFT of Pin\$^-\$ Surfaces},
  author={Arun Debray and Sam Gunningham},
  journal={arXiv: Mathematical Physics},
  year={2018}
}
The Arf-Brown invariant $\mathit{AB}(\Sigma)$ is an 8th root of unity associated to a surface $\Sigma$ equipped with a pin$^-$ structure. In this note we investigate a certain fully extended, invertible, topological quantum field theory (TQFT) whose partition function is the Arf-Brown invariant. Our motivation comes from the recent work of Freed-Hopkins on the classification of topological phases, of which the Arf-Brown TQFT provides a nice example of the general theory; physically, it can be… 

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References

SHOWING 1-10 OF 96 REFERENCES

Stable Postnikov data of Picard 2-categories

Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1-, and 2-cells. The classifying space of a Picard 2-category $\mathcal{D}$ is an infinite loop space, the zeroth space of

Topological Quantum Field Theories from Compact Lie Groups

It is a long-standing question to extend the definition of 3-dimensional Chern-Simons theory to one which associates values to 1-manifolds with boundary and to 0-manifolds. We provide a solution in

Connective Real $K$-Theory of Finite Groups

This book is about equivariant real and complex topological $K$-theory for finite groups. Its main focus is on the study of real connective $K$-theory including $ko^*(BG)$ as a ring and $ko_*(BG)$ as

CLIFFORD MODULES

  • A.
  • Mathematics
  • 1964
The purpose of the paper is to undertake a detailed investigation of the role of Clifford algebras and spinors in the K&theory of real vector bundles. On the one hand the use of Clifford algebras

Spin, statistics, orientations, unitarity

A topological quantum field theory is Hermitian if it is both oriented and complex-valued, and orientation-reversal agrees with complex-conjugation. A field theory satisfies spin-statistics if it is

From gauge to higher gauge models of topological phases

A bstractWe consider exactly solvable models in (3+1)d whose ground states are described by topological lattice gauge theories. Using simplicial arguments, we emphasize how the consistency condition

Fermionic matrix product states and one-dimensional topological phases

We develop the formalism of fermionic matrix product states (fMPS) and show how irreducible fMPS fall in two different classes, related to the different types of simple ${\mathbb{Z}}_{2}$ graded

Fermionic symmetry protected topological phases and cobordisms

A bstractIt has been proposed recently that interacting Symmetry Protected Topological Phases can be classified using cobordism theory. We test this proposal in the case of Fermionic SPT phases with

International Journal of Modern Physics a C World Scientific Publishing Company

We review and extend the progress made over the past few years in understanding the structure of toric quiver gauge theories; those which are induced on the world-volume of a stack of D3-branes

Topological gauge theories and group cohomology

We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH4(BG,Z). In a
...