## Algebraic & Geometric Topology

- Algebr. Geom. Topol.
- Volume 17, Number 2 (2017), 917-956.

### Spin, statistics, orientations, unitarity

#### Abstract

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 both spin and super, and $36{0}^{\circ}$–rotation of the spin structure agrees with the operation of flipping the signs of all fermions. We set up a framework in which these two notions are precisely analogous. In this framework, field theories are defined over ${\mathrm{\text{Vect}}}_{\mathbb{R}}$, but rather than being defined in terms of a single tangential structure, they are defined in terms of a bundle of tangential structures over $Spec\left(\mathbb{R}\right)$. Bundles of tangential structures may be étale-locally equivalent without being equivalent, and hermitian field theories are nothing but the field theories controlled by the unique nontrivial bundle of tangential structures that is étale-locally equivalent to Orientations. This bundle owes its existence to the fact that ${\pi}_{1}^{\text{\xe9t}}\left(Spec\left(\mathbb{R}\right)\right)={\pi}_{1}BO\left(\infty \right)$. We interpret Deligne’s “existence of super fiber functors” theorem as implying that ${\pi}_{2}^{\text{\xe9t}}\left(Spec\left(\mathbb{R}\right)\right)={\pi}_{2}\phantom{\rule{0.3em}{0ex}}BO\left(\infty \right)$ in a categorification of algebraic geometry in which symmetric monoidal categories replace commutative rings. One finds that there are eight bundles of tangential structures étale-locally equivalent to Spins, one of which is distinguished; upon unpacking the meaning of a field theory with that distinguished tangential structure, one arrives at a field theory that is both hermitian and satisfies spin-statistics. Finally, we formulate in our framework a notion of reflection-positivity and prove that if an étale-locally-oriented field theory is reflection-positive then it is necessarily hermitian, and if an étale-locally-spin field theory is reflection-positive then it necessarily both satisfies spin-statistics and is hermitian. The latter result is a topological version of the famous spin-statistics theorem.

#### Article information

**Source**

Algebr. Geom. Topol., Volume 17, Number 2 (2017), 917-956.

**Dates**

Received: 28 October 2015

Revised: 6 June 2016

Accepted: 24 June 2016

First available in Project Euclid: 19 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.agt/1508431449

**Digital Object Identifier**

doi:10.2140/agt.2017.17.917

**Mathematical Reviews number (MathSciNet)**

MR3623677

**Zentralblatt MATH identifier**

1361.81142

**Subjects**

Primary: 14A22: Noncommutative algebraic geometry [See also 16S38] 57R56: Topological quantum field theories 81T50: Anomalies

**Keywords**

TQFT spin super categorification torsors Galois theory

#### Citation

Johnson-Freyd, Theo. Spin, statistics, orientations, unitarity. Algebr. Geom. Topol. 17 (2017), no. 2, 917--956. doi:10.2140/agt.2017.17.917. https://projecteuclid.org/euclid.agt/1508431449