## Algebraic & Geometric Topology

### Spin, statistics, orientations, unitarity

Theo Johnson-Freyd

#### 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 $360∘$–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 , but rather than being defined in terms of a single tangential structure, they are defined in terms of a bundle of tangential structures over $Spec(ℝ)$. 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 . We interpret Deligne’s “existence of super fiber functors” theorem as implying that 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

#### 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

#### References

• M Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68 (1988) 175–186
• D Ayala, Geometric cobordism categories, PhD thesis, Stanford University (2009) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/304999097 {\unhbox0
• D Ayala, J Francis, Factorization homology of topological manifolds, J. Topol. 8 (2015) 1045–1084
• J,C Baez, J Dolan, From finite sets to Feynman diagrams, from “Mathematics unlimited–-2001 and beyond” (B Engquist, W Schmid, editors), Springer, Berlin (2001) 29–50
• G,J Bird, Limits in $2$–categories of locally presentable categories, PhD thesis, University of Sydney (1984) Available at \setbox0\makeatletter\@url http://maths.mq.edu.au/~street/BirdPhD.pdf {\unhbox0
• M Brandenburg, A Chirvasitu, T Johnson-Freyd, Reflexivity and dualizability in categorified linear algebra, Theory Appl. Categ. 30 (2015) 808–835
• D Calaque, C Scheimbauer, A note on the $(\infty,n)$–category of cobordisms, preprint (2015)
• A Chirvasitu, T Johnson-Freyd, The fundamental pro-groupoid of an affine $2$–scheme, Appl. Categ. Structures 21 (2013) 469–522
• P Deligne, Catégories tensorielles, Mosc. Math. J. 2 (2002) 227–248
• C,L Douglas, C Schommer-Pries, N Snyder, The balanced tensor product of module categories, preprint (2014)
• S Eilenberg, Abstract description of some basic functors, J. Indian Math. Soc. 24 (1960) 231–234
• D,S Freed, M,J Hopkins, Reflection positivity and invertible topological phases, preprint (2016)
• N Ganter, M Kapranov, Symmetric and exterior powers of categories, Transform. Groups 19 (2014) 57–103
• R Haugseng, Iterated spans and “classical” topological field theories, preprint (2014)
• M Kapranov, Supergeometry in mathematics and physics, preprint (2015)
• G,M Kelly, Basic concepts of enriched category theory, London Math. Soc. Lecture Note Ser. 64, Cambridge University Press (1982)
• A Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics 163, Cambridge University Press (2005)
• L,G Lewis, Jr, When projective does not imply flat, and other homological anomalies, Theory Appl. Categ. 5 (1999) 202–250
• D Li-Bland, The stack of higher internal categories and stacks of iterated spans, preprint (2015)
• J Lurie, On the classification of topological field theories, from “Current developments in mathematics” (D Jerison, B Mazur, T Mrowka, W Schmid, R Stanley, S-T Yau, editors), Int. Press, Somerville, MA (2009) 129–280
• V Ostrik, On symmetric fusion categories in positive characteristic, preprint (2015)
• C,J Schommer-Pries, The classification of two-dimensional extended topological field theories, PhD thesis, University of California, Berkeley (2009) Available at \setbox0\makeatletter\@url \unhbox0
• G Segal, The definition of conformal field theory, from “Topology, geometry and quantum field theory” (U Tillmann, editor), London Math. Soc. Lecture Note Ser. 308, Cambridge University Press (2004) 421–577
• S Stolz, P Teichner, Supersymmetric field theories and generalized cohomology, from “Mathematical foundations of quantum field theory and perturbative string theory” (H Sati, U Schreiber, editors), Proc. Sympos. Pure Math. 83, Amer. Math. Soc., Providence, RI (2011) 279–340
• R,F Streater, A,S Wightman, PCT, spin and statistics, and all that, W,A Benjamin, New York (1964)
• C,E Watts, Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc. 11 (1960) 5–8