# The 576-fold Bott Periodicity of the Majorana fermions

### APA

Henriques, A. (2016). The 576-fold Bott Periodicity of the Majorana fermions. Perimeter Institute. https://pirsa.org/16040053

### MLA

Henriques, Andre. The 576-fold Bott Periodicity of the Majorana fermions. Perimeter Institute, Apr. 06, 2016, https://pirsa.org/16040053

### BibTex

@misc{ pirsa_PIRSA:16040053, doi = {10.48660/16040053}, url = {https://pirsa.org/16040053}, author = {Henriques, Andre}, keywords = {Other}, language = {en}, title = {The 576-fold Bott Periodicity of the Majorana fermions}, publisher = {Perimeter Institute}, year = {2016}, month = {apr}, note = {PIRSA:16040053 see, \url{https://pirsa.org}} }

Andre Henriques University of Oxford

## Abstract

Bott periodicity (1956) is a classical and old result in mathematics.

Its easiest incarnation of which concerns Clifford algebras. It says

that, up to Morita equivalence, the real Clifford algebras Cl_1(R),

Cl_2(R), Cl_3(R), etc. repeat with period 8. A similar result holds

for complex Clifford algebras, where the period is now 2. The modern

way of phrasing Bott periodicity in is terms of K-theory: I will

explain how one computes K-theory, and we will see the 8-fold Bott

periodicity emerge from the computations.

Elliptic cohomology is a fancy version of K-theory which can be

thought of as the K-theory of the loop space. A useful slogan is that

K-theory is to quantum mechanics, what elliptic cohomology is to

string theory. This cohomology theory satisfies a version of Bott

periodicity, with period 576. I will explain where that number 576

comes from, and what physical significance this might have.

I conjecture that the above 576-fold periodicity reflects itself in

the classification of 3d TQFTs. Here, the relevant TQFTs are the ones

associated to the chiral Majorana fermion (a type of abelian

Chern-Simons theory of central charge c=1/2). The claim is that the

theory becomes trivial once the central charge reaches 576·1/2 = 288.

The classification of abelian Chern-Simons theories has been

considered by Belov-Moore (2005), who claimed that the periodicity was

reached at c = 24 and later by Kapustin-Saulina (2010), who claimed

that the periodicity was never reached. Our proposal lies strictly in

between those of Belov-Moore and Kapustin-Saulina.