The 576-fold Bott Periodicity of the Majorana fermions


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


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


          @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

Talk Type Scientific Series


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.