Abstract
We investigate the origins and implications of the duality between topological insulators and topological superconductors in three and four spacetime dimensions. In the latter, the duality transformation can be made at the level of the path integral in the standard way, while in three dimensions, it takes the form of “self-duality in odd dimensions”. In this sense, it is closely related to the particle-vortex duality of planar systems. In particular, we use this to elaborate on Son’s conjecture that a three dimensional Dirac fermion that can be thought of as the surface mode of a four dimensional topological insulator is dual to a composite fermion.
Highlights
On the other hand, a topological superconductor is a superconductor with fully gapped quasi-particle excitations in the bulk — the Cooper pairs responsible for superconductivity — but has topologically protected, gapless quasi-particle states on the boundary
We investigate the origins and implications of the duality between topological insulators and topological superconductors in three and four spacetime dimensions
It is closely related to the particle-vortex duality of planar systems. We use this to elaborate on Son’s conjecture that a three dimensional Dirac fermion that can be thought of as the surface mode of a four dimensional topological insulator is dual to a composite fermion
Summary
To summarise our introductory comments, the actions for a topological insulator and a topological superconductor are given by (1.1) and (1.2) respectively. At suffienctly large energies where we can ignore the mass term for the photon Aμ, this relation is nothing but the usual Maxwell duality,. The θ-term is topological and so does not contribute to the equation of motion. On the other hand, varying the master action with respect to Fμν, obtains the equation of motion. Which is, the action (1.1) for the topological insulator. The relation between the two fields is the usual Maxwell duality. This confirms our statement that in 3+1 dimensions, topological insulators and topological superconductors are related through Maxwell electric-magnetic duality. Note that we needed to be at sufficiently large energies for this
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