A time-reversal invariant vortex in topological superconductors and gravitational Z_2 topology

We study a class DIII topological superconductor in the presence of a time-reversal invariant

vortex. The eigenmodes of the Bogoliubov-de-Genne (BdG) Hamiltonian show a Z_2 topology:

the time-reversal invariant vortex with odd winding number supports a helical Majorana zero-

modes at the vortex, while there is no such zero-modes when the winding number is even. We find

that this Z_2 structure can be interpreted as an emergent gravitational effect. Identifying the gap

function as spatial components of the vielbein in the theory of gravity, we can explicitly convert

the BdG equation into the Dirac equation coupled to a nontrivial gravitational background. Then

the gravitational curvature is induced at the vortex core, with its total flux quantized in integer

multiples of pi, reflecting the Z_2 topological structure. Although the curvature vanishes everywhere

except at the vortex core, the fermionic spectrum remains sensitive to the total curvature flux, owing

to the gravitational Aharonov-Bohm effect.