Abstract
For a spacetime of odd dimensions endowed with a unit vector field, we introduce a new topological current that is identically conserved and whose charge is equal to the Euler character of the even dimensional spacelike foliations. The existence of this current allows us to introduce new Chern-Simons-type terms in the effective field theories describing relativistic quantum Hall states and (2+1) dimensional superfluids. Using effective field theory, we calculate various correlation functions and identify transport coefficients. In the quantum Hall case, this current provides the natural relativistic generalization of the Wen-Zee term, required to characterize the shift and Hall viscosity in quantum Hall systems. For the superfluid case this term is required to have nonzero Hall viscosity and to describe superfluids with non s-wave pairing.
Highlights
A electromagnetic Chern-Simons term ǫμνρAμ∂νAρ as well as a mixed ChernSimons term ǫμνρAμ∂νωρ (ω being the spin connection of the spatial manifold), the latter often called the Wen-Zee coupling
For a spacetime of odd dimensions endowed with a unit vector field, we introduce a new topological current that is identically conserved and whose charge is equal to the Euler character of the even dimensional spacelike foliations
We have constructed a new current in odd dimensions, whose charge is the Euler characteristic of the codimension one hypersurface on which it is calculated. We showed that it is identically conserved but its construction requires the existence of a vector field of unit norm
Summary
∇νuμ is at each point constrained in the two dimensional surface perpendicular to u. Where in the first equality the first term vanishes as we have three vectors ∇u perpendicular to u contracted with an epsilon, and the last term vanishes by the second Bianchi identity. We see that this current is identically conserved. We emphasize that the arguments above require the vector field to be of constant norm. What this means is that we require a nowhere vanishing vector field that we can normalize. We can consider small fluctuations about a large magnetic field. We will consider this example in detail
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