Abstract
We study the coupling of a 2 + 1 dimensional non-relativistic spin 1/2 fermion to a curved Newton-Cartan geometry, using null reduction from an extra-dimensional relativistic Dirac action in curved spacetime. We analyze Weyl invariance in detail: we show that at the classical level it is preserved in an arbitrary curved background, whereas at the quantum level it is broken by anomalies. We compute the trace anomaly using the Heat Kernel method and we show that the anomaly coefficients a, c are proportional to the relativistic ones for a Dirac fermion in 3 + 1 dimensions. As for the previously studied scalar case, these coefficents are proportional to 1/m, where m is the non-relativistic mass of the particle.
Highlights
Degrees of freedom coming from a given ultraviolet (UV) description
We compute the trace anomaly using the Heat Kernel method and we show that the anomaly coefficients a, c are proportional to the relativistic ones for a Dirac fermion in 3 + 1 dimensions
Since in our conventions the charge associated to the particle number symmetry is q = m, we find a gyromagnetic ratio g = 1
Summary
We will consider the coupling of non-relativistic fermions in 2 + 1 dimensions to a background NC geometry. The velocity vector is not unique (it is only required to satisfy nμvμ = 1) and the ambiguity in the choice of v is related to the last ingredient of the NC geometry: a non-dynamical gauge field Aμ, whose presence is necessary to guarantee Milne boost invariance. This gauge field will act as a source for the particle number symmetry. Since we are dealing with spinors, the covariant derivative contains the spin connection term and it is necessary to introduce an orthonormal frame field (vielbein) which relates the metric in the curved spacetime with the flat tangent space.
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