Abstract
We describe the physics of fermionic Lifschitz theories once the anisotropic scaling exponent is made arbitrarily small. In this limit the system acquires an enhanced (Carrollian) boost symmetry. We show, both through the explicit computation of the path integral Jacobian and through the solution of the Wess-Zumino consistency conditions, that the translation symmetry in the anisotropic direction becomes anomalous. This turns out to be a mixed anomaly between boosts and translations. In a Newton-Cartan formulation of the space-time geometry such anomaly is sourced by torsion. We use these results to give an effective field theory description of the anomalous transport coefficients, which were originally computed through Kubo formulas in [1]. Along the way we provide a link with warped CFTs.
Highlights
We describe the physics of fermionic Lifschitz theories once the anisotropic scaling exponent is made arbitrarily small
In this limit the system acquires an enhanced (Carrollian) boost symmetry. Both through the explicit computation of the path integral Jacobian and through the solution of the Wess-Zumino consistency conditions, that the translation symmetry in the anisotropic direction becomes anomalous
We can use a wider class of regulators than in the isotropic case, we have shown that the independent regulatory parameters vanish as z → 0, leaving behind only powers of q fixed by the spurionic symmetry
Summary
We study the Fujikawa regularization of translations in the anisotropic direction for the action (1.6), which for clarity we racall here. To find the anomalous variation of the effective action we first need to couple this system to the relevant external gauge fields. In our case this means introducing the appropriate background geometry for the Lifschitz system. In [1] this was realized by considering a Newton-Cartan setup without any boost symmetry nor U(1) gauge field The spin connection has the usual form in terms of the vielbein eaμ In this geometry one may decompose any vector field ξμ = θvμ + ξaEaμ, with θ , ξa well defined parmaters due to the absence of boost symmetry. We will be interested in understanding how the Ward identity for πμ may be violated in the presence of nontrivial background torsion
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