Abstract
Holography for Lifshitz space-times corresponds to dual field theories on a fixed torsional Newton-Cartan (TNC) background. We examine the coupling of non-relativistic field theories to TNC backgrounds and uncover a novel mechanism by which a global U(1) can become local. This involves the TNC vector $M_\mu$ which sources a particle number current, and which for flat NC space-time satisfies $M_{\mu}=\partial_{\mu}M$ with a Schroedinger symmetry realized on $M$. We discuss various toy model field theories on flat NC space-time for which the new mechanism leads to extra global space-time symmetries beyond the generic Lifshitz symmetry, allowing for an enhancement to Schroedinger symmetry. On the holographic side, the source $M$ also appears in the Lifshitz vacuum with exactly the same properties as for flat NC space-time. In particular, the bulk diffeomorphisms that preserve the boundary conditions realize a Schroedinger algebra on $M$, allowing for a conserved particle number current. Finally, we present a probe action for a complex scalar field on the Lifshitz vacuum, which exhibits Schroedinger invariance in the same manner as seen in the field theory models.
Highlights
Extending holography to settings that go beyond the original AdS-setup has received considerable attention in recent years
The resulting local transformations of the sources are given in (2.43) which is in agreement with the way background fields transform in torsional Newton-Cartan (TNC) geometry [17]
The resulting TNC geometry on the boundary is discussed in subsection 2.4 and readers who are not interested in the holographic origin of this geometry may immediately jump to this subsection
Summary
Extending holography to settings that go beyond the original AdS-setup has received considerable attention in recent years This has been motivated in part by applying holographic ideas to the study of strongly coupled condensed matter systems, which often exhibit non-relativistic scaling, and necessitate the consideration of bulk space-times with asymptotics different from AdS [1,2,3,4]. The work of [17] was used in the holographic context to show [15, 16] that for Lifshitz space-times there is an underlying Schrodinger symmetry that acts on the sources and vevs, strongly suggesting that the boundary theory can have a global Schrodinger invariance This observation was supported in the Letter [19] by a complimentary analysis of bulk versus boundary Killing symmetries (employing the TNC analogue of a conformal Killing vector [14]), by considering the conditions for the boundary theory to admit conserved currents. See the works [41,42,43] for a different approach to NC geometry
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