We discuss a particular non-relativistic limit of NS-NS gravity that can be taken at the level of the action and equations of motion, without imposing any geometric constraints by hand. This relies on the fact that terms that diverge in the limit and that come from the Vielbein in the Einstein-Hilbert term and from the kinetic term of the Kalb-Ramond two-form field cancel against each other. This cancelling of divergences is the target space analogue of a similar cancellation that takes place at the level of the string sigma model between the Vielbein in the kinetic term and the Kalb-Ramond field in the Wess-Zumino term. The limit of the equations of motion leads to one equation more than the limit of the action, due to the emergence of a local target space scale invariance in the limit. Some of the equations of motion can be solved by scale invariant geometric constraints. These constraints define a so-called Dilatation invariant String Newton-Cartan geometry.

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A non-relativistic limit of NS-NS gravity.

Symmetry plays a central role in quantum ﬁeld theory. Recent developments include symmetries that act on defects and other subsystems, and symmetries that are categorical rather than group-like. These generalized notions of symmetry allow for new kinds of anomalies that constrain dynamics. We review some transformative instances of these novel aspects of symmetry in quantum ﬁeld theory, and give a broad-brush overview of recent applications.

Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond

We reformulate known exotic theories (including theories of fractons) on a Euclidean spacetime lattice. We write them using the Villain approach, and then we modify them to a convenient range of parameters. The new lattice models are closer to the continuum limit than the original lattice versions. In particular, they exhibit many of the recently found properties of the continuum theories, including emergent global symmetries and surprising dualities. In addition, these new models provide a clear and rigorous formulation to the continuum models and their singularities. In Appendixes A–C, we use this approach to review the well-studied lattice models and their continuum limits. These include the XY-model, the ZN clock-model, and various gauge theories in diverse dimensions. This paper clarifies the relation between the condensed-matter and the high-energy views of these systems. It emphasizes the role of symmetries associated with the topology of field space, duality, and various anomalies.

A modified Villain formulation of fractons and other exotic theories

We study field theories with global dipole symmetries and gauge dipole symmetries. The famous Lifshitz theory is an example of a theory with a global dipole symmetry. We study in detail its 1+1d version with a compact field. When this global symmetry is promoted to a U(1) dipole gauge symmetry, the corresponding gauge field is a tensor gauge field. This theory is known to lead to fractons. In order to resolve various subtleties in the precise meaning of these global or gauge symmetries, we place these 1+1d theories on a lattice and then take the continuum limit. Interestingly, the continuum limit is not unique. Different limits lead to different continuum theories, whose operators, defects, global symmetries, etc. are different. We also consider a lattice gauge theory with a ZN dipole gauge group. Surprisingly, several physical observables, such as the ground state degeneracy and the mobility of defects depend sensitively on the number of sites in the lattice. Our analysis forces us to think carefully about global symmetries that do not act on the standard Hilbert space of the theory, but only on the Hilbert space in the presence of defects. We refer to them as time-like global symmetries and discuss them in detail. These time-like global symmetries allow us to phrase the mobility restrictions of defects (including those of fractons) as a consequence of a global symmetry. ar X iv :2 20 1. 10 58 9v 2 [ co nd -m at .s tr -e l] 1 7 Ju n 20 22

Global dipole symmetry, compact Lifshitz theory, tensor gauge theory, and fractons

/pdf/a-non-relativistic-limit-of-ns-ns-gravity-3cvru5fo9v.pdf

A Non-Relativistic Limit of NS-NS Gravity

We construct field theories in 2 + 1 dimensions with multiple conformal symmetries acting on only one of the spatial directions. These can be considered a conformal extension to “subsystem scale invariances”, borrowing the language often used for fractons.

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Reduced conformal symmetry

We consider multi-energy level distributions in the SYK model, and in particular, the role of global fluctuations in the density of states of the SYK model. The connected contributions to the moments of the density of states go to zero as N → ∞, however, they are much larger than the standard RMT correlations. We provide a diagrammatic description of the leading behavior of these connected moments, showing that the dominant diagrams are given by 1PI cactus graphs, and derive a vector model of the couplings which reproduces these results. We generalize these results to the first subleading corrections, and to fluctuations of correlation functions. In either case, the new set of correlations between traces (i.e. between boundaries) are not associated with, and are much larger than, the ones given by topological wormholes. The connected contributions that we discuss are the beginning of an infinite series of terms, associated with more and more information about the ensemble of couplings, which hints towards the dual of a single realization. In particular, we suggest that incorporating them in the gravity description requires the introduction of new, lighter and lighter, fields in the bulk with fluctuating boundary couplings.

/pdf/multi-trace-correlators-in-the-syk-model-and-non-geometric-a5ivna5lgt.pdf

Multi-trace correlators in the SYK model and non-geometric wormholes

We construct field theories in $2+1$ dimensions with multiple conformal symmetries acting on only one of the spatial directions. These can be considered a conformal extension to "subsystem scale invariances", borrowing the language often used for fractons.

/pdf/reduced-conformal-symmetry-39u0wmnt0t.pdf

Reduced Conformal Symmetry

We investigate the spontaneous breaking of subsystem symmetries directly in the context of continuum field theories by calculating the correlation function of charged operators. Our methods confirm the lack of spontaneous symmetry breaking in some of the existing continuum field theories with subsystem symmetries, as had previously been established based on a careful analysis of the spectrum. We present some novel continuum field theory constructions that do exhibit spontaneous symmetry breaking whenever allowed by general principles. These interesting patterns of symmetry breaking occur despite the fact that all the theories we study are non-interacting.

Spontaneously Broken Subsystem Symmetries

We consider multi-energy level distributions in the SYK model, and in particular, the role of global fluctuations in the density of states of the SYK model The connected contributions to the moments of the density of states go to zero as $N \to \infty$, however, they are much larger than the standard RMT correlations We provide a diagrammatic description of the leading behavior of these connected moments, showing that the dominant diagrams are given by 1PI cactus graphs, and derive a vector model of the couplings which reproduces these results We generalize these results to the first subleading corrections, and to fluctuations of correlation functions In either case, the new set of correlations between traces (ie between boundaries) are not associated with, and are much larger than, the ones given by topological wormholes The connected contributions that we discuss are the beginning of an infinite series of terms, associated with more and more information about the ensemble of couplings, which hints towards the dual of a single realization In particular, we suggest that incorporating them in the gravity description requires the introduction of new, lighter and lighter, fields in the bulk with fluctuating boundary couplings

Amir Raz

Papers

Reduced conformal symmetry

Multi-trace correlators in the SYK model and non-geometric wormholes

Reduced Conformal Symmetry

Spontaneously Broken Subsystem Symmetries

Multi-trace Correlators in the SYK Model and Non-geometric Wormholes