# Fractons, dipole symmetries and curved spacetime

### Submission summary

 As Contributors: Emil Have Preprint link: scipost_202201_00038v2 Date accepted: 2022-06-10 Date submitted: 2022-06-02 12:13 Submitted by: Have, Emil Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

We study complex scalar theories with dipole symmetry and uncover a no-go theorem that governs the structure of such theories and which, in particular, reveals that a Gaussian theory with linearly realised dipole symmetry must be Carrollian. The gauging of the dipole symmetry via the Noether procedure gives rise to a scalar gauge field and a spatial symmetric tensor gauge field. We construct a worldline theory of mobile objects that couple gauge invariantly to these gauge fields. We systematically develop the canonical theory of a dynamical symmetric tensor gauge field and arrive at scalar charge gauge theories in both Hamiltonian and Lagrangian formalism. We compute the dispersion relation of the modes of this gauge theory, and we point out an analogy with partially massless gravitons. It is then shown that these fractonic theories couple to Aristotelian geometry, which is a non-Lorentzian geometry characterised by the absence of boost symmetries. We generalise previous results by coupling fracton theories to curved space and time. We demonstrate that complex scalar theories with dipole symmetry can be coupled to general Aristotelian geometries as long as the symmetric tensor gauge field remains a background field. The coupling of the scalar charge gauge theory requires a Lagrange multiplier that restricts the Aristotelian geometries.

Published as SciPost Phys. 12, 205 (2022)

### List of changes

In addition to the changes mentioned in the replies to the referees, we have made the following improvements to the manuscript:

* The paragraph on notation has been updated to include our conventions for the Riemann and Ricci tensors.
* We have added a footnote and comment that point out that the worldline actions in Section 3 describe mobile dipoles.
* The third bullet point in the list of conditions imposed on the Hamiltonian in Sec. 4.3 has been updated to: "Quadratic in :math:E_{ij}, so that we can integrate out :math:E_{ij} and obtain a Lagrangian that is second order in time derivatives."
* We have included an analysis of the bounds on the coupling constants that make the Hamiltonian of the scalar charge gauge theory bounded from below in any :math:d\geq 3.
* We have emphasised that the traceless scalar charge gauge theory as we define it is only non-trivial for :math:d\geq 3.
* The argument in Sec. 5.2 that shows that the requirement of minimal torsion implies that :math:C_{\mu\nu}^\rho has been rephrased.
* Sect. 7 has been significantly refined. Highlights of the changes made include:
* Referring to the condition~(7.10) as Einstein'' in dimensions different from $d=3$ was inaccurate, and we have modified the language to reflect this.
* We have added a new section (Sec. 7.1.3) that details the coupling of the traceless scalar charge theory to curved space.
* When :math:d=2, there exists a traceless Chern--Simons-like theory that was discussed in Ref. [32]. We have added a new section (Sec. 7.1.4) that discusses the coupling of this theory to curved space, which reproduces the results of Refs. [31,32].
* As an aid to the reader, we have included tables 1 and 2 which summarise the coupling of various scalar charge gauge theories to both curved space and spacetime in various dimensions.
* The condition (7.39) that the Aristotelian background must satisfy previously included an arbitrary function :math:f. However, our choice of connection implies that this must be zero, and we have corrected the new version to take this into account.
* We have specified the condition for the electric theory alone to be coupled to curved spacetime, which works out to be :math:v^\mu R_{\mu\nu\rho}{^\sigma} = 0 (see Section 7.2.2). As we now show, this amounts to the requirement that the Riemann tensor is entirely spatial.
* The introduction, summary and discussion have all been changed to reflect the changes we have listed above.
* What used to be appendix A has become appendix C to reflect the fact that it is not referred to before we refer to appendices A and B.