Generalised global symmetries in states with dynamical defects: the case of the transverse sound in field theory and holography

TL;DR

The paper develops a symmetry-based framework for states with conserved dynamical defects using generalised global (one-form) symmetries, focusing on a 2+1D viscoelastic-like state with two perpendicular sets of line defects. An effective field theory built from two-form currents and stress-energy recovers transverse sound and mirrors elasticity, establishing an equivalence between a two-form symmetry EFT and conventional elasticity at linear order. A holographic dual with two-form bulk fields is constructed, and the thermodynamics, hydrodynamics, and full spectrum (including zero and finite density) are analyzed, revealing RG-scale–controlled mass gaps and light-like modes consistent with the EFT. The work demonstrates a coherent EFT-holography correspondence for systems governed by higher-form symmetries and highlights how mixed boundary conditions and running couplings shape the IR/UV structure. This provides a robust framework for exploring line-defect fluids and could inform studies of novel viscoelastic phases in condensed matter and holographic contexts.

Abstract

In this work, we show how states with conserved numbers of dynamical defects (strings, domain walls, etc.) can be understood as possessing generalised global symmetries even when the microscopic origins of these symmetries are unknown. Using this philosophy, we build an effective theory of a $2+1$-dimensional fluid state with two perpendicular sets of immersed elastic line defects. When the number of defects is independently conserved in each set, then the state possesses two one-form symmetries. Normally, such viscoelastic states are described as fluids coupled to Goldstone bosons associated with spontaneous breaking of translational symmetry caused by the underlying microscopic structure---the principle feature of which is a transverse sound mode. At the linear, non-dissipative level, we verify that our theory, based entirely on symmetry principles, is equivalent to a viscoelastic theory. We then build a simple holographic dual of such a state containing dynamical gravity and two two-form gauge fields, and use it to study its hydrodynamic and higher-energy spectral properties characterised by non-hydrodynamic, gapped modes. Based on the holographic analysis of transverse two-point functions, we study consistency between low-energy predictions of the bulk theory and the effective boundary theory. Various new features of the holographic dictionary are explained in theories with higher-form symmetries, such as the mixed-boundary-condition modification of the quasinormal mode prescription that depends on the running coupling of the boundary double-trace deformations. Furthermore, we examine details of low- and high-energy parts of the spectrum that depend on temperature, line defect densities and the renormalisation group scale.

Figures (6)

• Figure 1: The equilibrium configuration with two sets of perpendicular lines. The number of these fluctuating extended objects (defects) is conserved in each of the two mutually perpendicular sets of strings, which gives rise to two independent generalised global symmetries.
• Figure 2: A depiction of a transverse deformation of the line defect structures (a transverse lattice displacement) used to construct our theory in Section \ref{['sec:Introduction']}, which is parametrised by $\delta h^y_1$. In the language of conventional elasticity theory, this a shear deformation of the lattice, i.e. the $\delta U_{xy}$ perturbation.
• Figure 3: Plots of imaginary and real parts of the hydrodynamic and the non-hydrodynamic dispersion relations at $\bar{\mathcal{M}} = 5$, with $\bar{\omega} \equiv \omega / r_h$ and $\bar{k} \equiv k / r_h$. Crosses depict the numerically computed poles of the retarded transverse part of the $\langle J^{\mu\nu}_1 J^{\rho\sigma}_1\rangle_R$ correlator at zero density, which follow from the prescription \ref{['QNMModZ']}. The solid lines show the analytical approximation \ref{['Full2solutions']} to the dispersion relations. The dashed lines represent the linear dispersion relation $\bar{\omega} = \pm \bar{k}$.
• Figure 4: Imaginary parts of the hydrodynamic and the non-hydrodynamic dispersion relations of the retarded transverse $\langle J^{\mu\nu}_1 J^{\rho\sigma}_1\rangle_R$ correlator at zero density, plotted at $\bar{\mathcal{M}} = 10$ up to the point of the collision. Crosses depict the full numerically obtained poles and the solid lines are their analytical approximations from Eq. \ref{['Full2solutions']}.
• Figure 5: Comparison between numerically computed speeds of transverse sound waves for $\bar{\mathcal{M}} = \{2,5,10,15,20\}$ and the hydrodynamic, analytical result of Eq. \ref{['eq:VAinAdS']}. The hydrodynamic prediction deviates from numerics for large values of $\bar{m}$. As is apparent from the plots, the agreement is better for larger values of the dimensionless parameter $\bar{\mathcal{M}}$.