Dissipative hydrodynamics with higher-form symmetry

Abstract

A theory of parity-invariant dissipative fluids with q-form symmetry is formulated to first order in a derivative expansion. The fluid is anisotropic with symmetry SO(D − 1 − q) × SO(q) and carries dissolved q-dimensional charged objects that couple to a (q + 1)-form background gauge field. The case q = 1 for which the fluid carries string charge is related to magnetohydrodynamics in D = 4 spacetime dimensions. We identify q+7 parity-even independent transport coefficients at first order in derivatives for q > 1. In particular, compared to the q = 1 case under the assumption of parity and charge conjugation invariance, fluids with q > 1 are characterised by q extra transport coefficients with the physical interpretation of shear viscosity in the SO(q) sector and current resistivities. We discuss certain issues related to the existence of a hydrostatic sector for fluids with higher-form symmetry for any q ≥ 1. We extend these results in order to include an interface separating different fluid phases and study the dispersion relation of capillary waves finding clear signatures of anisotropy. The formalism developed here can be easily adapted to study hydrodynamics with multiple higher-form symmetries.fluid dynamics [4][5][6][7][8][9]; the establishment of a framework for describing interfaces between different fluid phases [10-12]; a new formalism for studying non-relativistic fluids [13][14][15][16]; and the development of hydrodynamic theories with generalised global 1-form symmetries and their connections to magnetohydrodynamics [17][18][19][20] as well as their role in the understanding of effective theories with translational symmetry breaking and states with dynamical defects [21]. This paper introduces a framework for building effective hydrodynamic theories of dissipative fluids with q-form symmetries, generalising previous work for q = 0, 1. These effective theories correspond to the hydrodynamic limit of microscopic descriptions whose underlying fundamental charged objects are q-dimensional (i.e. q-dimensional branes). These q-dimensional objects couple to a background gauge field A q+1 . In the language of [22], these fluids describe microscopic systems with a generalised q-form global symmetry. Associated with the q-form symmetry is a (q + 1)-form current J whose integral over a (D − q − 1) dimensional hypersurface M Γ yields a conserved dipole charge( 1.1) where the operator is the Hodge dual operator in D-dimensional spacetime. This dipole charge counts the number of q-dimensional objects that cross the (D − q − 1)-dimensional hypersurface M Γ . 1 The hydrodynamic theories constructed here capture the collective excitations of these charged q-dimensional objects around a state of thermal equilibrium.

Viscoelastic hydrodynamics and holography

“…The case of d = 2 involving two one-form symmetries was considered in [22], albeit in a very restrictive case and ignoring the issues that require partial symmetry breaking. The understanding of partial symmetry breaking is essential for consistency of higher-form hydrodynamics with thermal equilibrium partition functions, as has been previously observed in [25,26].…”

“…In app. C.3 we provide the comparison between the ideal order higher-form hydrodynamics of this section with that of [25].…”

“…We can also define a projector This can be compared directly to section 2 of [25] with Q = q 11 √ γ 11 and µ = √ γ 11 . In principle, we can extend this comparison to include one-derivative corrections as well.…”

“…In this section, we compare our construction of viscoelastic fluids in terms of higher-form symmetries given in section 4 to the higher-form hydrodynamics formulated in [25]. The work of [25] only involves a single higher-form symmetry as opposed to the multiple copies generically required in viscoelastic hydrodynamics. Therefore, a precise comparison is only possible for k = 1.…”

“…Recently, it has been argued that viscoelastic hydrodynamics can be recast as a theory of higherform hydrodynamics making its global symmetries manifest and avoiding the need to introduce microscopic dynamical fields [22]. This follows the recent line of research where hydrodynamic systems with dynamical fields, such as magnetohydrodynamics with dynamical gauge fields, are recast in terms of dual hydrodynamic systems with higher-form symmetries [23][24][25][26][27][28]. Building up on this idea, here we provide a completely equivalent description of viscoelastic hydrodynamics of anisotropic (liquid) crystals in arbitrary spacetime dimensions by identifying the correct degrees of freedom of higher-form hydrodynamics.…”