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2-group global symmetries, hydrodynamics and holography

Nabil Iqbal, Napat Poovuttikul

TL;DR

This work extends hydrodynamics to systems with a 2-group global symmetry built from a $U(1)$ zero-form and a $U(1)$ one-form, deriving ideal constitutive relations and thermodynamics governed by Ward identities that couple the two form degrees of freedom. The authors develop an EFT based on Stueckelberg fields to encode the 2-group structure, analyze equilibrium partition functions, and identify novel transport features including an equilibrium current along magnetic flux lines and a chiral-like sound mode. They also propose a minimalist holographic dual with bulk fields $\mathcal{A}_a$ and $\mathcal{B}_{ab}$, showing that holography reproduces the equilibrium current and the chiral mode, and clarifies how the 2-group data are encoded geometrically. The results illuminate how higher-form and ordinary symmetries fuse to yield new collective behavior at finite temperature and density, and open avenues for exploring dissipative corrections, phase structure, and richer higher-group generalizations in holography and field theory.

Abstract

2-group global symmetries are a particular example of how higher-form and conventional global symmetries can fuse together into a larger structure. We construct a theory of hydrodynamics describing the finite-temperature realization of a 2-group global symmetry composed out of $U(1)$ zero-form and $U(1)$ one-form symmetries. We study aspects of the thermodynamics from a Euclidean partition function and derive constitutive relations for ideal hydrodynamics from various points of view. Novel features of the resulting theory include an analogue of the chiral magnetic effect and a chiral sound mode propagating along magnetic field lines. We also discuss a minimalist holographic description of a theory dual to 2-group global symmetry and verify predictions from hydrodynamic descriptions. Along the way we clarify some aspects of symmetry breaking in higher-form theories at finite temperature.

2-group global symmetries, hydrodynamics and holography

TL;DR

This work extends hydrodynamics to systems with a 2-group global symmetry built from a zero-form and a one-form, deriving ideal constitutive relations and thermodynamics governed by Ward identities that couple the two form degrees of freedom. The authors develop an EFT based on Stueckelberg fields to encode the 2-group structure, analyze equilibrium partition functions, and identify novel transport features including an equilibrium current along magnetic flux lines and a chiral-like sound mode. They also propose a minimalist holographic dual with bulk fields and , showing that holography reproduces the equilibrium current and the chiral mode, and clarifies how the 2-group data are encoded geometrically. The results illuminate how higher-form and ordinary symmetries fuse to yield new collective behavior at finite temperature and density, and open avenues for exploring dissipative corrections, phase structure, and richer higher-group generalizations in holography and field theory.

Abstract

2-group global symmetries are a particular example of how higher-form and conventional global symmetries can fuse together into a larger structure. We construct a theory of hydrodynamics describing the finite-temperature realization of a 2-group global symmetry composed out of zero-form and one-form symmetries. We study aspects of the thermodynamics from a Euclidean partition function and derive constitutive relations for ideal hydrodynamics from various points of view. Novel features of the resulting theory include an analogue of the chiral magnetic effect and a chiral sound mode propagating along magnetic field lines. We also discuss a minimalist holographic description of a theory dual to 2-group global symmetry and verify predictions from hydrodynamic descriptions. Along the way we clarify some aspects of symmetry breaking in higher-form theories at finite temperature.

Paper Structure

This paper contains 17 sections, 107 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Summary and outline
  3. 2-group background field and equilibrium partition function
  4. 1-form symmetries in thermal equilibrium
  5. 2-group global symmetries in thermal equilibrium
  6. Hydrodynamic effective action
  7. Internal symmetries of Stueckelberg fields
  8. Retarded 2-point functions and hydrodynamic modes
  9. Minimalist' holographic dual
  10. Holographic deconstruction of 2-group gauge theory
  11. Equilibrium current and chiral sound modes from holography
  12. Current one-point function
  13. Quasinormal modes
  14. Discussions and outlook
  15. Entropy production analysis
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Illustration of $\rho_b$ strings per unit area pointing along the direction of a spatial unit vector $h^\mu$ combined with $\rho_a$ particle densities. This ensemble arrange itself in such a way that the 2-group Ward identities are satisfied.
  • Figure 2: Summary of the procedure which reduced the dependence of bulk fields $\mathcal{A}_{a},\mathcal{B}_{ab}$ to thermodynamic quantities $\{ \mu_a,\mu_b h_i\}$. From left to right, we list all the bulk fields and the Stueckelberg fields in the dual QFT. Upon imposing the radial gauge, the bulk theory can only depends on $\mathcal{A}^{(2)}_\mu,\mathcal{B}^{(2)}_{\mu\nu}$ in \ref{['eq:bulkgauge-1']},\ref{['eq:bulkgauge-2']} dual to $A_\mu,B_{\mu\nu}$ in \ref{['eq:defGAB']}. Lastly, we identify the residual gauge transformation as the shift symmetry in \ref{['eq:ZeroFormShift-1']}-\ref{['eq:ZeroFormShift-2']} and \ref{['eq:OneFormShift']}. Requiring that the physical quantities is independent of the residual gauge transformation, we conclude that the effective action of the dual QFT can only depends on $\mu_a$ and $\mu_b h_\mu$ in \ref{['def:mua']} and \ref{['def:mub']}.
  • Figure 3: Illustration of the Schwinger-Keldysh time contour for real-time evolution of the thermal state and its corresponding geometry constructing with the procedure in Skenderis:2008dhSkenderis:2008dg, see also deBoer:2018qqm for more explicit computations. The bulk geometry is obtained by sewing the two Lorentzian AdS spaces at $t_L = t_R = 0$ with the Euclidean AdS at $\tau = 0$ and $\tau = \beta$ along the dashed line and at $t_R = -t_L = \infty$. This way, the time evolution of the boundary of the space on the right corresponds to the time contour on the left.
  • Figure 4: (LEFT) The real part of $\mathfrak{w} = \omega/(2\pi T)$ as a function of $\mathfrak{q} = q_z/(2\pi T)$. The $\bullet$ denotes the numerical result while the dashed line corresponds to the predicted dispersion relation in \ref{['eq:chiralSound']}$\text{Re}\, \omega = -c_s q_z$ with the value of $c_s$ obtained via \ref{['eq:speedOfSoundHolo']} for $\mathcal{K} = \{1,1/2,1/8\}$. (RIGHT) The numerical result of quasinormal mode in complex $\mathfrak{w}-$plane showing the behaviour $\text{Im}\, \mathfrak{w} \sim \mathfrak{q}^2$ at small $\text{Re }\mathfrak{w}$ and/or $\mathfrak{q}$ as we vary ${\bf q}$ from positive to negative values.