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Hydrodynamics of two-dimensional CFTs

Kevin Nguyen

TL;DR

The paper addresses how gravity in $AdS_3$ and its $CFT_2$ dual emerge from hydrodynamics, proposing the Virasoro geometric action on coadjoint orbits as the effective theory for a non-dissipative $2d$ conformal fluid via the conserved tensor $T_{\mu\nu}$. It provides the explicit action $S[F]$ with $b_0=-\frac{c}{48\pi}$ and $T = b_0[F](\partial F)^2 - \frac{c}{24\pi} S[F]$, where $S[F]=\frac{\partial^3 F}{\partial F} - \frac{3}{2} (\frac{\partial^2 F}{\partial F})^2$, and shows how entropy conservation $\partial J_S = -\beta^z \bar{\partial} T$ arises in this framework. The results reproduce the leading perfect-fluid constitutive relations and render the reparametrization mode as fluid-velocity fluctuations, thereby unifying the AdS$_3$/CFT$_2$ holographic description with a hydrodynamic EFT. The work thus offers an EFT perspective on holography where gravity and hydrodynamics are two faces of the same Virasoro geometric-action description.

Abstract

We demonstrate that the geometric action on a coadjoint orbit of the Virasoro group appropriately describes non-dissipative two-dimensional conformal fluids. While this action had already appeared in the context of AdS$_3$ gravity, the hydrodynamical interpretation given here is new. We use this to argue that the geometric action manifestly controls both sides of the fluid/gravity correspondence, where the gravitational `hologram' gives an effective hydrodynamical description of the dual CFT. As a byproduct, our work sheds light on the nature of the AdS$_3$ reparametrization theory used to effectively compute Virasoro vacuum blocks at large central charge, as the reparametrization mode is now understood as a fluctuation of the fluid velocity.

Hydrodynamics of two-dimensional CFTs

TL;DR

The paper addresses how gravity in and its dual emerge from hydrodynamics, proposing the Virasoro geometric action on coadjoint orbits as the effective theory for a non-dissipative conformal fluid via the conserved tensor . It provides the explicit action with and , where , and shows how entropy conservation arises in this framework. The results reproduce the leading perfect-fluid constitutive relations and render the reparametrization mode as fluid-velocity fluctuations, thereby unifying the AdS/CFT holographic description with a hydrodynamic EFT. The work thus offers an EFT perspective on holography where gravity and hydrodynamics are two faces of the same Virasoro geometric-action description.

Abstract

We demonstrate that the geometric action on a coadjoint orbit of the Virasoro group appropriately describes non-dissipative two-dimensional conformal fluids. While this action had already appeared in the context of AdS gravity, the hydrodynamical interpretation given here is new. We use this to argue that the geometric action manifestly controls both sides of the fluid/gravity correspondence, where the gravitational `hologram' gives an effective hydrodynamical description of the dual CFT. As a byproduct, our work sheds light on the nature of the AdS reparametrization theory used to effectively compute Virasoro vacuum blocks at large central charge, as the reparametrization mode is now understood as a fluctuation of the fluid velocity.
Paper Structure (6 sections, 52 equations, 1 figure)

This paper contains 6 sections, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hydrodynamics from Virasoro geometric action
  3. Reparametrization theory revisited
  4. Conclusions
  5. Acknowledgments
  6. Appendix: Hydrodynamics of CFT$_2$

Figures (1)

  • Figure 1: Feynman diagram corresponding to stress tensor exchanges between two pairs of identical operators, borrowed from Nguyen:2021jja. The exchange of the stress tensor operator is effectively computed by exchanging the reparametrization mode $\epsilon(z,\bar{z})$ using a set of effective Feynman rules.