Area terms and entanglement entropy in the $c=1$ string theory

TL;DR

The paper addresses whether a dilaton-dependent area term appears in the generalized entropy of subregions in the $c=1$ string theory. It analyzes both the target-space EFT and the dual matrix quantum mechanics, showing that the nonlocal leg-pole transformation in the singlet sector does not produce a leading area term, and that any such term would likely reside in non-singlet sectors or in more subtle subregion constructions. Through covariant phase-space methods, UV-finiteness considerations, and targeted numerics in the singlet sector, the work argues that the area term is not captured by singlet MQM entanglement and highlights the potential role of non-singlets or generalized entanglement wedges in realizing the gravitational entropy. Overall, the results clarify how gravitational entropy notions map to microscopic degrees of freedom in the $c=1$ duality and guide future efforts to identify area-law contributions in low-dimensional holographic-like systems.

Abstract

We study entanglement entropy in the low-energy effective field theory of two-dimensional string theory as well as in the singlet sector of the dual $c=1$ matrix quantum mechanics. From the target space perspective, we argue that a generic bulk subregion is expected to have an associated generalized entanglement entropy combining a dilaton-dependent gravitational term and a matter contribution coming from the tachyon. Given that the gravitational area-like term is absent in previous analyses of entanglement entropy in the $c=1$ model, we examine several possible mechanisms for its emergence. We show that the nonlocal transformation induced by the leg-pole factor that relates the target space tachyon and the matrix model collective excitations cannot account for the area-like term, and we comment on its possible origin in the non-singlet sectors of the theory.

Figures (3)

• Figure 1: Entanglement entropy for an interval $[\bar{\phi}-\Delta\phi,\bar{\phi}+\Delta\phi]$ in target space. Far away from the tachyon wall at $\phi=0$ the entanglement entropy scales logarithmically with both $\Delta\phi$ and $\bar{\phi}$. For intervals where the endpoint $\bar{\phi}+\Delta\phi$ comes close to the tachyon wall, the entanglement entropy starts oscillating. The computation used a lattice of 801 points distributed symmetrically around $\phi=0$ with lattice spacing $a=0.025$.
• Figure 2: Entanglement entropy at fixed $\bar{\phi}$ and varying $\Delta\phi$ (LHS) and at fixed $\Delta\phi$ and varying $\bar{\phi}$ (RHS). The logarithmic scaling of the entanglement entropy for small intervals far away from the tachyon wall is clearly visible on the left hand side (note the logarithmic scale of the $\Delta\phi$ axis). The computation used a lattice of 801 points distributed symmetrically around $\phi=0$ with lattice spacing $a=0.025$.
• Figure 3: Plot of the nonlocal transformation $K(\tau)$. The function falls off exponentially to the left in the red-shaded region while it oscillates with increasing frequency to the right in the green-shaded region. Only the region close to $\tau=0$ contributes significantly as other contributions are either exponentially small or average out. Thus, the transformation $K(\tau)$ only introduces a small amount of nonlocality with the largest contributions coming from close to the point where the eigenvalue and target space coordinates are equal.