On $dS_4$ extremal surfaces and entanglement entropy in some ghost CFTs

TL;DR

This work probes how negative central charge theories, exemplified by bc-ghosts with $c=-2$ and a complex anti-commuting scalar CFT, can yield a formally defined, negative entanglement entropy via the replica trick, illuminating the negative areas found for complex extremal surfaces in $dS_4$ within the $dS/CFT$ framework. By analyzing the $SL(2)$ vacuum versus ghost-ground state, the role of background charge, and zero modes, the paper shows that the replica construction produces $S_A = -{2\over 3}\log {l\over \epsilon}$ for a single interval, with twist operators carrying negative dimensions. Extending to matter+ghost systems reveals an effective cancellation of entanglement, $S_A = {c_m-2\over 3}\log {l\over \epsilon}$, while a logarithmic $\chi\bar{\chi}$ CFT yields analogous results, reinforcing that negative EE is a formal, nonunitary feature. A simple ghost-spin toy model demonstrates how non-positive norms can give reduced density matrices with negative eigenvalues and complex entropies. Collectively, these toy models provide a controlled arena to interpret the negative areas of $dS_4$ complex extremal surfaces and spur further investigation into nonunitary holography and possible duals to de Sitter spaces.

Abstract

In arXiv:1501.03019 [hep-th], the areas of certain complex extremal surfaces in de Sitter space were found to have resemblance with entanglement entropy in appropriate dual Euclidean non-unitary CFTs, with the area being real and negative in $dS_4$. In this paper, we study some toy models of 2-dim ghost conformal field theories with negative central charge with a view to exploring this further from the CFT point of view. In particular we consider $bc$-ghost systems with central charge $c=-2$ and study the replica formulation for entanglement entropy for a single interval, and associated issues arising in this case, notably pertaining to (i) the $SL(2)$ vacuum coinciding with the ghost ground state, and (ii) the background charge inherent in these systems which leads to particular forms for the norms of states (involving zero modes). This eventually gives rise to negative entanglement entropy. We also discuss a (logarithmic) CFT of anti-commuting scalars, with similarities in some features. Finally we discuss a simple toy model of two "ghost-spins" which mimics some of these features.