Gravity and the Crossed Product

Abstract

Recently Leutheusser and Liu [1,2] identified an emergent algebra of Type III${}_1$ in the operator algebra of $\mathcal{N}=4$ super Yang-Mills theory for large $N$. Here we describe some $1/N$ corrections to this picture and show that the emergent Type III${}_1$ algebra becomes an algebra of Type II${}_{\infty }$. The Type II${}_{\infty }$ algebra is the crossed product of the Type III${}_1$ algebra by its modular automorphism group. In the context of the emergent Type II${}_{\infty }$ algebra, the entropy of a black hole state is well-defined up to an additive constant, independent of the state. This is somewhat analogous to entropy in classical physics.

28 pp, a number of corrections in v. 2, minor corrections in v. 3,4,5

References in corpus (2)

• Relative entropy and the Bekenstein bound
• Gravity Dual of Connes Cocycle Flow

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