Algebraic perturbation theory: traversable wormholes and generalized entropy beyond subleading order

TL;DR

The paper develops a perturbative crossed-product framework to study unitary deformations of type III algebras, producing semifinite type II algebras that admit a trace and well-defined entropy. It shows that expectation values in the crossed product acquire a perturbation-dependent weighting determined by the modular charge, enabling systematic computation of entropy corrections to arbitrary order. Applying this to a traversable wormhole in AdS/CFT, the work extends Gao–Jafferis–Wall by yielding new contributions to the generalized entropy beyond linear order, including changes to the area term via QES dynamics. The results illustrate how algebraic methods can address nonlocal modular structure in quantum gravity while highlighting conceptual and technical challenges in extending the framework to dynamical spacetime regions.

Abstract

The crossed product has recently emerged as an important ingredient in describing algebras of observables for quantum field theory and gravity. We combine this with perturbation theory, and study perturbative crossed product algebras obtained from a unitary deformation of the original system. Motivated by the problem of black hole evaporation, we propose an abstract framework in which black hole information can be transferred to Hawking radiation by passing to a perturbative crossed product exhibiting a degree of non-locality in its modular structure. As both a concrete example and a toy model for evaporation, we analyze the algebra of observables of the traversable wormhole in anti-de-Sitter space. We obtain new contributions to the generalized entropy beyond subleading order relative to the original work by Gao, Jafferis, and Wall. We close with some comments on the potential applicability of the algebraic approach to quantum gravity.

Figures (2)

• Figure 1: (Left) Unperturbed TFD state, with the spacetime regions to which the bulk exterior algebras are associated shaded in blue. These are holographically dual to the boundary algebras $\frak{A}_{L,0}$, $\frak{A}_{R,0}$. (Right) TFD after the double-trace deformation, which injects negative energy into the black hole, thereby decreasing the area; the new horizons are shown in red, along with the corresponding exterior regions dual to the boundary algebras $\frak{A}_L$, $\frak{A}_R$. Importantly, each of these will contain degrees of freedom from both$\frak{A}_{L,0}$ and $\frak{A}_{R,0}$ due to the traversability of the wormhole, i.e., the causal connectivity between the original and deformed exteriors.
• Figure 2: Close-up of the center region in the right panel of fig. \ref{['fig:bulkpic']}, showing various intersections of horizons. The QES begins at the original bifurcation surface labeled 0 in the unperturbed TFD. After the deformation, the QES may either move directly upwards to the point labeled 2, or split into two solutions at the points labeled $1_\mathrm{l,r}$. Note that in the latter case, the left and right boundary algebras overlap in the central diamond region. From the perspective of black hole complementarity Susskind:1993if, both of these scenarios are consistent with the symmetry of the TFD. However, while the latter has an interesting geometric interpretation in terms of modular inclusions explored in Jefferson:2018ksk, here we follow GJW Gao:2016bin in assuming that the QES moves directly upwards to point 2, which is more obviously consistent with the purity of the TFD. We note however that this does not preclude corrections to the area of the horizon, as we will see below.