Real-time gravitational replicas: formalism and a variational principle

This work is the first step in a two-part investigation of real-time replica wormholes. Here we study the associated real-time gravitational path integral and construct the variational principle that will define its saddle-points. We also describe the general structure of the resulting real-time replica wormhole saddles, setting the stage for construction of explicit examples. These saddles necessarily involve complex metrics, and thus are accessed by deforming the original real contour of integration. However, the construction of these saddles need not rely on analytic continuation, and our formulation can be used even in the presence of non-analytic boundary-sources. Furthermore, at least for replica- and CPT-symmetric saddles we show that the metrics may be taken to be real in regions spacelike separated from a so-called ‘splitting surface’. This feature is an important hallmark of unitarity in a field theory dual.

Continuity of the Restriction Maps on Smirnov Classes

We prove the restriction maps define continuous linear operators on the Smirnov classes for some certain domain with analytic boundary.

On the Asymptotic Behavior of p-Faber Polynomials for Domains with Piecewise Analytic Boundary

Let be a domain bounded by a piecewise analytic Jordan's curve L, and let denote the p-Faber polynomials associated with G. We derive estimates of the form , for , where depends on geometric properties of L and the parameter p. Also, we show that O cannot be replaced by o in the relation given above.