Boundary and bulk notions of transport in the AdS$_3$/CFT$_2$ correspondence

TL;DR

This paper constructs operators in AdS$_3$/CFT$_2$ that act as bulk spacelike vector fields within the boundary theory, revealing insights into boundary-bulk relations and holographic complexity.

Contribution

It introduces a method to realize bulk spacelike vector fields as boundary operators using parallel transport, and discusses extending this to timelike vectors and other bulk quantities.

Findings

Operators for bulk spacelike vectors constructed in boundary CFT

Challenges identified in extending to timelike vectors

Applications to boundary expressions of bulk geometric quantities

Abstract

We construct operators in holographic two-dimensional conformal field theory, which act locally in the code subspace as arbitrary bulk spacelike vector fields. Key to the construction is an interplay between parallel transport in the bulk spacetime and in kinematic space. We outline challenges, which arise when the same construction is extended to timelike vector fields. We also sketch several applications, including boundary formulations of the bulk Riemann tensor, dreibein, and spin connection, as well as an application to holographic complexity.