Pure and twisted holography

Abstract

We analyze a simple example of a holographically dual pair in which we topologically twist both theories. The holography is based on the two-dimensional N = 2 supersymmetric Liouville conformal field theory that defines a unitary bulk quantum supergravity theory in three-dimensional anti-de Sitter space. The supersymmetric version of three-dimensional Liouville quantum gravity allows for a topological twist on the boundary and in the bulk. We define the topological bulk supergravity theory in terms of twisted boundary conditions. We corroborate the duality by calculating the chiral configurations in the bulk supergravity theory and by quantizing the solution space. Moreover, we note that the boundary calculation of the structure constants of the chiral ring carries over to the bulk theory as well. We thus construct a topological AdS/CFT duality in which the bulk theory is independent of the boundary metric.

Twisted string theory in anti-de Sitter space

“…More work needs to be done to understand the bulk topological string theory well. Firstly, one would like to compute the G + 0 cohomology directly from the bulk perspective (as was done in the Chern-Simons supergravity in [18]). One can be hopeful that this is possible to all orders in the string length over the curvature radius, in a generic conformal field theory describing a N = 2 superconformal string background.…”

“…To obtain a topological string theory (in the sense that it will be independent of the boundary values of the metric), we allow the boundary metric to couple to other degrees of freedom in the theory [18]. Drawing inspiration from the topological twist of ordinary quantum field theories [2], we expect that the boundary metric will also couple to the derivative of the R-symmetry current.…”

“…We introduce a dependence of the U(1) R gauge field on the asymptotic boundary metric g (0) ij in order to couple the boundary metric non-trivially to the R-current. We draw inspiration from the analogous exercise performed in supergroup Chern-Simons theory [18] as well as from the literature on topological quantum field theories [2]. We wish to introduce a coupling (J R ±J R ) µ ω µ between the R-currents and the spin connection one-form ω = ω µ dx µ on the boundary, where the relative sign depends on the twist we perform.…”

“…In this paper, we follow the conceptual road laid out in [18], where we exploited a holographic duality between Chern-Simons supergravity in the bulk and a boundary conformal field theory with extended supersymmetry. In this simplified model of quantum holography, we showed how to obtain a topologically twisted bulk theory.…”

“…These twisted asymptotic boundary conditions, combined with a suitably defined cohomology of physical states, define a bulk theory independent of the boundary metric. In that sense, it is a topologically twisted theory of quantum gravity [18].…”