Airy function and 4d quantum gravity

Abstract

We study four-dimensional quantum gravity with negative cosmological constant in the minisuperspace approximation and compute the partition function for the S 3 boundary geometry. In this approximation scheme the path integrals become dominated by a class of asymptotically AdS "microstate geometries." Despite the fact that the theory is pure Einstein gravity without supersymmetry, the result precisely reproduces, up to higher curvature corrections, the Airy function in the S 3 partition function of the maximally supersymmetric Chern-Simons-matter (CSM) theory which sums up all perturbative 1/N corrections. We also show that this can be interpreted as a concrete realization of the idea that the CFT partition function is a solution to the Wheeler-DeWitt equation as advocated in the holographic renormalization group. Furthermore, the agreement persists upon the inclusion of a string probe and it reproduces the Airy function in the vev of half-BPS Wilson loops in the CSM theory. These results may suggest that the supergravity path integrals localize to the minisuperspace in certain cases and the use of the minisuperspace approximation in AdS/CFT may be a viable approach to study 1/N corrections to large N CFTs.

Quantum holographic entanglement entropy to all orders in 1/N expansion

“…As shown in our previous work [17], remarkably, the 4d pure Einstein gravity in the minisuperspace approximation suffices to reproduce the Airy function in the S 3 partition function of the ABJM theory at strong coupling [22,23]. 2 From the recap in Section 2.1, the sphere partition function in the minisuperspace approximation is given by…”

“…The computation of quantum HEE in this paper is a generalization of our previous work on the sphere partition function [17]. We thus first review the path integral method in our previous work used to calculate the S 3 partition function.…”

“…As the main result of [17], it should be stressed that the minisuperspace partition function (2.15) is precisely the Airy function [22,23] that appears in the S 3 partition function of the ABJM theory at large λ via the AdS/CFT dictionary, 3V 3 2…”

“…We consider the AdS 4 /CFT 3 correspondence as a specific example and adopt the path integral method, in conjunction with the Wheeler-DeWitt equation, to four-dimensional quantum gravity with negative cosmological constant. This is based on a similar method employed in the earlier works in the context of quantum cosmology [15,16] and the same approach was applied to our previous work on the S 3 partition function [17]. In order to compute quantum corrections to HEE, we use the replica trick and thereby generalize our previous work to the case of replica geometries.…”

“…The organization of this paper is as follows: In Section 2 we review the path integral method, in particular, the evaluation of the path integrals in the minisuperspace approxi-mation, used to compute the S 3 partition function in our previous work [17]. Within this framework, we also provide an alternative and complementary approach using the Wheeler-DeWitt equation to bolster the weakness of the path integral method.…”