Abstract
We compute the quantum gravity partition function of M-theory on $AdS_4 \times X_7 $ by using localization techniques in four-dimensional gauged supergravity obtained by a consistent truncation on the Sasaki-Einstein manifold $X_{7}$. The supergravity path integral reduces to a finite dimensional integral over two collective coordinates that parametrize the localizing instanton solutions. The renormalized action of the off-shell instanton solutions depends linearly and holomorphically on the "square root" prepotential evaluated at the center of $AdS_{4}$. The partition function resembles the Laplace transform of the wave function of a topological string and with an assumption about the measure for the localization integral yields an Airy function in precise agreement with the computation from the boundary ABJM theory on a 3-sphere. Our bulk quantum gravity computation is nonperturbatively exact in four-dimensional Planck length but ignores corrections due to brane-instantons.
Highlights
Given the centrality of the notion of holography for quantum gravity, it is clearly important to go beyond the classical limit and study quantum effects in the bulk
We compute the quantum gravity partition function of M-theory on AdS4 ×X7 by using localization techniques in four-dimensional gauged supergravity obtained by a consistent truncation on the Sasaki-Einstein manifold X7
An important advantage of the AdS/CF T correspondence is that the boundary theory can guide the computation in the bulk. This is especially useful given the notorious difficulties in making sense of the functional integral of quantum gravity. Such a study of finite N effects has met with considerable success in the context of AdS2/CF T1 holography which arises near the horizon of dyonic supersymmetric black holes in string theory
Summary
We describe the computation of the partition function of a large class of models described by a Chern-Simons-Matter gauge theory on the S3 boundary of Euclidean AdS4. In the case of ABJM theory, the matrix model is very similar to that of the lens space Chern-Simons matrix model for which the spectral curve is known [21,22,23] This solution was used in [24] to calculate the expectation value of Wilson loops and in [16] to calculate the S3 partition function. Each gauge group has a Chern-Simons term at level k(a) = n(a)k with n(a) = 0, bifundamental chiral multiplets, and in addition an arbitrary number Nf(a) of chiral multiplets in the fundamentals of the gauge groups In this case the matrix model of [7] is. In the case of ABJM theory the constants are c = 2/π2 and n0 = k/24 − 1/3k, which reproduces (1.1) This Airy function expression is corrected by non-perturbative O(e−N ) terms, which we will not study.
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