The holographic dual to supergravity instantons in AdS5× S5/ℤk

Abstract

We investigate the holographic dual to supergravity instanton solutions in AdS 5 × S 5 /Z k , which are described entirely in terms of geodesics on the AdS moduli space. These instantons are expected to be holographically dual to instantons in an N = 2 necklace quiver gauge theory in four dimensions with k gauge nodes, at large N. For the supersymmetric instantons we find a precise match between the on-shell actions and the vevs of Tr[F 2 ] and Tr[F ∧F ] computed on both sides of the duality. This correspondence requires an exact identification between the massless supergravity scalars and the dual gauge couplings which we give in detail. We also find a candidate for the supergravity dual of a quasi-instanton in the quiver theory, which is a non-supersymmetric extremal solution.manifold (1.1) has exactly k Abelian isometries that act as shifts of k real scalars. This fixes the Wick-rotation uniquely to [2]If one restricts to instanton solutions with spherical symmetry (i.e. respecting the rotational SO(5) symmetry of Euclidean AdS 5 ) for simplicity, then the scalar field equations of motion reduce to geodesic equations on M moduli and the Einstein equations of motion decouple from the scalar fields into a universal form, [2,6,7]. The metric is then only sensitive to the constant velocity of the geodesics (in an affine parametrization), which we denote as cwhere φ I denote the moduli, G IJ is the metric on (1.2) and a dot is a derivative with respect to the affine coordinate. Not surprisingly the description of geodesics on a coset space like (1.2) can be understood entirely using group theory [1]. The properties and meaning of geodesics depend strongly on whether they are time-like (c > 0), space-like (c < 0) or null (c = 0). When c < 0 the instantons are called super-extremal and the geometry describes a smooth two-sided wormhole that asymptotes to AdS on both sides [8]. A very explicit and concrete embedding of such wormholes was found inside AdS 5 × S 5 /Z k when k > 1 [2]. When k = 1 the scalars are singular and the corresponding wormholes are not considered as physical [9]. These Euclidean "axionic" wormholes have a long history in cosmology, QCD, holography and quantum gravity (see for instance [2, 10-17] ) which was reviewed in the comprehensive paper [18].When c > 0 the instantons are called sub-extremal and the geometry corresponds to a singular "spiky" deformation of AdS. The holographic description of these instantons is unclear but for c small enough a suggestion was made in [9] in the case of AdS 5 × S 5 . In this paper we find that the same interpretation holds water as well in AdS 5 × S 5 /Z k .The focus of our paper is on the extremal instantons, for c = 0. When k > 1 it was found that these instantons come into two families, supersymmetric or not, whereas for k = 1 there are only supersymmetric solutions. One can expect that the supersymmetric instantons are the easiest to understand. When k = 1 these instantons are simply the D(-1)/D3 bound states zoomed in near the horizon of the D3. Their hol...

Instantons, Euclidean wormholes and AdS/CFT

“…All the explicit geodesic curves and their properties were studied in [23,22]. Such an explicit analysis is possible because geodesics on symmetric coset spaces can be found using the exponential map [7].…”

“…An easy explanation of this using a formal path integral method can be found in the introduction of [6] Holography (from string theory) could provide a concrete test for these "old school" semi-classical quantum gravity ideas [16,6,24]. But exactly the property of being overextremal causes trouble in the holographic context since the BPS bound in the dual field theories cannot be violated [5,22]. If one ignores the issue of a double boundary or alternatively, one ignores the effect of cutting the wormhole in half, one can perform the usual holographic computation of n-point functions in a wormhole background.…”

“…Strictly speaking the computation in [5] was not valid as the wormhole regularity criterion was not satisfied. Nonetheless the explicit regular embedding of [17] gives the identical problem in the dual N = 2 quiver theory [22].…”

“…But they fail to be supersymmetric since each gauge node preserves a different part of the 4D SUSY generators. We symbolically represent the difference between SUSY and quasi instantons in The holographic "match" in [22] only involved the computation of the one-point functions, the quantisation of charges, and the on-shell actions, athough there is a non-trivial subtlety if one wishes to match the on-shell actions [22].…”

“…However, the above is still unclear and in fact a suggestion of a possible holographic dual was made in [5] for k = 1 and extends trivially to k > 1 [22]. It is in spirit of the quasi instantons of [32].…”