Instantons from geodesics in AdS moduli spaces

Abstract

Abstract:We investigate supergravity instantons in Euclidean AdS 5 × S 5 /Z k . These solutions are expected to be dual to instantons of N = 2 quiver gauge theories. On the supergravity side the (extremal) instanton solutions are neatly described by the (lightlike) geodesics on the AdS moduli space for which we find the explicit expression and compute the on-shell actions in terms of the quantised charges. The lightlike geodesics fall into two categories depending on the degree of nilpotency of the Noether charge matrix carried by the geodesic: for degree 2 the instantons preserve 8 supercharges and for degree 3 they are non-SUSY. We expect that these findings should apply to more general situations in the sense that there is a map between geodesics on moduli-spaces of Euclidean AdS vacua and instantons with holographic counterparts.

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

“…In contrast, the metric on the moduli space of Lorentzian AdS which (somewhat confusingly) has Euclidean signature, can be obtained trivially from the above metric by analytically continuing N and (dζ i ) to imaginary values, thus flipping the negative signs in the metric (2.9). 3 We fix α and β such that for the spherically symmetric harmonic we simply have H = r in the gauge f = a 4 4 With respect [1], we make, for notational convenience, the following rescaling…”

“…The geodesic solutions can most easily be constructed using the exponential map M = M (0) exp(2Qτ ), with Q an element of the Lie algebra of the coset, τ the affine coordinate and M a matrix, built from the coset representative L in the solvable gauge. The details can be found in [1]. Here we simply present the solutions in terms of the scalars.…”

“…, k − 1. The condition c = 0 implies that the Noether charge matrix Q is nilpotent either of degree 2 (Q 2 = 0) or degree 3 (Q 3 = 0) [1]. The degree 2 solutions are supersymmetric and imply that either all p's vanish or all m's vanish.…”

“…We now present the results of [1] for the the Euclidean on-shell action, which is generically complex. The imaginary part of the on-shell supergravity action, for all instantons, is particularly simple and only a function of the quantised charges (and not the number c):…”

“…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).…”