In this paper we examine a small but detailed test of the emergent gravity picture with explicit solutions in gravity and gauge theory. We first derive symplectic U (1) gauge fields starting from the Eguchi-Hanson metric in four-dimensional Euclidean gravity. The result precisely reproduces the U (1) gauge fields of the Nekrasov-Schwarz instanton previously derived from the top-down approach. In order to clarify the role of noncommutative spacetime, we take the Braden-Nekrasov U (1) instanton defined in ordinary commutative spacetime and derive a corresponding gravitational metric. We show that the Kähler manifold determined by the Braden-Nekrasov instanton exhibits a spacetime singularity while the Nekrasov-Schwarz instanton gives rise to a regular geometrythe Eguchi-Hanson space. This result implies that the noncommutativity of spacetime plays an important role for the resolution of spacetime singularities in general relativity. We also discuss how the topological invariants associated with noncommutative U (1) instantons are related to those of emergent four-dimensional Riemannian manifolds according to the emergent gravity picture.

We reconsider the sub-leading quantum perturbative corrections to N = 2 cubic special Kähler geometries. Imposing the invariance under axion-shifts, all such corrections (but the imaginary constant one) can be introduced or removed through suitable, lower unitriangular symplectic transformations, dubbed PecceiQuinn (PQ) transformations.Since PQ transformations do not belong to the d = 4 U -duality group G 4 , in symmetric cases they generally have a non-trivial action on the unique quartic invariant polynomial I 4 of the charge representation R of G 4 . This leads to interesting phenomena in relation to theory of extremal black hole attractors; namely, the possibility to make transitions between different charge orbits of R, with corresponding change of the supersymmetry properties of the supported attractor solutions. Furthermore, a suitable action of PQ transformations can also set I 4 to zero, or vice versa it can generate a non-vanishing I 4 : this corresponds to transitions between "large" and "small" charge orbits, which we classify in some detail within the "special coordinates" symplectic frame.Finally, after a brief account of the action of PQ transformations on the recently established correspondence between Cayley's hyperdeterminant and elliptic curves, we derive an equivalent, alternative expression of I 4 , with relevant application to black hole entropy.

We compute the effective black hole potential V BH of the most general N = 2,d = 4 (local ) special Kähler geometry with quantum perturbative corrections, consistent with axion-shift Peccei-Quinn symmetry and with cubic leading order behavior.We determine the charge configurations supporting axion-free attractors, and explain the differences among various configurations in relations to the presence of "flat" directions of V BH at its critical points.Furthermore, we elucidate the role of the sectional curvature at the non-supersymmetric critical points of V BH , and compute the Riemann tensor (and related quantities), as well as the so-called E-tensor. The latter expresses the non-symmetricity of the considered quantum perturbative special Kähler geometry.

Emergent gravity is aimed at constructing a Riemannian geometry from U(1) gauge fields on a noncommutative spacetime. But this construction can be inverted to find corresponding U(1) gauge fields on a (generalized) Poisson manifold given a Riemannian metric (M, g). We examine this bottom-up approach with the LeBrun metric which is the most general scalar-flat Kähler metric with a U(1) isometry and contains the Gibbons-Hawking metric, the real heaven as well as the multi-blown up Burns metric which is a scalar-flat Kähler metric on C 2 with n points blown up. The bottom-up approach clarifies some important issues in emergent gravity.