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 study the effect of the Dp-brane gas in string cosmology. When one kind of Dp-brane gas dominates , we find that the cosmology is equivalent to that of the Brans-Dicke theory with the perfect fluid type matter. We obtain γ, the equation of state parameter, in terms of p and the space-time dimension.
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.
We study the condensation of localized tachyon in non-supersymmetric orbifold C 2 /Z n . We first show that the G-parities of chiral primaries are preserved under the condensation of localized tachyon(CLT) given by the chiral primaries. Using this, we finalize the proof of the conjecture that the lowest-tachyon-mass-squared increases under CLT at the level of type II string with full consideration of GSO projection. We also show the equivalence between the G-parity given by G = [jk 1 /n] + [jk 2 /n] coming from partition function and that given by G = {jk 1 /n}k 2 −{jk 2 /n}k 1 coming from the monomial construction for the chiral primaires in the dual mirror picture.
Emergent gravity is based on the Darboux theorem or the Moser lemma in symplectic geometry stating that the electromagnetic force can always be eliminated by a local coordinate transformation as far as U(1) gauge theory is defined on a spacetime with symplectic structure. In this approach, the spacetime geometry is defined by U(1) gauge fields on noncommutative (NC) spacetime. Accordingly the topology of spacetime is determined by the topology of NC U(1) gauge fields. We show that the topology change of spacetime is ample in emergent gravity and the subsequent resolution of spacetime singularity is possible in NC spacetime. Therefore the emergent gravity approach provides a well-defined mechanism for the topology change of spacetime which does not suffer any spacetime singularity in sharp contrast to general relativity.The general theory of relativity predicts the existence of spacetime singularity at the center of black holes and the very beginning of our universe. The singularity theorem [1] debunks that classical general relativity cannot be an ultimate theory of space and time. In order to avoid the spacetime singularities, one would have to resort to a viable quantum theory of gravity which requires to consider fluctuations not only in geometry but also in topology. The topology of spacetime enters general relativity through the fundamental assumption that spacetime is organized as a (pseudo-)Riemannian manifold. But it was shown [2] that generic topology changing spacetimes are singular and so topology change does not seem to be allowed in classical general relativity. So far this issue has been discussed largely in the context of Euclidean quantum gravity [3] which is hard to justify from first principles and also difficult to do reliable calculations since the Euclidean Einstein action is unbounded from below.But a folklore in quantum gravity is that the description of spacetime using commutative coordinates is not valid below a certain fundamental scale, e.g., the Planck length and beyond that scale the spacetime has a noncommutative (NC) structure leading to a resolution of spacetime singularities. Recently string theory has been successful to resolve certain types of geometrical singularities using the fuzziness of geometry on string length scales. See, for example, a recent review [4]. The reason that a vanishing cycle can be nonsingular in string theory is that strings can sense not only the volume of the cycle but also the flux of B fields. (See section 3 in [4] for this feature and section 4 for the significance of NC geometry for the topology change and singularity resolution.) Since the emergent gravity is also based on a gauge theory with B-field backgrounds [5], viz., NC spacetime, it will be interesting to see whether a similar physics to string theory can arise in the emergent gravity approach.The basic picture of emergent gravity in [5] is that gravity and spacetime are collective manifestations of U(1) gauge fields on a NC spacetime. In this approach, the spacetime geometry is defined by U...
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