Instantons and emergent geometry

Abstract

We show that self-dual electromagnetism in noncommutative spacetime is equivalent to self-dual Einstein gravity.

Noncommutative electromagnetism as a large N gauge theory

“…This proves, as first shown in [15,16], that selfdual NC electromagnetism is equivalent to self-dual Einstein gravity. Note that the Einstein gravity described by the metric (3.4) arises from the commutative, i.e., O(θ) limit.…”

“…In this work we will find a sound realization of this idea. It turns out that the emergent gravity recently found in [13,14,15,16] can be elegantly understood in this framework. Therefore the correspondence between NC field theory and gravity [3] is certainly akin to the gauge/gravity duality in large N limit [10,11,12].…”

“…Recently we showed in [15,16] for the d = 0 and m = 4 case that self-dual electromagnetism in NC spacetime is equivalent to self-dual Einstein gravity and the metric is precisely given by Eq.(3.4). A key observation [16] was that the self-dual system (3.5) defined by vector fields on M can be derived from the action (2.2) or (2.13) for slowly varying fields, where all ⋆-commutators between NC fields are approximated by the Poisson bracket (1.1).…”

“…A key observation [16] was that the self-dual system (3.5) defined by vector fields on M can be derived from the action (2.2) or (2.13) for slowly varying fields, where all ⋆-commutators between NC fields are approximated by the Poisson bracket (1.1). An important point in NC geometry is that the adjoint action of (covariant) coordinates with respect to star product can be identified with (generalized) vector fields on some (curved) manifold [15,16], as the trivial case was already used in Eq.(2.7). In the end, a D-dimensional manifold described by the metric (3.4) corresponds to an emergent geometry arising from the gauge-Higgs system in Eq.(3.3).…”

“…In this section we will further elucidate the emergent gravity arising from gauge fields by showing that the gauge-Higgs system (A µ , Φ a ) in half-BPS configurations describes self-dual Einstein gravity. Since the case for D = 4 and n = 2 has been extensively discussed in [14,15,16], we will consider the other cases for D ≥ 4. For simplicity, the metrics in the action (2.13) are supposed to be the form already used in Eq.…”