We present an exposition on the geometrization of the electromagnetic force. We show that, in noncommutative (NC) spacetime, there always exists a coordinate transformation to locally eliminate the electromagnetic force, which is precisely the Darboux theorem in symplectic geometry. As a consequence, the electromagnetism can be realized as a geometrical property of spacetime like gravity. We show that the geometrization of the electromagnetic force in NC spacetime is the origin of gravity, dubbed as the emergent gravity. We discuss how the emergent gravity reveals a novel, radically different picture about the origin of spacetime. In particular, the emergent gravity naturally explains the dynamical origin of flat spacetime, which is absent in Einstein gravity. This spacetime picture turns out to be crucial for a tenable solution of the cosmological constant problem.

We showed before that self-dual electromagnetism in noncommutative (NC) space–time is equivalent to self-dual Einstein gravity. This result implies a striking picture about gravity: gravity can emerge from electromagnetism in NC space–time. Gravity is then a collective phenomenon emerging from gauge fields living in fuzzy space–time. We elucidate in some detail why electromagnetism in NC space–time should be a theory of gravity. In particular, we show that NC electromagnetism is realized through the Darboux theorem as a diffeomorphism symmetry G which is spontaneously broken to symplectomorphism H due to a background symplectic two-form Bμν = (1/θ)μν, giving rise to NC space–time. This leads to a natural speculation that the emergent gravity from NC electromagnetism corresponds to a nonlinear realization G/H of the diffeomorphism group, more generally its NC deformation. We also find some evidences that the emergent gravity contains the structures of generalized complex geometry and NC gravity. To illuminate the emergent gravity, we illustrate how self-dual NC electromagnetism nicely fits with the twistor space describing curved self-dual space–time. We also discuss derivative corrections of Seiberg–Witten map which give rise to higher-order gravity.

Emergent gravity is based on a novel form of the equivalence principle known as 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 spacetime admits a symplectic structure, in other words, a microscopic spacetime becomes noncommutative (NC). If gravity emerges from U(1) gauge theory on NC spacetime, this picture of emergent gravity suggests a completely new quantization scheme where quantum gravity is defined by quantizing spacetime itself, leading to a dynamical NC spacetime. Therefore the quantization of emergent gravity is radically different from the conventional approach trying to quantize a phase space of metric fields. This approach for quantum gravity allows a background independent formulation where spacetime as well as matter fields is equally emergent from a universal vacuum of quantum gravity.

We map noncommutative (NC) U(1) gauge theory on IR C leads to an emergent geometry in the (d + 2n)-dimensional spacetime whose metric was determined by Ward a long time ago. In particular, the 10-dimensional gravity for d = 4 and n = 3 corresponds to the emergent geometry arising from the 4-dimensional N = 4 vector multiplet in the AdS/CFT duality. We further elucidate the emergent gravity by showing that the gauge-Higgs system (A µ , Φ a ) in half-BPS configurations describes selfdual Einstein gravity.

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