Instantons and Emergent Geometry

Gravity is a mysterious force; we still don’t know what gravity is. But recent developments from string and M theories gradually reveal an intriguing picture on gravity (for recent reviews, see, for example, [1, 2]): Gravity may be a collective phenomenon emergent from gauge fields. That is, the spin-two graviton might arise as a composite of two spin-one gauge bosons. Although the emergent gravity is a fascinating approach suggesting that gravity itself may not be “fundamental physics”, it is not easy to realize it from ordinary quantum field theories due to a general no-go theorem known as the Weinberg-Witten theorem [3], stating that an interacting graviton cannot emerge from an ordinary quantum field theory in the same spacetime. One has to notice, however, that Weinberg and Witten introduced two basic assumptions to prove this theorem. The first hidden assumption is that gravitons and gauge fields live in the same spacetime. The second assumption is the existence of a Lorentz-covariant stress-energy tensor. Thus, to realize the emergent gravity, one has to relax one of the assumptions in some ways. One also has to recall that gravity has no local gaugeinvariant degrees of freedom. So one cannot start with a theory containing local gauge-invariant operators to generate Einstein gravity as an emergent phenomenon in the same spacetime. The clue to relax the first assumption comes from the holographic principle [4]: Gravitons live in a higher dimensional spacetime than gauge fields. In this description known as the anti-de Sitter space and conformal field theory (AdS/CFT) duality [5], gravity in higher dimensions is an emergent phenomenon arising from particle interactions in a gravityless, lower-dimensional world. The bulk geometry, i.e., AdS space, has to contain a holographic screen on which the boundary theory, typically super Yang-Mills theory, is defined. Since the Yang-Mills theory contains many local gauge-invariant observables and the diffeomorphism only appears in the bulk, the AdS/CFT duality does not provide any information on lower-dimensional gravity in the same spacetime where the boundary theory is defined. One may try to relax the second assumption instead. This assumption can be relaxed, for example, by introducing a symplectic structure of spacetime