My research plan encompasses Non-commutative geometry and emergent gravity and is aimed at the ultimate unification between matter and geometry. The trinity of instantons namely between NC U(1) instantons, gravitational instantons and SU(2) Yang-Mills instantons may pave the path for such a novel unification scheme. A better understanding of the integrable systems in 4 dimension may be useful in the sense that, self dual 2 forms can be defined only in 4 dimensions and all physical forces eg. electromagnetic force, strong and weak force and gravitational force are represented by 2 forms on some orientable manifolds. According to Ward conjecture, self-dual systems in d=4 is the parent theory of (all) integrable systems in lower dimensions for eg. Hitchin systems in 2d, KdV etc. According to a theorem by Hitchin any hyperkahler manifold (kaher and Ricci flat) has one-to-one correspondence with a twistor space, so twistor space descriptions of instantons are also worth pursuing topics. Matrix model descriptions of gravitational instantons (which are hyper kahler manifolds) will lead to background independent theory of quantum gravity and emergent concept of spacetime will be a key concept in this framework. A better understanding of integrable systems or twistor space construction in terms of ADHM prescription is also an important subject for further study.
News published in Agência FAPESP Newsletter about the scholarship: