On Deformation Theory and Quantization

Deformation theory requires solving Maurer-Cartan equation (MCE) associated to an DGLA (L-infinity algebra). The universal solution of [HS] is obtained iteratively, as a fixed point of a contraction, analogous to the Picard method. The role of the Kuranishi functor in this construction is emphasized. The parallel with Lie theory suggests that deformation theory is a higher “dimensional” version. The deformation determined by the solution of the Maurer-Cartan equation associated to a contraction, splits the epimorphism, leading to a “doubling and gluing” interpretation. The *-operator associated to a contraction is introduced, and the connection with Hodge structures and generalized complex structures ( dd∗ -lemma) is established. The relations with bialgebra deformation quantization on one hand and ConnesKreimer renormalization on the other, are suggested.