Proofs of the integral identity conjecture over algebraically closed fields

Recently, it has become well known that the conjectural integral identity is of crucial importance in the motivic Donaldson–Thomas invariants theory for noncommutative Calabi–Yau threefolds. The purpose of this article is to consider different versions of the identity, for regular functions and formal functions, and to give them the positive answer for the algebraically closed ground fields. Technically, the result on the motivic Milnor fiber by Hrushovski–Loeser using Hrushovski–Kazhdan’s motivic integration and Nicaise’s computations on motivic integrals on special formal schemes are main tools.