Proofs of the integral identity conjecture over algebraically closed fields

TL;DR

This paper proves the integral identity conjecture over algebraically closed fields, which is vital for motivic Donaldson-Thomas invariants in non-commutative Calabi-Yau threefolds, using advanced motivic integration techniques.

Contribution

It establishes the validity of the integral identity conjecture for regular and formal functions over algebraically closed fields, expanding its applicability.

Findings

Confirmed the integral identity conjecture over algebraically closed fields.

Extended the validity to both regular and formal functions.

Utilized motivic Milnor fiber techniques and motivic integration methods.

Abstract

Recently, it is well known that the conjectural integral identity is of crucial importance in the motivic Donaldson-Thomas invariants theory for non-commutative 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 ground field algebraically closed. Technically, the result on 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.