On a conjecture of Kontsevich and Soibelman

Abstract

We consider a conjecture of Kontsevich and Soibelman which is regarded as a foundation of their theory of motivic Donaldson-Thomas invariants for noncommutative 3d Calabi-Yau varieties. We will show that, in some certain cases, the answer to this conjecture is positive.

A proof of the integral identity conjecture, II

“…The conjecture was first proved in [5] in the case where f is either a function of Steenbrink type or the composition of a pair of regular functions with a polynomial in two variables. In [6, Theorem 1.2], we show that, if the field k is algebraically closed, Conjecture 1.1 holds in M μ loc .…”

“…The latter has been a challenging problem, and the previous attempts [5], [6] and [7] for solving it had to use certain additional assumptions. Now we write W n,m for the set of (ϕ 1 , ϕ 2 , ϕ 3 ) in W n with ord t ϕ 1 + ord t ϕ 2 = m, and let us observe that it is still stable under the canonical μ n -action.…”