Let \(S\) be a variety over a field \(k\) of characteristic \(0\), \(\mu_ n=\mathrm{Spec }k[t]/(t^n-1)\), and let \(\hat{\mu}=\lim_{\leftarrow n}\mu_n\) be the limit of the projective system \((\mu_n)_{n\in\mathbb N^\ast}\). Take the \(\hat{\mu}\)-equivariant Grothendieck ring of \(S\)-varieties \(K^{\hat{\mu}}_0(\mathrm{Var}_S)\), let \(\mathcal M^{\hat{\mu}}_S\) be its localization with respect to \(\mathbb L\), and \(\mathcal M_{\mathrm{loc}}^{\hat{\mu}}\) its localization with respect to \((1-\mathbb L^{-n})_{n\in\mathbb N^\ast}\). Furthermore, for \(X\) a smooth purely dimensional \(k\)-variety, and \(f\) a regular function on it with nonempty zero locus \(X_0\), there are notions of motivic nearby cycles \(\mathcal S_f\) of \(f\), defined as a limit of the motivic zeta function of \(f\), and for a closed point \(x\in X\), the motivic nearby fiber \(\mathcal S_{f,x}\) of \(f\) at \(x\).
Kontsevich and Soibelman introduced in [M. Kontsevich and I. Soibelman, “Stability structures, motivic Donaldson-Thomas invariants and cluster transformations”, Preprint, arXiv:0811.24352] motivic Donaldson-Thomas invariants for three dimensional non-commutative Calabi-Yau varieties. Their existence is based on the following integral identity conjecture in \(\mathcal M_k^{\hat{\mu}}\), stated below for regular functions (see [loc. cit.], sec. 4.4).
Conjecture (Kontsevich-Soibelman). Let \(f\in k[x,y,z]\) with \(f(0,0,0)=0\), such that \(f(\lambda x,\lambda^{-1}y,z)=f(x,y,z)\) for \(\lambda\in\mathbb G_m\), where \((x,y,z)\) are the coordinates in \(k^{d_1}\times k^{d_2}\times k^{d_3}\). Then in \(\mathcal M_k^{\hat{\mu}}\) holds \(\int_{\mathbb A^{d_1}}i^\ast\mathcal S_f=\mathbb L^{d_1}\mathcal S_{\tilde f,0}\), where \(\tilde f=\mathrm{res}_{\mathbb A^{d_3}}(f)\) and \(i:\mathbb A^{d_1}_k\hookrightarrow f^{-1}(0)\).
Here \(\mathbb A^{d_1}_k\simeq\mathbb A^{d_1}_k\times\{0\}\times\{0\}\), and by the homogenity condition it becomes a subvariety of \(f^{-1}(0)\).
This conjecture has been proved in a few cases, including the case of function \(f\) either of Steebrink type, or being composition of a couple of regular functions with polynomial in two variables [Lê Quy Thuong, Algebra Number Theory 6, No. 2, 389–404 (2012; Zbl 1263.14015)], and in the ring \(\mathcal M_{\mathrm{loc}}^{\hat{\mu}}\), supposed \(k\) is algebraically closed [L. Q. Thuong, Duke Math. J. 164, No. 1, 157–194 (2015; Zbl 1370.14017)]. It has been proved also in \(\mathcal M_k^{\hat{\mu}}\) over a field containing all roots of unity [J. Nicaisse and S. Payne, “A tropical motivic Fubini theorem with applications to Donaldson-Thomas theory”, Preprint, arXiv:1703.102285]. The main result in this article is that the conjecture above holds over an arbitrary field \(k\) of characteristic 0 (not necessary algebraically closed) in the ring \(\mathcal M_{\mathrm{loc}}^{\hat{\mu}}\). A crucial role in its proof plays the motivic integration of constructible motivic functions [R. Cluckers and F. Loeser, Invent. Math. 173, No. 1, 23–121 (2008; Zbl 1179.14011)]. After some notions from it, as definable subassignments and the corresponding \(\hat{\mu}\)-equivariant Grothendieck ring have been briefly discussed, in the final section is given a sketch of the proof of the main result.