The Integral Identity Conjecture in Motivic Homotopy Theory

Abstract The integral identity conjecture of Kontsevich and Soibelman plays an important role in proving the existence of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau manifolds. There are a number of different formulations of this conjecture in different contexts, and accordingly, there are corresponding solutions to them. The methods devoted to solving this conjecture are diverse, ranging from script l ℓ -adic cohomology of rigid analytic varieties to Hrushovski-Kazhdan motivic integration and motivic Fubini theorem for tropicalization maps,. . . In Ivo24, Ivorra deduces a functorial version of the integral identity in the motivic stable homotopy categories of schemes, from the Braden hyperbolic localization theorem. This functorial version concerns Ayoub’s nearby cycles functor associated with a upper G Subscript m Gₘ G m -equivariant function f colon double struck upper V left parenthesis script upper E right parenthesis long right arrow upper A Superscript 1 f V (E) A¹ f: V (E) ⟶ A 1 on a vector bundle double struck upper V left parenthesis script upper E right parenthesis V (E) V (E) over a field of characteristic zero. In the present work, we follow the functorial approach from Ivo24 and extend the scope of the original conjecture by Kontsevich and Soibelman by studying more generally the case of upper G Subscript m Gₘ G m -equivariant functions on algebraic S -spaces with a tau τ -locally linearizable action of upper G Subscript m Gₘ <mml: math xmlns: xlink="http: //www. w3. org/1999/xlink" xmlns: mnf="http: //cambridge. org/core/manifest" xmlns: cup="http: //contentservices. cambridge