The integral identity conjecture in motivic homotopy theory
Pith reviewed 2026-05-23 08:32 UTC · model grok-4.3
The pith
The integral identity conjecture extends to G_m-equivariant functions on algebraic S-spaces with tau-locally linearizable actions over noetherian bases in motivic stable homotopy categories.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We extend the scope of the original conjecture by studying more generally the case of G_m-equivariant functions on algebraic S-spaces with a tau-locally linearizable action of G_m over a noetherian base scheme S, deducing a functorial version in the motivic stable homotopy categories.
What carries the argument
The tau-locally linearizable G_m-action on algebraic S-spaces, which permits direct application of Braden's hyperbolic localization theorem to obtain the functorial integral identity for Ayoub's nearby cycles functor.
If this is right
- The functorial integral identity applies to algebraic S-spaces beyond the vector-bundle case treated by Ivorra.
- The identity holds over arbitrary noetherian base schemes rather than only fields of characteristic zero.
- Motivic Donaldson-Thomas invariants become available for a larger class of three-dimensional noncommutative Calabi-Yau manifolds defined over such bases.
- The nearby cycles functor satisfies the identity functorially inside the motivic stable homotopy category of S-spaces.
Where Pith is reading between the lines
- The extension opens the possibility of treating families of such functions parametrized by the base scheme S in a uniform way.
- Analogous localization arguments might yield similar identities for other group actions in motivic homotopy theory.
- Explicit checks on toric varieties or other spaces with linearizable actions would provide direct tests of the new scope.
Load-bearing premise
The tau-locally linearizable condition on the G_m-action together with the noetherian base scheme S allows the direct application of Ivorra's functorial version derived from Braden's hyperbolic localization theorem.
What would settle it
A concrete counterexample consisting of an algebraic S-space carrying a tau-locally linearizable G_m-action and an equivariant function f for which the integral identity fails to hold for the nearby cycles functor inside the motivic stable homotopy category over S.
read the original 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 $\ell$-adic cohomology of rigid analytic varieties to Hrushovski-Kazhdan motivic integration and motivic Fubini theorem for tropicalization maps,... In a recent work, 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 $\mathbb{G}_m$-equivariant function $f \colon \mathbb{V}(\mathcal{E}) \longrightarrow \mathbb{A}^1$ on a vector bundle $\mathbb{V}(\mathcal{E})$ over a field of characteristic zero. In the present work, we follow the functorial approach of Ivorra and extend the scope of the original conjecture by Kontsevich and Soibelman by studying more generally the case of $\mathbb{G}_m$-equivariant functions on algebraic $S$-spaces with a $\tau$-locally linearizable action of $\mathbb{G}_m$ over a noetherian base scheme $S$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper extends the integral identity conjecture of Kontsevich and Soibelman to the case of G_m-equivariant functions on algebraic S-spaces equipped with a tau-locally linearizable G_m-action over a noetherian base scheme S. Following Ivorra's functorial approach derived from Braden's hyperbolic localization theorem, it deduces a corresponding functorial version of the identity in the motivic stable homotopy categories.
Significance. If the extension is rigorously established, the result broadens the geometric scope of the integral identity beyond vector bundles over fields of characteristic zero, which could support motivic Donaldson-Thomas invariants in more general settings involving algebraic spaces. The explicit reliance on Ivorra's deduction from Braden localization is a methodological strength that maintains functoriality.
major comments (1)
- [Introduction / main deduction section] The central deduction asserts that the tau-locally linearizable condition on the G_m-action together with the noetherian base S permits direct application of Ivorra's functorial version (originally for Ayoub's nearby cycles on G_m-equivariant functions on vector bundles V(E) over a field k of char 0). However, the manuscript does not supply an explicit reduction of the algebraic S-space case to the scheme/vector-bundle case or a verification that Braden hyperbolic localization extends while preserving motivic homotopy functoriality; this reduction is load-bearing for the claimed extension.
Simulated Author's Rebuttal
We thank the referee for their careful reading and for identifying the need for greater explicitness in the central deduction. We address the major comment below and will revise the manuscript to strengthen the exposition.
read point-by-point responses
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Referee: [Introduction / main deduction section] The central deduction asserts that the tau-locally linearizable condition on the G_m-action together with the noetherian base S permits direct application of Ivorra's functorial version (originally for Ayoub's nearby cycles on G_m-equivariant functions on vector bundles V(E) over a field k of char 0). However, the manuscript does not supply an explicit reduction of the algebraic S-space case to the scheme/vector-bundle case or a verification that Braden hyperbolic localization extends while preserving motivic homotopy functoriality; this reduction is load-bearing for the claimed extension.
Authors: We agree that the reduction step merits a more explicit treatment to make the argument fully self-contained. In the revised manuscript we will add a dedicated paragraph (or short subsection) immediately following the statement of the main theorem. This paragraph will recall the definition of a tau-locally linearizable G_m-action, note that algebraic S-spaces admit tau-covers by schemes on which the action linearizes to a vector bundle, and explain that the motivic stable homotopy category satisfies the requisite descent (Nisnevich or etale) so that the nearby-cycles functor and the hyperbolic localization isomorphism descend from the scheme case. We will also record the relevant generalizations of Braden's theorem to algebraic spaces over noetherian bases that appear in the motivic homotopy literature, thereby confirming that functoriality is preserved. These additions will render the deduction load-bearing step fully transparent without altering the overall strategy. revision: yes
Circularity Check
No circularity: extension follows external Ivorra derivation from Braden theorem
full rationale
The paper states it follows Ivorra's functorial version (derived from Braden hyperbolic localization) and extends the setting to algebraic S-spaces under the tau-locally linearizable G_m-action over noetherian S. No self-citations, self-definitional steps, or fitted inputs renamed as predictions appear in the abstract or description. The central deduction is presented as building directly on the cited external result rather than reducing to the paper's own inputs by construction. The tau-locally linearizable condition is asserted to permit the extension, but without evidence of internal reduction this remains an independent claim.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Ayoub's nearby cycles functor behaves functorially for G_m-equivariant functions in the motivic stable homotopy categories of schemes
- domain assumption Braden's hyperbolic localization theorem applies in the relevant motivic categories
discussion (0)
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