Smooth-to-holonomic equivalence conjecture over the Novikov field

From papers

Let XX be a symplectic manifold and c>0c>0. Let μsm(X;u<c)\mu^{\mathrm{sm}}(X;u<c) be the subcategory of smooth holonomic objects in the microlocal category μ(X;u<c)\mu(X;u<c), and let μhol(X;u<c)\mu^{\mathrm{hol}}(X;u<c) be the subcategory of holonomic objects. Let Λ0\Lambda_0 be the Novikov ring and Λ\Lambda its Novikov field.

Smooth-to-holonomic equivalence conjecture. The inclusion

μsm(X;u<c)μhol(X;u<c)\mu^{\mathrm{sm}}(X;u<c)\hookrightarrow \mu^{\mathrm{hol}}(X;u<c)

induces an equivalence over the Novikov field

μsm(X;u<c)Λ0Λμhol(X;u<c)Λ0Λ.\mu^{\mathrm{sm}}(X;u<c)\otimes_{\Lambda_0}\Lambda\xrightarrow{\cong} \mu^{\mathrm{hol}}(X;u<c)\otimes_{\Lambda_0}\Lambda.

This asserts that after extending scalars from the Novikov ring to its field, holonomic microlocal objects are generated by smooth ones. The supplied text gives no evidence that the statement has been resolved.

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Sources & referencesView supporting material

Primary source

Yuichi Ike and Tatsuki Kuwagaki, “Microlocal categories over Novikov rings”, arXiv:2307.01561 (2026).

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