Normed Calabi–Yau structure of Shklyarov’s ∞-trace on MF(W_Δ)

Establish that for every Delzant polytope Δ, the ∞-trace Θ of Shklyarov on the matrix factorization category MF(W_Δ) defines a normed chain-level Calabi–Yau structure in the sense of the paper’s Definition of a normed ∞-trace (cohomologically unital, gapped, valuation-preserving, and inducing a non-degenerate pairing).

Background

From any Delzant polytope Δ, the authors construct a valued Landau–Ginzburg model LG(Δ) and equip MF(W_Δ) with Shklyarov’s trace Θ, a refinement of the Kapustin–Li trace. They prove Θ has dimension equal to dim Δ and is cohomologically unital when dim Δ is odd, and they also show Θ is normed for the standard simplex Δ^n with n odd.

The conjecture seeks to extend these properties to all Delzant polytopes, asserting that Θ satisfies the full normed Calabi–Yau axioms (including non-degeneracy and valuation compatibility) across MF(W_Δ). This would provide the categorical foundation needed for the enumerative mirror constructions proposed in the paper.