Joyce's natural morphism conjecture for oriented Lagrangian morphisms

Let XX be a (1)(-1)-shifted symplectic stack and let f ⁣:LXf\colon L\rightarrow X be a Lagrangian morphism. Assume that both XX and LXL\rightarrow X are equipped with orientation data.

Joyce's conjecture. There is a natural morphism

μL ⁣:Qt0(L)[dimL]f!ϕX.\mu_L\colon \mathbf{Q}_{t_0(L)}[\dim L]\longrightarrow f^!\phi_X.

This conjecture, attributed to Joyce, proposes the morphism needed to define the sheaf-theoretic Lagrangian contribution associated with oriented (1)(-1)-shifted symplectic geometry. The supplied passage does not state whether the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Florian Naef and Pavel Safronov, “Torsion volume forms”, arXiv:2308.08369 (2023).

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