Relative Calabi-Yau structure on microlocalization

For an oriented manifold $M$ and a compact subanalytic Legendrian $\mathrm{\Lambda }\subseteq S^{\ast }M$, we construct a canonical strong smooth relative Calabi–Yau structure on the microlocalization at infinity and its left adjoint $m_{\mathrm{\Lambda }}^l:{\mathrm{\mu sh}}_{\mathrm{\Lambda }}(\mathrm{\Lambda })\rightleftharpoons {\mathrm{Sh}}_{\mathrm{\Lambda }}(M{)}_0:m_{\mathrm{\Lambda }}$ between compactly supported sheaves on $M$ with singular support on $\mathrm{\Lambda }$ and microsheaves on $\mathrm{\Lambda }$. We also construct a canonical strong Calabi-Yau structure on microsheaves ${\mathrm{\mu sh}}_{\mathrm{\Lambda }}(\mathrm{\Lambda })$. Our approach does not require local properness and hence does not depend on arborealization. We thus obtain a canonical smooth relative Calabi-Yau structure on the Orlov functor for wrapped Fukaya categories of cotangent bundles with Weinstein stops, such that the wrap-once functor is the inverse dualizing bimodule.