Relative Calabi-Yau structure on microlocalization

Abstract

For an oriented manifold $M$ and a compact subanalytic Legendrian $\hat{I}›\subseteq S^{\ast }M$, we construct a canonical strong smooth relative Calabi--Yau structure on the microlocalization at infinity and its left adjoint $m_{\hat{I}}{›}^l:{\hat{\mathrm{I}}¼\mathrm{sh}}_{\hat{I}}›(\hat{I}›)\rightleftharpoons {\mathrm{Sh}}_{\hat{I}}›(M{)}_0:m_{\hat{I}}›$ between compactly supported sheaves on $M$ with singular support on $\hat{I}›$ and microsheaves on $\hat{I}›$. We also construct a canonical strong Calabi-Yau structure on microsheaves ${\hat{\mathrm{I}}¼\mathrm{sh}}_{\hat{I}}›(\hat{I}›)$. 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.

50 pages, 1 figure. Comments are welcome!