The theory of microlocal sheaves, developed by Kashiwara--Schapira, has found many remarkable applications in the study of symplectic topology. For a smooth Lagrangian L in a cotangent bundle of a smooth manifold and a commutative ring spectrum k, one can associate a sheaf of microlocal categories, which is locally constant with fiber equivalent to Mod(k). It admits a classifying map L--->BPic(k) as a fiber bundle does. We will show that the classifying map factors through the Gauss map L--->U/O and the delooping of the J-homomorphism U/O--->BPic(S), where S is the sphere spectrum. As an application, combining with previous results of Guillermou, we show that if L is a compact smooth exact Lagrangian, then the classifying map is homotopically trivial, recovering a result of Abouzaid--Kragh. 

Conference ID: 660 7704 3025

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