Microlocal sheaf categories and the J-homomorphism

For an exact Lagrangian $L$ in a cotangent bundle, one can define a sheaf of stable-infinity categories on it, called the Kashiwara--Schapira stack. Assuming the Lagrangian is smooth, the sheaf of categories is a local system with fiber equivalent to Mod(k), where k is the coefficient ring spectrum (at least E_2). I will show that the classifying map for the local system of categories factors through the stable Gauss map L--->U/O and the delooping of the J-homomorphism U/O-->BPic(S), where S is the sphere spectrum. Part of the proof employs the (\infty,2)-category of correspondences developed by Gaitsgory--Rozenblyum. If time permits, I will also talk about some applications of this result.

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