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Register a new userLagrangian cobordism functor in microlocal sheaf theory
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Wenyuan Li
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Given a Lagrangian cobordism $L$ of Legendrian submanifolds from $Lambda_-$ to $Lambda_+$, we construct a functor $Phi_L^*: Sh^c_{Lambda_+}(M) rightarrow Sh^c_{Lambda_-}(M) otimes_{C_{-*}(Omega_*Lambda_-)} C_{-*}(Omega_*L)$ between sheaf categories of compact objects with singular support on $Lambda_pm$ and its adjoint on sheaf categories of proper objects, using Nadler-Shendes work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high dimensional Legendrian submanifolds.
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We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.
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