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5. Structures in arboreal geometry

The theory of arboreal spaces (or "arborifolds") offers a rich new geometry with many interesting questions. The theory also offers us a new language for presenting already useful or important structures from symplectic, differential, and complex geometry.

    1.     By splitting a closed $(n+1)$-dimensional arboreal space along an $n$-dimensional arboreal space, we can treat it as a relation between two $n$-dimensional arboreal spaces. This geometry motivates the following question:

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      Problem 5.1.

      Is there a theory of cobordism for closed arboreal spaces?

        •     The data of a vector bundle with flat connection on a smooth manifold $M$ is the same as the data of a microlocal sheaf on $T^*M$ microsupported on $M$. This suggests that there should be a notion of "microlocal vector bundles" on an arboreal space $L$ so that "microlocal vector bundles with flat connection" are precisely microlocal sheaves on $L$.

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          Problem 5.2.

          Give the correct definition of "microlocal vector bundles" on an arboreal space.
            1. Remark. The resulting category should contain an object called the "microlocal tangent bundle" of $L$, which is the same thing as a ribbon for $L$. We should be able to recover invariants like Chern classes from index theory.

                •     Boundary conditions in Floer theory are given by smooth Lagrangians equipped with local systems. As in the above question, the analogous data on an arboreal Lagrangian is a microlocal sheaf. We might hope to understand this analogy in a Floer-theoretic way.

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                  Problem 5.3.

                  Is it possible to set up Floer theory with boundary on an arboreal skeleton? What are arboreal Lagrangian boundary conditions?


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                      Problem 5.4.

                      Suppose a Weinstein manifold $W$, with arboreal skeleton $L$, has a holomorphic symplectic structure. How can we use this structure to enhance the arboreal space $L$ with extra data?


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                          Problem 5.5.

                          Is there a reasonable notion of string topology on an arboreal space?

                              Cite this as: AimPL: Arborealization of singularities of Lagrangian skeleta, available at http://aimpl.org/arborlagrange.