Pseudoholomorphic curves with boundary: Can you count them? Can you really?

Pseudoholomorphic curves with boundary: Can you count them? Can you really? - Sara Tukachinsky

Sara Tukachinsky
Member, School of Mathematics
November 4, 2019

Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from curves with boundary to a symplectic manifold, with various constraints on boundary and interior marked points. The presence of boundary poses an obstacle to invariance. In a joint work with J. Solomon (2016-2017), we defined genus zero OGW invariants under cohomological conditions.

    The construction is rather abstract. Nonetheless, in a recent work, also joint with J. Solomon, we find that the generating function of OGW has many properties that enable explicit calculations. Most notably, it satisfies a system of PDE called open WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equation. For the projective space, this PDE generates recursion relations that allow the computation of all invariants.

      Moreover, the open WDVV can be reinterpreted as an associativity of a suitable version of a quantum product.

        No prior knowledge of any of the above notions will be assumed.

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