Categorified BPS invariants for the category of coherent sheaves on a CY3

This talk is on joint work with Sven Meinhardt.  The (refined) BPS invariants for a CY3 category C (equipped with orientation data) are defined as plethystic logarithms of partition functions recording the weights on the total hypercohomology of the vanishing cycle-type sheaf defined by Ben-Bassat, Brav, Bussi, Dupont and Joyce.  In the case that C=Coh(X), for X a CY3 variety, the integrality conjecture states that these plethystic logarithms produce Laurent polynomials, as opposed to Laurent formal power series.  Then, setting q^{1/2}=1 we recover (more down to earth!) BPS/DT invariants.  

Heuristically, these polynomials are the weight polynomials for the "space of BPS states" - although there is no definition of such a space in the picture, hence the inverted commas.  We have shown that this picture can be categorified, in the sense that the above-mentioned hypercohomology is isomorphic to the symmetric algebra generated by an explicit mixed Hodge structure, which we can then define to be the mixed Hodge structure on the cohomology of the space of BPS states.  This produces the following strengthening of the integrality conjecture: for a fixed stability condition and slope, there is a Lie algebra g, endowed with a mixed Hodge structure, graded by Chern classes of the same fixed slope, and for which each graded piece is finite dimensional, whose graded components recover the refined BPS invariants as their weight polynomials.

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