CMSA Algebraic Geometry in String Theory Seminar: Stacky small resolutions of determinantal octic double solids and noncommutative Gopakumar-Vafa invariants

SEMINARS, CMSA EVENTS

Speaker:

Sheldon Katz - UIUC

A determinantal octic double solid is the double cover X of P^3 branched along the degree 8 determinant of a symmetric matrix of homogeneous forms on P^3. These X are nodal CY threefolds which do not admit a projective small resolution. B-model techniques can be applied to compute GV invariants up to g \le 32. This raises the question: what is the geometric meaning of these invariants?

Evidence suggests that these enumerative invariants are associated with moduli stacks of coherent sheaves of modules over a sheaf B of noncommutative algebras on X constructed by Kuznetsov. One of these moduli stacks is a stacky small resolution X’ of X itself. This leads to another geometric interpretation of the invariants as being associated with moduli of sheaves on X’ twisted by a Brauer class. Geometric computations based on these working definitions always agree with the B-model computations.

This talk is based on joint work with Albrecht Klemm, Thorsten Schimannek, and Eric Sharpe