Motivic homotopy theory of group scheme actions

We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic $K$-theory is representable in the resulting homotopy category. Additionally, we establish homotopical purity and blow-up theorems for finite abelian groups.Comment: Final version, to appear in Journal of Topology. arXiv admin note: text overlap with arXiv:1403.191

BPS states of curves in Calabi--Yau 3--folds

The Gopakumar-Vafa conjecture is defined and studied for the local geometry of a curve in a Calabi-Yau 3-fold. The integrality predicted in Gromov-Witten theory by the Gopakumar-Vafa BPS count is verified in a natural series of cases in this local geometry. The method involves Gromov-Witten computations, Mobius inversion, and a combinatorial analysis of the numbers of etale covers of a curve.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper9.abs.html Version 3 is GT version 2 and has corrections to eq (2) on p 295, to 1st eq in Prop 2.1 and the tables on p 39