Relative Lefschetz Action and BPS State Counting

Abstract

In this paper, we propose a mathematical definition of a new ``numerical\ninvariants" of Calabi--Yau 3-folds from stable sheaves of dimension one, which\nis motivated by the Gopakumar-Vafa conjecture in M-theory. Moreover, we show\nthat for any projective morphism $f:X -> Y$ of normal projective varieties,\nthere exists a natural $sl_2 \\times sl_2$ action on the intersection cohomology\ngroup $IH(X, \\Q)$ which fits into the perverse Leray spectral sequence.\n