Relative Lefschetz Action and BPS State Counting

Abstract

In this paper, we propose a mathematical definition of a new ``numerical invariants" of Calabi--Yau 3-folds from stable sheaves of dimension one, which is motivated by the Gopakumar-Vafa conjecture in M-theory. Moreover, we show that for any projective morphism $f:X->Y$ of normal projective varieties, there exists a natural $s{l}_2\times s{l}_2$ action on the intersection cohomology group $IH(X, \Q)$ which fits into the perverse Leray spectral sequence.

latex file, 25 pages, minor corrections, typos

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