International Congress of Mathematicians

Three Questions in Gromov-Witten Theory 507The degree 2 variables {t a }aeD 2 are formally suppressed in q via the divisor equation:8 TT h 7a t„ 2 for all i.(ii) n + j ß ci(X) = YJi=i deg(7 Qi )-The invariants will be defined to satisfy the divisor equation (which allows for theextraction of degree 2 classes 7 a ) and defined to vanish if degree 0 or 1 classes areinserted or if condition (ii) is violated. If L ci (X) = 0, then nî is well-definedwithout cohomology insertions.The new invariants nî (7 Ql ,..., 7 Q „ ) are defined via Gromov-Witten theory bythe following equation:F X = E E »s A*- 2 s 1 (sm(dX/2)y-" dßqd I A/2 Ig>0 ß^to, f ßCl (X)=0 d>0v'7E E^y E t*~ •••**!g>0 ß^o, f ß ci(X)>0 »>0 ai,...,o„G-D>2g. , x2a_ 2 /sin(A/2)V g - 2+ W x)•n ß (la 1 ,--- ,la n )X 9 I I ^.The above equation uniquely determines the invariants n|(7 Ql ,... ,7 Q „).Conjecture 2. For all nonsingular projective threefolds X,(i) the invariants nî (7 Ql ,..., 7 Q „ ) are integers,(ii) for fixed ß, the invariants nî (7 Ql ,..., 7 Q „ ) vanish for all sufficiently largegenera g.If X is a Calabi-Yau threefold, the Gopakumar-Vafa conjecture is recovered[15], [16]. Here, the invariants nî arise as BPS state counts in a study of TypeIIA string theory on X via M-theory. The outcome is a physical deduction of theconjecture in the Calabi-Yau case.Gopakumar and Vafa further propose a mathematical construction of theCalabi-Yau invariants nî using moduli spaces of sheaves on X. The invariantsnî should arise as multiplicities of special representations of SI2 in the cohomologyof the moduli space of sheaves. The local Calabi-Yau threefold consisting of a curveG together with a rank 2 normal bundle N satisfying ci(N) = OJC should be themost basic case. Here the BPS states n s d should be found in the cohomology of anappropriate moduli space of rank d bundles on G. A mathematical development of