Genus Zero BPS Invariants for Local ℙ1

Abstract

Abstract. We study the equivariant version of the genus zero BPS invariants of the total space of a rank 2 bundle on P 1 whose determinant is O P 1 (−2). We define the equivariant genus zero BPS invariants by the residue integrals on the moduli space of stable sheaves of dimension one as proposed by Sheldon Katz [11]. We compute these invariants for low degrees by counting the torus fixed stable sheaves. The results agree with the prediction in local Gromov-Witten theory studied in [3].

Rank 2 Sheaves on Toric 3-Folds: Classical and Virtual Counts

“…He used his methods to calculate (1) for X = P 2 and r = 2 obtaining the holomorphic part of a quasi-modular form of weight 3/2 [Kly2,VW]. Other localization calculations on (stacky/ordinary) toric surfaces appear in [Per2,Cho,GJK,Koo2].…”

Torus action on the moduli spaces of torsion plane sheaves of multiplicity four

“…Sheaves supported on a line. If the sheaf is supported on a line, the problem is the same as the problem on local P 1 with k = 1 studied in [6]. By the discussion in [6, Section 5.4 and ( 16)], we have 7 equivariant sheaves supported on a fixed line.…”

“…XY, XZ, YZ)14,11,12,8,4,3,13,7,5,10,9,6 (XY, XZ, Y 2 ) 12, 15, 11, 13, 8, 4, 14, 7, 5, 10, 9, 6 (YZ, XZ, Y 2 ) X 2 , Y 2 ) 13,17,9,14,12,16,6,8,15,7,10,11 …”

Cohomology bounds for sheaves of dimension one

“…By computing slopes, we see that F is semistable. For more details on the torus equivariant sheaves, see [1, §2.3] and [2].…”

“…Hence, part (1) of Proposition 2.4 must be satisfied. For part (2), we use the semistability of F. Suppose the generalized spectrum of F does not satisfy the condition in part (2). Then we can write π * F = G ′ ⊕G ′′ such that Hom(G ′ , G ′′ (1)) = 0.…”