$O(-2)$ Blow-up Formula via Instanton Calculus on $\hat{C^2/Z_2}$ and Weil Conjecture

Details $O(-2)$ Blow-up Formula via Instanton Calculus on $\hat{C^2/Z_2}$ and Weil Conjecture $O(-2)$ Blow-up Formula via Instanton Calculus on $\hat{C^2/Z_2}$ and Weil Conjecture

2006-03-21

arxiv.org/abs/hep-th/0603162v2

Physics

High Energy Physics

High Energy Physics - Theory

31 pages, reference and appendix added

Scientific paper

We calculate Betti numbers of the framed moduli space of instantons on $\hat{{\bf C}^2/{\bf Z}_2}$, under the assumption that the corresponding torsion free sheaves $E$ have vanishing properties ($Hom(E,E(-l_\infty))=Ext^2(E,E(-l_\infty))=0$). Moreover we derive the generating function of Betti numbers and obtain closed formulas. On the other hand, we derive a universal relation between the generating function of Betti numbers of the moduli spaces of stable sheaves on $X$ with an $A_1$-singularity and that on $\hat{X}$ blow-uped at the singularity, by using Weil conjecture. We call this the $O(-2)$ blow-up formula. Applying this to $X={\bf C}^2/{\bf Z}_2$ case, we reproduce the formula given by instanton calculus.

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