We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincar\'e polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the mathematical framework for the counting of D4-D2-D0 branes bound states on total spaces of the bundles $\cO_{\PP^1}(-p)$.
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机译：我们通过使用定位技术对Hirzebruch表面上框架无扭力槽轮的模量进行了研究。我们通过在框架模块的Huybrechts-Lehn理论的框架内对其进行研究，讨论该模空间的一些一般性质。我们对在模空间上的复曲面作用下的不动点进行分类，并使用它来计算后者的庞加莱多项式。这意味着我们正在考虑的模空间是不可约的。我们还考虑了分数的第一类Chern类，这意味着我们正在将计算扩展到Hirzebruch表面的堆叠变形。从物理角度来看，我们的结果提供了计算束$ \ cO _ {\ PP ^ 1}（-p）$上D4-D2-D0 branes束缚态的数学框架。
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