We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vacuum expectation value of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey–Kirwan residue formula) leads to the Bradlow bounds ( upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss the properties of the moduli space volume in these theories. Our formulae are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with $\mathbb{C} P^N$ target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

中文翻译：

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颤动涡模空间的体积

我们研究了紧凑黎曼曲面上颤振规范理论中 BPS 涡旋的模空间体积。BPS 涡流的存在对颤振规范理论施加了约束。我们表明，模空间体积是由超对称颤动规范理论中合适的上同调算子（体积算子）的真空期望值给出的，其中嵌入了涡旋的 BPS 方程。在超对称规范理论中，模空间体积通过局部化被精确地评估为轮廓积分。图论有助于构建超对称颤振规范理论和推导体积公式。体积的等高线积分公式（Jeffrey-Kirwan 剩余公式的推广）导致 Bradlow 界（涡度的上限除以黎曼曲面的面积除以涡旋的固有尺寸）。我们给出了各种颤抖规范理论的一些例子，并讨论了这些理论中模空间体积的性质。我们的公式适用于具有 $\mathbb{C} P^N$ 目标空间的规范非线性 sigma 模型中的涡量模空间的体积，该模型是通过父颤动规范理论的强耦合极限获得的。我们还讨论了颤抖规范理论的非阿贝尔推广和体积公式的“阿贝尔化”。我们的公式适用于具有 $\mathbb{C} P^N$ 目标空间的规范非线性 sigma 模型中的涡量模空间的体积，该模型是通过父颤动规范理论的强耦合极限获得的。我们还讨论了颤抖规范理论的非阿贝尔推广和体积公式的“阿贝尔化”。我们的公式适用于具有 $\mathbb{C} P^N$ 目标空间的规范非线性 sigma 模型中的涡量模空间的体积，该模型是通过父颤动规范理论的强耦合极限获得的。我们还讨论了颤抖规范理论的非阿贝尔推广和体积公式的“阿贝尔化”。