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Vafa–Witten Invariants from Exceptional Collections
Communications in Mathematical Physics ( IF 2.4 ) Pub Date : 2021-06-12 , DOI: 10.1007/s00220-021-04074-2
Guillaume Beaujard , Jan Manschot , Boris Pioline

Supersymmetric D-branes supported on the complex two-dimensional base S of the local Calabi–Yau threefold \(K_S\) are described by semi-stable coherent sheaves on S. Under suitable conditions, the BPS indices counting these objects (known as generalized Donaldson–Thomas invariants) coincide with the Vafa–Witten invariants of S (which encode the Betti numbers of the moduli space of semi-stable sheaves). For surfaces which admit a strong collection of exceptional sheaves, we develop a general method for computing these invariants by exploiting the isomorphism between the derived category of coherent sheaves and the derived category of representations of a suitable quiver with potential (QW) constructed from the exceptional collection. We spell out the dictionary between the Chern class \(\gamma \) and polarization J on S versus the dimension vector \(\vec {N}\) and stability parameters \(\vec {\zeta }\) on the quiver side. For all examples that we consider, which include all del Pezzo and Hirzebruch surfaces, we find that the BPS indices \(\Omega _\star (\gamma )\) at the attractor point (or self-stability condition) vanish, except for dimension vectors corresponding to simple representations and pure D0-branes. This opens up the possibility to compute the BPS indices in any chamber using either the flow tree or the Coulomb branch formula. In all cases we find precise agreement with independent computations of Vafa–Witten invariants based on wall-crossing and blow-up formulae. This agreement suggests that (1) generating functions of DT invariants for a large class of quivers coming from strong exceptional collections are mock modular functions of higher depth and (2) non-trivial single-centered black holes and scaling solutions do not exist quantum mechanically in such local Calabi–Yau geometries.



中文翻译:

来自特殊集合的 Vafa-Witten 不变量

支撑在复杂的二维基超对称d-branes小号本地卡拉比-丘的三倍\(K_S \)由上半稳定相干滑轮描述小号。在合适的条件下,计算这些对象的 BPS 指数(称为广义唐纳森-托马斯不变量)与S的 Vafa-Witten 不变量(编码半稳定滑轮模空间的 Betti 数)一致。对于允许大量异常滑轮集合的表面,我们开发了一种通用方法来计算这些不变量,方法是利用相干滑轮的派生类别与具有潜力的合适颤动的表示的派生类别之间的同构(QW) 从特殊集合构造。我们拼出S上的陈类\(\gamma \)和极化J与颤动侧的维向量\(\vec {N}\)和稳定性参数\(\vec {\zeta }\)之间的字典. 对于我们考虑的所有例子,包括所有 del Pezzo 和 Hirzebruch 表面,我们发现 BPS 指数\(\Omega _\star (\gamma )\)在吸引点(或自稳定条件)处消失,除了对应于简单表示和纯 D0-膜的维向量。这开辟了使用流动树或库仑分支公式计算任何腔室中 BPS 指数的可能性。在所有情况下,我们都发现与基于穿墙和爆破公式的 Vafa-Witten 不变量的独立计算精确一致。该协议表明 (1) 来自强大异常集合的一大类颤动的 DT 不变量的生成函数是更高深度的模拟模块化函数和 (2) 非平凡的单中心黑洞和缩放解决方案在量子力学上不存在在这种局部的 Calabi-Yau 几何中。

更新日期:2021-06-13
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