Vafa-Witten Invariants from Exceptional Collections

Supersymmetric D-branes supported on the complex two-dimensional base S of the local Calabi-Yau threefold KS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_S$$\end{document} 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 (Q, W) constructed from the exceptional collection. We spell out the dictionary between the Chern class gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} and polarization J on S versus the dimension vector N ->\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {N}$$\end{document} and stability parameters zeta ->\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {\zeta }$$\end{document} 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(gamma)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\star (\gamma )$$\end{document} 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.

Automated summary

This paragraph discusses supersymmetric D-branes supported on local Calabi-Yau threefolds, describing them using semi-stable coherent sheaves on a complex two-dimensional base, and explores the relationship between BPS indices and Vafa-Witten invariants. A general method for computing these invariants is developed by exploiting the isomorphism between coherent sheaves and representations of a specific quiver with potential. The paragraph also elaborates on the dictionary between different concepts and finds precise agreement with independent computations of Vafa-Witten invariants.