A mathematical theory of D-string world-sheet instantons, I: Compactness of the stack of $Z$-semistable Fourier-Mukai transforms from a compact family of nodal curves to a projective Calabi-Yau 3-fold

Abstract

In a suitable regime of superstring theory, D-branes in a Calabi-Yau space and their most fundamental behaviors can be nicely described mathematically through morphisms from Azumaya spaces with a fundamental module to that Calabi-Yau space. In the earlier work [L-L-S-Y] (D(2): arXiv:0809.2121 [math.AG], with Si Li and Ruifang Song) from the project, we explored this notion for the case of D1-branes (i.e. D-strings) and laid down some basic ingredients toward understanding the notion of D-string world-sheet instantons in this context. In this continuation, D(10), of D(2), we move on to construct a moduli stack of semistable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold $Y$. In this Part I of the note, D(10.1), we define the notion of twisted central charge $Z$ for Fourier-Mukai transforms of dimension 1 and width [0] from nodal curves and the associated stability condition on such transforms and prove that for a given compact stack of nodal curves $C_{\cal M}/{\cal M}$, the stack $FM^{1,[0];Zss}_{C_{\cal M}/{\cal M}}(Y,c)$ of $Z$-semistable Fourier-Mukai transforms of dimension 1 and width [0] from nodal curves in the family $C_{\cal M}/{\cal M}$ to $Y$ of fixed twisted central charge $c$ is compact. For the application in the sequel D(10.2), $C_{\cal M}/{\cal M}$ will contain $C_{\bar{\cal M}_g}/\bar{\cal M}_g$ as a substack and $FM^{1,[0];Zss}_{C_{\cal M}/{\cal M}}(Y,c)$ in this case will play a key role in defining stability condition for morphisms from arbitrary Azumaya nodal curves (with the underlying nodal curves not necessary in the family $C_{\cal M}/{\cal M}$) to $Y$.