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 $C_{\mathcal{M}}/\mathcal{M}$, the stack $F{M}_{C_{\mathcal{M}}/\mathcal{M}}^{1,[0];Zss}(Y,c)$ of $Z$-semistable Fourier-Mukai transforms of dimension 1 and width [0] from nodal curves in the family $C_{\mathcal{M}}/\mathcal{M}$ to $Y$ of fixed twisted central charge $c$ is compact. For the application in the sequel D(10.2), $C_{\mathcal{M}}/\mathcal{M}$ will contain $C_{{\bar{\mathcal{M}}}_g}/{\bar{\mathcal{M}}}_g$ as a substack and $F{M}_{C_{\mathcal{M}}/\mathcal{M}}^{1,[0];Zss}(Y,c)$ 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 $C_{\mathcal{M}}/\mathcal{M}$) to $Y$.

arXiv comment25+2 pages

References in corpus (21)

• Fivebranes, Membranes and Non-Perturbative String Theory
• TASI Lectures on D-Branes
• I-Brane Inflow and Anomalous Couplings on D-Branes
• D-Branes on Calabi-Yau Spaces and Their Mirrors
• D-brane Instantons in Type II String Theory
• Open string instantons and superpotentials
• Lectures on D-branes
• Dirichlet branes, homological mirror symmetry, and stability
• Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair
• Noncommutative Geometry and D-Branes
• Lectures on Anomalies
• TASI 2003 Lectures on Anomalies
• Superstring Dualities, Dirichlet Branes and the Small Scale Structure of Space
• Moduli of PT-Semistable Objects I
• Azumaya-type noncommutative spaces and morphisms therefrom: Polchinski's D-branes in string theory from Grothendieck's viewpoint
• Morphisms from Azumaya prestable curves with a fundamental module to a projective variety: Topological D-strings as a master object for curves
• D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck
• Stability data and t-structures on a triangulated category
• Azumaya noncommutative geometry and D-branes - an origin of the master nature of D-branes
• Nontrivial Azumaya noncommutative schemes, morphisms therefrom, and their extension by the sheaf of algebras of differential operators: D-branes in a $B$-field background à la Polchinski-Grothendieck Ansatz
• D-branes of A-type, their deformations, and Morse cobordism of A-branes on Calabi-Yau 3-folds under a split attractor flow: Donaldson/Alexander-Hilden-Lozano-Montesinos-Thurston/Hurwitz/Denef-Joyce meeting Polchinski-Grothendieck

Cited by in corpus (3)

• D-branes and Azumaya/matrix noncommutative differential geometry, I: D-branes as fundamental objects in string theory and differentiable maps from Azumaya/matrix manifolds with a fundamental module to real manifolds
• Dynamics of D-branes II. The standard action --- an analogue of the Polyakov action for (fundamental, stacked) D-branes
• A mathematical theory of D-string world-sheet instantons, II: Moduli stack of $Z$-(semi)stable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold