[Paper Review] Morphisms from Azumaya prestable curves with a fundamental module to a projective variety: Topological D-strings as a master object for curves

Si Li, Chien‐Hao Liu+2 more|ArXiv.org|Sep 12, 2008

Black Holes and Theoretical Physics16 references19 citations

TL;DR

This paper establishes a mathematical framework for topological D-string world-sheet instantons by constructing the moduli stack of morphisms from Azumaya prestable curves with a fundamental module to a projective variety, using the Polchinski-Grothendieck Ansatz. The key contribution is a geometric realization of B-type D-strings as morphisms that unify D-brane moduli with stable and prestable curve moduli via forgetful functors and surrogates.

ABSTRACT

This is a continuation of our study of the foundations of D-branes from the viewpoint of Grothendieck in the region of the related Wilson's theory-space where "branes" are still branes. In this work, we focus on D-strings and construct the moduli stack of morphisms from Azumaya prestable curves $C^{Az}$ with a fundamental module ${\cal E}$ to a fixed target $Y$ of a given combinatorial type. Such a morphism gives a prototype for a Wick-rotated D-string of B-type on $Y$, following the Polchinski-Grothendieck Ansatz, and this stack serves as a ground toward a mathematical theory of topological D-string world-sheet instantons.

Motivation & Objective

• To develop a mathematical theory of topological D-string world-sheet instantons in string theory using algebraic geometry.
• To extend the Polchinski-Grothendieck Ansatz to D-strings by modeling them as morphisms from Azumaya prestable curves with fundamental modules.
• To establish a moduli stack of such morphisms that serves as a master object unifying standard moduli spaces of curves and vector bundles.
• To prove boundedness of families of morphisms and provide a concrete presentation when the target is projective space.
• To embed classical moduli stacks like $̅{M}_g$ and $̅{Bun}_{(g,r,χ)}$ into the D-string moduli stack via forgetful functors and stabilizations.

Proposed method

• Construct the moduli stack $Τ_{̅{Az}(g,r,χ)^f}(Y,\beta)$ parametrizing morphisms from Azumaya prestable curves with a fundamental module to a fixed projective variety $Y$ of combinatorial type $(g,r,\chi\mid\beta)$.
• Use the surrogate construction to recast the noncommutative Azumaya setting into a purely commutative-geometric problem via quotients of sheaves on $C_{g,n} \times Y$.
• Apply boundedness theorems for families of morphisms by leveraging the recast commutative formulation and relative ampleness of line bundles.
• Present morphisms to $\mathbb{P}^k$ using Quot schemes of sheaves on $C_{g,n} \times Y$ with fixed rank and Euler characteristic.
• Equip the moduli stack with a decorated atlas using relative Quot schemes and ample line bundles to ensure algebraic stack structure.
• Utilize forgetful functors and stabilization procedures to relate the D-string stack to classical moduli stacks of stable curves and vector bundles.

Experimental results

Research questions

• RQ1 How can the Polchinski-Grothendieck Ansatz be extended to model B-type D-strings as morphisms from Azumaya prestable curves with a fundamental module?
• RQ2 What is the structure of the moduli stack of such morphisms, and how can it be constructed algebraically for a projective target variety?
• RQ3 Can the noncommutative geometry of Azumaya structures be recast into a commutative-geometric framework to ensure boundedness and constructibility?
• RQ4 How does the D-string moduli stack relate to standard moduli spaces of curves and vector bundles via forgetful and stabilization functors?
• RQ5 In what way does the D-string moduli stack serve as a master object that unifies different moduli problems in algebraic geometry?

Key findings

• The moduli stack $Τ_{̅{Az}(g,r,χ)^f}(Y,\beta)$ is constructed as a well-defined algebraic stack with a decorated atlas using relative Quot schemes and ample line bundles.
• The noncommutative setting of Azumaya prestable curves is shown to be equivalent to a commutative-geometric problem via the surrogate construction, enabling boundedness proofs.
• Boundedness of families of morphisms is established by reducing the problem to a commutative setting and applying standard boundedness theorems.
• When the target is $\mathbb{P}^k$, morphisms are parametrized by Quot schemes of sheaves on $C_{g,n} \times Y$ with fixed rank $r$ and Euler characteristic $\chi_{n'}$.
• The D-string moduli stack admits forgetful maps to $\overline{M}_g$ and $\mathrm{Bun}_{(g,r,\chi)}$, embedding classical moduli spaces into the D-string framework.
• The stack serves as a universal master object that unifies moduli of curves, vector bundles, and Hurwitz schemes via surrogates and stabilization, supporting a mathematical theory of topological D-string instantons.