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

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 $\mathcal{E}$ 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.

30 pages

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Cited by in corpus (18)

• D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck
• 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
• Azumaya structure on D-branes and resolution of ADE orbifold singularities revisited: Douglas-Moore vs. Polchinski-Grothendieck
• Azumaya noncommutative geometry and D-branes - an origin of the master nature of D-branes
• D-branes and Azumaya/matrix noncommutative differential geometry,II: Azumaya/matrix supermanifolds and differentiable maps therefrom -- with a view toward dynamical fermionic D-branes in string theory
• 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
• Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its first variation, and the equations of motion for D-branes --- with remarks on the non-Abelian Chern-Simons/Wess-Zumino term
• Azumaya structure on D-branes and deformations and resolutions of a conifold revisited: Klebanov-Strassler-Witten vs. Polchinski-Grothendieck
• 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
• Further studies on the notion of differentiable maps from Azumaya/matrix manifolds, I. The smooth case
• 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, 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
• D0-brane realizations of the resolution of a reduced singular curve
• 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
• Further studies of the notion of differentiable maps from Azumaya/matrix supermanifolds I. The smooth case: Ramond-Neveu-Schwarz and Green-Schwarz meeting Grothendieck
• $N=1$ fermionic D3-branes in RNS formulation I. $C^{\infty }$-Algebrogeometric foundations of $d=4$, $N=1$ supersymmetry, SUSY-rep compatible hybrid connections, and $\hat{D}$-chiral maps from a $d=4$ $N=1$ Azumaya/matrix superspace
• Soft noncommutative schemes via toric geometry and morphisms from an Azumaya scheme with a fundamental module thereto -- (Dynamical, complex algebraic) D-branes on a soft noncommutative space
• D-branes and synthetic/$C^{\infty }$-algebraic symplectic/calibrated geometry, I: Lemma on a finite algebraicness property of smooth maps from Azumaya/matrix manifolds