D0-brane realizations of the resolution of a reduced singular curve

Abstract

Based on examples from superstring/D-brane theory since the work of Douglas and Moore on resolution of singularities of a superstring target-space $Y$ via a D-brane probe, the richness and the complexity of the stack of punctual D0-branes on a variety, and as a guiding question, we lay down a conjecture that any resolution $Y^{\prime }\to Y$ of a variety $Y$ over $\mathbb{C}$ can be factored through an embedding of $Y^{\prime }$ into the stack ${\mathfrak{M}}_r^{0_p^{A{z}^f}}(Y)$ of punctual D0-branes of rank $r$ on $Y$ for $r\ge r_0$ in $\mathbb{N}$, where $r_0$ depends on the germ of singularities of $Y$. We prove that this conjecture holds for the resolution $\rho :C^{\prime }\to C$ of a reduced singular curve $C$ over $\mathbb{C}$. In string-theoretical language, this says that the resolution $C^{\prime }$ of a singular curve $C$ always arises from an appropriate D0-brane aggregation on $C$ and that the rank of the Chan-Paton module of the D0-branes involved can be chosen to be arbitrarily large.

9+2 pages

References in corpus (6)

• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• 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
• Azumaya structure on D-branes and resolution of ADE orbifold singularities revisited: Douglas-Moore vs. Polchinski-Grothendieck
• Azumaya structure on D-branes and deformations and resolutions of a conifold revisited: Klebanov-Strassler-Witten vs. Polchinski-Grothendieck

Cited by in corpus (2)

• 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 noncommutative geometry and D-branes - an origin of the master nature of D-branes