约化奇异曲线分解的D0-brane实现
D0-brane realizations of the resolution of a reduced singular curve
摘要：
基于超弦/D膜理论的例子，自从Douglas和Moore通过D膜探针解决超弦目标空间$Y$的奇点问题以来，各种点状D 0膜堆叠的丰富性和复杂性，作为一个指导性问题，我们提出了一个猜想，即通过将$Y^{\prime}$嵌入到${\Bbb C}$上，可以分解出各种$Y$的任何分辨率$Y^{\prime}\rightarrow Y$。对于${\Bbb N}$中的$r\ge r_0$，在$Y$上秩为$r$的点D 0-膜的栈${\frak M}^{0^{A z^f}_{\;p}}_r (Y)$，其中$r_0$依赖于$Y$的奇异胚。我们证明了这个猜想适用于${\Bbb C}$上的约化奇异曲线$C$的分辨率$ρ: C^{\prime}\rightarrow C$。用弦理论的语言来说，这就是说奇异曲线$C$的分解$C^{\prime}$总是由$C$上的一个适当的D 0-膜聚集产生，并且所涉及的D 0-膜的Chan-Paton模的秩可以选择为任意大。
英文摘要：
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 $Y^{\prime}\rightarrow Y$ of a variety $Y$ over ${\Bbb C}$ can be factored through an embedding of $Y^{\prime}$ into the stack ${\frak M}^{0^{A z^f}_{\;p}}_r (Y)$ of punctual D0-branes of rank $r$ on $Y$ for $r\ge r_0$ in ${\Bbb N}$, where $r_0$ depends on the germ of singularities of $Y$. We prove that this conjecture holds for the resolution $ρ: C^{\prime}\rightarrow C$ of a reduced singular curve $C$ over ${\Bbb C}$. 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.
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