D-膜与$K$ -同调
D-branes and $K$-homology
摘要：
本文考察了弦理论时空流形$X$的拓扑$K$ -同调群与弦理论D-膜之间的密切关系。$K$ -同调群的一个元素由$K$ -圈$[M,E,ϕ]$的等价类给出，其中$M$是闭自旋$^c$流形，$E$是$M$上的复向量丛，$ϕ: M\rightarrow X$是连续映射.提出了一个$K$ -圈$[M,E,ϕ]$表示包裹子空间$ϕ(M)$的D-膜构型.因此，由$[M,E,ϕ]$定义的$K$ -同调元素代表一类具有相同物理电荷的D-膜构型。此外，D-膜的$K$ -循环表示类似于现代描述基本弦的方法，在这种方法中，弦被表示为二维表面，并带有到时空流形的映射。D-膜的这种分类也暗示了物理解释包裹时空奇异子空间的D-膜的可能性，扩大了弦理论可以处理的已知奇异性类型。
英文摘要：
In this thesis the close relationship between the topological $K$-homology group of the spacetime manifold $X$ of string theory and D-branes in string theory is examined. An element of the $K$-homology group is given by an equivalence class of $K$-cycles $[M,E,ϕ]$, where $M$ is a closed spin$^c$ manifold, $E$ is a complex vector bundle over $M$, and $ϕ: M\rightarrow X$ is a continuous map. It is proposed that a $K$-cycle $[M,E,ϕ]$ represents a D-brane configuration wrapping the subspace $ϕ(M)$. As a consequence, the $K$-homology element defined by $[M,E,ϕ]$ represents a class of D-brane configurations that have the same physical charge. Furthermore, the $K$-cycle representation of D-branes resembles the modern way of characterizing fundamental strings, in which the strings are represented as two-dimensional surfaces with maps into the spacetime manifold. This classification of D-branes also suggests the possibility of physically interpreting D-branes wrapping singular subspaces of spacetime, enlarging the known types of singularities that string theory can cope with.
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