Intersection Spaces, Spatial Homology Truncation, and String Theory
Intersection Spaces, Spatial Homology Truncation, and String Theory
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交集空间、空间同调截断和弦理论
DOI：
10.1007/978-3-642-12589-8
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发表时间：
2010
影响因子：
0.6
通讯作者：
Markus Banagl
中科院分区：
文献类型：
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作者：
Markus Banagl
对于一个分层奇异空间X，我们将新的空间I p <$X，它的反常p <$-交空间联系起来，使得当X是一个闭的定向伪流形时，如果p <$和q <$是互补的，则I p <$X的普通有理上同调是I q <$X的普通有理同调的庞加莱对偶。I p <$X的同调不同构于交同调，因此得到了伪流形的一个新的对偶理论，它解决了弦理论中与锥变换过程中无质量D-膜的存在及其作为上同调类的忠实表示有关的某些需要。相交同调正确地解释了IIA型弦论中所有的无质量D-膜，而相交空间的同调正确地解释了IIB型弦论中所有的无质量D-膜。事实上，对于奇异的卡-丘二次曲线，这两个理论在镜像对称的意义上是彼此的镜像。新理论还允许某些类型的帽产物，已知不存在交叉同源性。使用这些产品，我们表明，与对称的L-同调基本类的上限诱导的有理对称L-上同调的Im X和有理L-同调的In X之间的同构。X上的反常p-交向量丛可以定义为I p <$X上的实向量丛。在本专着中，I p X的构造是针对孤立奇点进行的，更一般地，针对具有平凡链丛的双层空间进行的。它是基于一个深入的和自治同伦理论分析的空间同调截断，其中重点放在调查函。
To a stratified singular space X, we associate new spaces I p̄X, its perversity p̄-intersection spaces, such that when X is a closed, oriented pseudomanifold, the ordinary rational cohomology of I p̄X is Poincaré dual to the ordinary rational homology of I q̄X if p̄ and q̄ are complementary perversities. The homology of I p̄X is not isomorphic to intersection homology so that a new duality theory for pseudomanifolds is obtained, which addresses certain needs in string theory related to the existence of massless D-branes in the course of conifold transitions and their faithful representation as cohomology classes. While intersection homology accounts correctly for all massless D-branes in type IIA string theory, the homology of intersection spaces accounts correctly for all massless D-branes in type IIB string theory. In fact, for singular Calabi-Yau conifolds, the two theories are mirrors of each other in the sense of mirror symmetry. The new theory also allows for certain types of cap products that are known not to exist for intersection homology. Using these products, we show that capping with the symmetric L-homology fundamental class induces an isomorphism between the rational symmetric L-cohomology of Im̄X and the rational L-homology of In̄X. Perversity p̄-intersection vector bundles on X may be defined as actual vector bundles on I p̄X. In the present monograph, the construction of I p̄X is carried out for isolated singularities and, more generally, for two-strata spaces with trivial link bundle. It is based on an in-depth and autonomous homotopy theoretic analysis of spatial homology truncation, where an emphasis was placed on investigating functoriality.