高阶非么正Quillen K '-理论、KK-对偶及其对拓扑$\mathbb{T}$ -对偶的应用
Higher nonunital Quillen K'-theory, KK-dualities and applications to topological $\mathbb{T}$-dualities
摘要：
奎伦介绍了一个新的$K'_0$ -理论的nonunital环，并表明，在某些假设（弱于存在的团结），这一新的理论同意通常的代数$K^{alg}_0$ -理论。对于特征$0$的域$k$，我们引入了$k$ -代数的高阶非单位$K$ -理论，记作$KQ$，它推广了Quillen关于$K'_0$函子的原始定义.证明了在幂等$k$ -代数范畴上，$KQ$ -理论是Morita不变的，并且在适当的意义下满足割连通性.利用这两个性质，我们证明了稳定$C^*$ -代数的$KQ$ -理论与拓扑$K$ -理论是一致的.该机器使我们能够产生一个DG的拓扑同调$\mathbb{T}$对偶使用双变$K$理论类的范畴形式主义。最后讨论了与$C^*$ -代数的强变形的联系以及在拓扑域论中的一些潜在应用。
英文摘要：
Quillen introduced a new $K'_0$-theory of nonunital rings and showed that, under some assumptions (weaker than the existence of unity), this new theory agrees with the usual algebraic $K^{alg}_0$-theory. For a field $k$ of characteristic $0$, we introduce higher nonunital $K$-theory of $k$-algebras, denoted $KQ$, which extends Quillen's original definition of the $K'_0$ functor. We show that the $KQ$-theory is Morita invariant and satisfies excision connectively, in a suitable sense, on the category of idempotent $k$-algebras. Using these two properties we show that the $KQ$-theory agrees with the topological $K$-theory of stable $C^*$-algebras. The machinery enables us to produce a DG categorical formalism of topological homological $\mathbb{T}$-duality using bivariant $K$-theory classes. A connection with strong deformations of $C^*$-algebras and some other potential applications to topological field theories are discussed towards the end.
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