论文标题
半圣经4维拓扑场理论无法检测到异国情调的平滑结构
Semisimple 4-dimensional topological field theories cannot detect exotic smooth structure
论文作者
论文摘要
我们证明，半神经4维的拓扑场理论会导致稳定的差异不变，因此无法区分同质封闭的封闭方向的平滑4个manifolds和同型同在简单地连接的封闭式定向的平滑4个manifolds。我们表明，目前所有已知的4维领域理论都是半imimple，包括统一的田野理论以及曾经延伸的字段理论，这些理论将代数或线性类别分配给2个manifolds。作为应用程序，我们根据其欧拉（Euler）的特征和签名来计算半简单场理论对简单连接的封闭的4个manifold的价值。此外，我们表明，在$ \ mathbb {c} p^2 $ - 稳定的差异性下，只有当时gluck扭曲的行为琐碎地起作用，半二维场理论是不变的。这可以解释为田间理论的“点粒子”之间没有费米子。这样的无费用场理论无法区分同等的4个manifolds。在整个过程中，我们通过与色带融合类别相关的Crane-toter-Kauffman场理论来说明我们的结果。作为我们的结果的代数推论，我们表明，当且仅当其高斯总和消失时，色带融合类别包含一个费米子对象。
We prove that semisimple 4-dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth 4-manifolds and homotopy equivalent simply connected closed oriented smooth 4-manifolds. We show that all currently known 4-dimensional field theories are semisimple, including unitary field theories, and once-extended field theories which assign algebras or linear categories to 2-manifolds. As an application, we compute the value of a semisimple field theory on a simply connected closed oriented 4-manifold in terms of its Euler characteristic and signature. Moreover, we show that a semisimple 4-dimensional field theory is invariant under $\mathbb{C}P^2$-stable diffeomorphisms if and only if the Gluck twist acts trivially. This may be interpreted as the absence of fermions amongst the `point particles' of the field theory. Such fermion-free field theories cannot distinguish homotopy equivalent 4-manifolds. Throughout, we illustrate our results with the Crane-Yetter-Kauffman field theory associated to a ribbon fusion category. As an algebraic corollary of our results applied to this field theory, we show that a ribbon fusion category contains a fermionic object if and only if its Gauss sums vanish.
