半简单四维拓扑场论无法检测 exotic 平滑结构
Semisimple 4-dimensional topological field theories cannot detect exotic smooth structure
AI总结：
该研究证明半简单四维拓扑场论无法区分同伦等价的4流形，并展示其与反物质的存在性相关。
摘要：
我们证明了半简单四维定向拓扑场论导致稳定的微分同胚不变量，因此不能区分同胚的闭定向4流形和同伦等价的单连通闭定向4流形。我们展示所有已知的四维场论都是半简单的，包括单位场论和一次扩展场论，后者将代数或线性范畴分配给2流形。作为应用，我们计算半简单场论在单连通闭定向4流形上的值，以它的欧拉示性与签名来表示。此外，我们显示半简单四维场论在$\mathbb{C}P^2$-稳定的微分同胚下不变当且仅当Gluck扭动作用平凡。这可以被解释为场论中`点粒子`中的反物质不存在。此类无反物质场论无法区分同伦等价的4流形。在整个过程中，我们用Crane-Yetter-Kauffman场论与关联的ribbon融合范畴来说明我们的结果。作为应用此场论的代数推论，我们显示ribbon融合范畴包含反物质对象当且仅当其Gauss和消失。
英文摘要：
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.
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