We prove that semisimple four-dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth four-manifolds and homotopy equivalent simply connected closed oriented smooth four-manifolds. We show that all currently known four-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 four-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, settling in the negative the question of whether it is sensitive to smooth structure. As a purely 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.

中文翻译：

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半简单四维拓扑场论无法检测奇异的光滑结构

我们证明了半单四维定向拓扑场论导致稳定的微分同胚不变量，因此不能区分同胚闭定向光滑四流形和同伦等价单连通闭定向光滑四流形。我们表明所有当前已知的四维场论都是半简单的，包括酉场论，以及将代数或线性范畴分配给 2-流形的曾经扩展的场论。作为一个应用程序，我们根据欧拉特征和签名计算单连通封闭定向 4 流形上半单场论的值。此外，我们证明半简单的四维场论在以下条件下是不变的$\mathbb{C}{P}^{\mathrm{2个}}$-稳定的微分同胚当且仅当 Gluck 扭曲作用微不足道时。这可以解释为场论的“点粒子”中不存在费米子。这种无费米子场论无法区分同伦等价 4-流形。在整个过程中，我们用与带状融合类别相关的 Crane-Yetter-Kauffman 场论来说明我们的结果，否定它是否对光滑结构敏感的问题。作为我们应用于该场论的结果的纯代数推论，我们表明当且仅当其高斯和消失时，带状融合类别包含费米子对象。