We show that for classical Liouville field theory, diffeomorphism invariance, Weyl invariance and locality cannot hold together. This is due to a genuine Virasoro center, present in the theory, that leads to an energy-momentum tensor with non-tensorial conformal transformations, in flat space, and with a non-vanishing trace, in curved space. Our focus is on a field-independent term, proportional to the square of the Weyl gauge field, , that makes the action Weyl-invariant and was disregarded in previous investigations of Weyl and conformal symmetry. We show this term to be related to the classical center of the Virasoro algebra. The mechanism uncovered here is a classical version of the quantum anomalous phenomenon: the generalization to curved space only allows to keep one of the two symmetries enjoyed by the flat space theory, either Lorentz (diffeomorphism) or conformal invariance.

中文翻译：

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刘维尔理论的经典引力异常


我们证明，对于经典的刘维尔场论，微分同胚不变性、Weyl 不变性和局部性不能同时成立。这是由于理论中存在真正的维拉索罗中心，它导致在平坦空间中具有非张量共角变换的能量动量张量，在弯曲空间中具有不消失的迹线。我们的重点是与场无关​​的项，它与外尔规范场的平方成正比，使得作用具有外尔不变性，并且在之前对外尔和共形对称性的研究中被忽略。我们证明这个术语与维拉索罗代数的经典中心相关。这里揭示的机制是量子反常现象的经典版本：推广到弯曲空间只允许保持平坦空间理论所享有的两种对称性之一，洛伦兹（微分同胚）或共形不变性。