一类广义共形场论的二次互易性
Quadratic reciprocity from a family of adelic conformal field theories
摘要：
我们考虑二维自由标量场理论的变形，通过提高拉普拉斯算子到一个正的实幂。结果表明，所得到的非局部广义自由作用量在整体共形对称代数的两个交换作用量下是不变的，尽管它在全Witt代数下不再是不变的.此外，有一个adelic版本的这个家庭的共形场论，参数化的选择一个数域，连同一个赫克字符。Tate的论文给出了这些理论的格林函数，并保证了这些格林函数满足一个广义乘积公式。特别是，当地的$L$ -因素有助于这些格林函数的前因子。二次互易性原来是一个结果的adelic版本的全纯因式分解属性的这个家庭的理论上的二次扩展$\mathbb{Q}$。我们解释说，在阿基米德的地方，所需的全纯因式分解如下的全球共形对称。
英文摘要：
We consider a deformation of the two-dimensional free scalar field theory by raising the Laplacian to a positive real power. It turns out that the resulting non-local generalized free action is invariant under two commuting actions of the global conformal symmetry algebra, although it is no longer invariant under the full Witt algebra. Furthermore, there is an adelic version of this family of conformal field theories, parameterized by the choice of a number field, together with a Hecke character. Tate's thesis gives the Green's functions of these theories, and ensures that these Green's functions satisfy an adelic product formula. In particular, the local $L$-factors contribute to the prefactors of these Green's functions. Quadratic reciprocity turns out to be a consequence of an adelic version of a holomorphic factorization property of this family of theories on a quadratic extension of $\mathbb{Q}$. We explain that at the Archimedean place, the desired holomorphic factorization follows from the global conformal symmetry.
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