论文标题
来自Adelic Sonpormal Field Theories的二次互惠
Quadratic reciprocity from a family of adelic conformal field theories
论文作者
论文摘要
我们通过将Laplacian提高到积极的真实力量来考虑二维自由标量场理论的变形。事实证明，在全球形式的对称代数的两个通勤动作下，由此产生的非本地广义自由行动是不变的，尽管在完整的Witt代数下不再不变。此外，这种同条场理论家族的Adelic版本是通过选择数字和Hecke字符的参数来参数的。泰特（Tate）的论文赋予了这些理论的绿色功能，并确保这些绿色的功能满足了Adelic产品公式。特别是，本地$ l $ factor为这些绿色功能的预成分做出了贡献。事实证明，二次互惠是由于$ \ mathbb {q} $的二次扩展，是该理论家族的全体形态分解属性的结果。我们解释说，在阿基米德的地方，所需的全体形态分解遵循全球形式的对称性。
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.
