Modular graph functions and odd cuspidal functions. Fourier and Poincaré series
Modular graph functions and odd cuspidal functions. Fourier and Poincaré series
复制标题
模图函数和奇尖函数。
DOI：
10.1007/jhep04(2019)136
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发表时间：
2019
影响因子：
5.4
通讯作者：
J. Kaidi
中科院分区：
文献类型：
--
作者：
E. D'hoker;J. Kaidi
模图函数是与模为τ的环面上的二维共形场论的Feynman图相联系的SL（2，λ）-不变函数。对于单圈图，它们归结为真实的解析艾森斯坦级数。我们得到了所有双环模图函数的Fourier级数，包括常数和非常数Fourier模，以及它们关于Γ∞\PSL（2，π）的Poincaré级数.傅立叶级数和庞加莱级数提供了使用Rankin-Selberg-Zagier方法计算双环模图函数的Petersson内积的工具。在τ → −τ <$$ \overline{\tau} $$下为奇数的模图函数是尖点函数，在尖点附近指数衰减，并且从两个循环开始存在。全纯子图约简和筛算法，在早期的工作中开发的，给出了一个下界的空间A$ \mathfrak{A} $$w的奇数双循环模图函数的重量w的维数。当w ≤ 11时，该界是饱和的，我们给出了A$ \mathfrak{A} $$w的基。
A bstractModular graph functions are SL(2, ℤ)-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus τ. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincaré series with respect to Γ∞\PSL(2, ℤ). The Fourier and Poincaré series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg-Zagier methods. Modular graph functions which are odd under τ → −τ¯$$ \overline{\tau} $$ are cuspidal functions, with exponential decay near the cusp, and exist starting at two loops. Holomorphic subgraph reduction and the sieve algorithm, developed in earlier work, are used to give a lower bound on the dimension of the space A$$ \mathfrak{A} $$w of odd two-loop modular graph functions of weight w. For w ≤ 11 the bound is saturated and we exhibit a basis for A$$ \mathfrak{A} $$w.