On the modular structure of the genus-one Type II superstring low energy expansion
On the modular structure of the genus-one Type II superstring low energy expansion
复制标题
属一II型超弦低能展开的模块化结构
DOI：
10.1007/jhep08(2015)041
复制
发表时间：
2015
影响因子：
5.4
通讯作者：
P. Vanhove
中科院分区：
文献类型：
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作者：
E. D'hoker;M. Green;P. Vanhove
BstractThe解析贡献的低能量扩展的II型弦振幅在genus-one是一个幂级数的时空导数的系数是由积分的模函数在复杂的结构模的世界片环面。这些模函数与世界片真空费曼图，并给出了多个总和的离散动量的环面。在本文中，我们展示了精确的微分和代数关系的某种无限类这样的模函数，证明他们满足拉普拉斯特征值方程的非齐次项是多项式的非全纯Eisenstein级数。此外，我们认为，一组模函数，有助于相互作用的系数到D 10 <$4 $$ {D}^{10}{\mathrm{\mathcal{R}^4 $$是线性和的功能在这类和二次多项式的爱森斯坦级数和奇黎曼zeta值。在复杂结构上的积分导致低能量展开的系数，其是奇数黎曼zeta值中的有理数乘以单项式。
A bstractThe analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order D10ℛ4$$ {D}^{10}{\mathrm{\mathcal{R}}}^4 $$ are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.