论文标题
一属属的多个未整合穿刺的开弦积分
Open-string integrals with multiple unintegrated punctures at genus one
论文作者
论文摘要
我们研究一环开放式弦振幅中的中间步骤中出现的积分，并在$ $ cycle of torus上有多个未集成的穿刺。我们构建了此类积分的向量，该向量在取得了总差异相对于$ n $不整合的穿刺和模块化参数$τ$后关闭。发现这些积分满足椭圆形的Knizhnik-Zamolodchikov-Bernard（KZB）方程，并且可以用$α$'（弦长平方）写成幂级系列，就椭圆形多个多聚群（EMPLS）而言。在$ n $ puncture案例中，KZB方程式显示了$ b_ {1，n} $的表示形式，$ n $ strands的编织组在圆环上，作用于其解决方案。我们写下这些编织组元素中最简单的元素 - 围绕另一个辫子的辫子穿刺 - 并获得了EMPL的分析延续的生成函数。所谓的通用案例中的KZB方程是根据属的drinfeld-kohno代数$ \ mathfrak {t} _ {1，n} \ rtimes \ rtimes \ mathfrak {d} $编写的。我们的构造确定了该代数的几个发电机的各个维度的矩阵表示，它们尊重其对通勤条款的评分。
We study integrals appearing in intermediate steps of one-loop open-string amplitudes, with multiple unintegrated punctures on the $A$-cycle of a torus. We construct a vector of such integrals which closes after taking a total differential with respect to the $N$ unintegrated punctures and the modular parameter $τ$. These integrals are found to satisfy the elliptic Knizhnik-Zamolodchikov-Bernard (KZB) equations, and can be written as a power series in $α$' -- the string length squared -- in terms of elliptic multiple polylogarithms (eMPLs). In the $N$-puncture case, the KZB equation reveals a representation of $B_{1,N}$, the braid group of $N$ strands on a torus, acting on its solutions. We write the simplest of these braid group elements -- the braiding one puncture around another -- and obtain generating functions of analytic continuations of eMPLs. The KZB equations in the so-called universal case is written in terms of the genus-one Drinfeld-Kohno algebra $\mathfrak{t}_{1,N} \rtimes \mathfrak{d}$, a graded algebra. Our construction determines matrix representations of various dimensions for several generators of this algebra which respect its grading up to commuting terms.
