We derive how to incorporate topological features of Riemann surfaces in string amplitudes by insertions of bi-local operators called handle operators. The resulting formalism is exact and globally well-defined in moduli space. After a detailed and pedagogical discussion of Riemann surfaces, complex structure deformations, global vs local aspects, boundary terms, an explicit choice of gluing-compatible and global (modulo U(1)) coordinates (termed `holomorphic normal coordinates'), finite changes in normal ordering, and factorisation of the path integral measure, we construct these handle operators explicitly. Adopting an offshell local coherent vertex operator basis for the latter, and gauge fixing invariance under Weyl transformations using holomorphic normal coordinates (developed by Polchinski), is particularly efficient. All loop amplitudes are gauge-invariant (BRST-exact terms decouple up to boundary terms in moduli space), and reparametrisation invariance is manifest, for arbitrary worldsheet curvature and topology (subject to the Euler number constraint). We provide a number of complementary viewpoints and consistency checks (including one-loop modular invariance, we compute all one- and two-point sphere amplitudes, glue two three-point sphere amplitudes to reproduce the exact four-point sphere amplitude, etc.).

中文翻译：

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处理弦论中的运算符

我们推导出如何通过插入称为句柄算子的双局部算子将黎曼曲面的拓扑特征合并到弦振幅中。由此产生的形式主义在模空间中是精确且全局定义的。经过对黎曼曲面、复杂结构变形、全局与局部方面、边界项、胶合兼容和全局（模 U(1)）坐标（称为“全纯法线坐标”）的明确选择、有限变化的详细和教学讨论在正常排序和路径积分度量的因式分解中，我们明确地构造了这些句柄运算符。后者采用离壳局部相干顶点算子基础，并使用全纯法线坐标（由 Polchinski 开发）在 Weyl 变换下规范固定不变性，特别有效。所有环振幅都是规范不变的（BRST 精确项解耦到模空间中的边界项），并且对于任意世界片曲率和拓扑（受欧拉数约束），重新参数化不变性是显而易见的。我们提供了许多互补的观点和一致性检查（包括单环模不变性，我们计算所有的一点和两点球体振幅，粘合两个三点球体振幅以重现精确的四点球体振幅等） .