Poisson equation for the three loop ladder diagram in string theory at genus one

摘要

The three loop ladder diagram is a graph with six links and four cubicvertices that contributes to the D^{12} R^4 amplitude at genus one in type IIstring theory. The vertices represent the insertion points of vertex operatorson the toroidal worldsheet and the links represent scalar Green functionsconnecting them. By using the properties of the Green function and manipulatingthe various expressions, we obtain a modular invariant Poisson equationsatisfied by this diagram, with source terms involving one, two and three loopdiagrams. Unlike the source terms in the Poisson equations for diagrams atlower orders in the momentum expansion or the Mercedes diagram, a particularsource term involves a five point function containing a holomorphic and aantiholomorphic worldsheet derivative acting on different Green functions. Wealso obtain simple equalities between topologically distinct diagrams, andconsider some elementary examples.