Recent developments on Feynman integrals and string amplitudes greatly benefitted from multiple polylogarithms and their elliptic analogues — iterated integrals on the sphere and the torus, respectively. In this talk, I will review the Brown-Levin construction of elliptic polylogarithms and propose a generalization to Riemann surfaces of arbitrary genus. In particular, iterated integrals on a higher-genus surface will be derived from a flat connection. The integration kernels of our flat connection consist of modular tensors, built from convolutions of Arakelov Green functions and their derivatives with holomorphic Abelian differentials. At genus one, these convolutions reproduce the Kronecker-Eisenstein kernels of elliptic polylogarithms and modular graph forms. I will conclude with an outlook on open problems and work in progress.

In this talk, I will describe new mathematical structures in the low-energy expansion of one-loop string amplitudes. The insertion of external states on the open- and closed-string worldsheets requires integration over punctures on a cylinder boundary and a torus, respectively. Suitable bases of such integrals will be shown to obey simple first-order differential equations in the modular parameter of the surface. These differential equations will be exploited to perform the integrals order by order in the inverse string tension, similar to modern strategies for dimensionally regulated Feynman integrals. Our method manifests the appearance of iterated integrals over holomorphic Eisenstein series in the low-energy expansion. Moreover, infinite families of Laplace equations can be generated for the modular forms in closed-string low-energy expansions.

In this talk, I will describe new mathematical structures in the low-energy expansion of one-loop string amplitudes. The insertion of external states on the open- and closed-string worldsheets requires integration over punctures on a cylinder boundary and a torus, respectively. Suitable bases of such integrals will be shown to obey simple first-order differential equations in the modular parameter of the surface. These differential equations will be exploited to perform the integrals order by order in the inverse string tension, similar to modern strategies for dimensionally regulated Feynman integrals. Our method manifests the appearance of iterated integrals over holomorphic Eisenstein series in the low-energy expansion. Moreover, infinite families of Laplace equations can be generated for the modular forms in closed-string low-energy expansions.

In this talk, I will review basic features of the pure spinor superstring as well as recent progress to compute and compactly represent scattering amplitudes in this framework. A string-inspired organization scheme for amplitudes in both field- and string-theories will be described where the non-linearities of ten-dimensional super Yang-Mills theory are encoded in so-called multiparticle superfields. They allow to efficiently capture the polarization dependence through cubic diagrams where the intuitive mapping as well as the composition rules for amplitudes are guided by BRST invariance.

We discuss tree level scattering of any number of massless open superstring states on a worldsheet of disk topology. The entire state dependence of the tree amplitude can be expressed in terms of gauge theory subamplitudes from the point particle limit. The string corrections entering through momentum dependent integrals over the disk boundary can be disentangled from the YM seeds and analyzed separately. Their power series expansion in the string length and momenta involves multiple zeta values (MZVs). We review some mathematical background on MZVs and the network of relations between them. The explicit form of any tree level string correction to YM theory is derived from the generating function of MZVs -- the Drinfeld associator. It interpolates between the worldsheet integrals in N-point and (N-1)-point scattering and leads to a recursive formula for the momentum expansion of any disk amplitude. Our results apply for any number of spacetime dimensions or supersymmetries and chosen helicity configurations.

I will discuss the mathematical structure of tree level amplitudes among massless superstring states. String corrections to these amplitudes take a compact and elegant form once the contributions from different classes of multiple zeta values (MZVs) are disentangled. The idea is to lift MZVs to their motivic versions endowed with a Hopf algebra structure: It induces an isomorphism which casts the amplitudes into a very symmetric form and represents the generalization of the symbol of a transcendental function. I will also comment on generalizations to loop amplitudes.