Carlos Rodriguez: Cohomologies for String Amplitudes

Abstract

In this thesis we cover methods useful for the low-energy-expansion, or α’-expansion of string amplitudes. The task of α’-expanding a string amplitude can be divided into two steps: decomposing your string amplitude into a family of integrals, and figuring out how to α’-expand each integral in these families. We review such integrals, which we call string integrals, in this thesis.

Related to string integrals, we also introduce versions of these integrals were some punctures are not integrated over. We call the resulting integrals stringy integrals, and these are going to be functions of these leftover punctures (and τ in the genus-one case). We characterize these family of stringy integrals at genus-zero and genus-one, which includes knowing what differential equations they satisfy and their α’-expansion. In fact, the differential equation of these integrals is crucial to obtain efficient α’-expansions, as generating functions of multiple polylogarithms or their elliptic counterparts, and highlight mathematical properties these integrals exhibit. We also study such generating functions of multiple polylogarithms in an abstract setting, to better understand properties of polylogarithms themselves.

We finalize with some discussion of twisted cohomology, and how it can be used to give a mathematical foundation for some of our families of stringy integrals, making them true bases of integrals. Hence, the title of the thesis. Moreover, we use twisted cohomology at genus-one to find a double-copy formula for some of our stringy integrals, and related them to generating functions of elliptic modular graph forms.