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The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of such functions. A general method is developed for analyzing the behavior of modular graph functions under non-separating degenerations in terms of a natural real parameter $t$. For arbitrary genus, the Arakelov Green function and the Kawazumi-Zhang invariant degenerate to a Laurent polynomial in $t$ of degree $(1,1)$ in the limit $ttoinfty$. For genus two, each coefficient of the low energy expansion of the string amplitude degenerates to a Laurent polynomial of degree $(w,w)$ in $t$, where $w+2$ is the degree of homogeneity in the kinematic invariants. These results are exact to all orders in $t$, up to exponentially suppressed corrections. The non-separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a Laurent polynomial in $t$ of bounded degree. The coefficients in the Laurent polynomial are generalized modular graph functions for a punctured Riemann surface of lower genus.
Higher genus modular graph tensors map Feynman graphs to functions on the Torelli space of genus-$h$ compact Riemann surfaces which transform as tensors under the modular group $Sp(2h , mathbb Z)$, thereby generalizing a construction of Kawazumi. An
In this thesis, we investigate the low-energy expansion of scattering amplitudes of closed strings at one-loop level (i.e. at genus one) in a ten-dimensional Minkowski background using a special class of functions called modular graph forms. These al
The integral of an arbitrary two-loop modular graph function over the fundamental domain for $SL(2,Z)$ in the upper half plane is evaluated using recent results on the Poincare series for these functions.
Modular graph forms are a class of modular covariant functions which appear in the genus-one contribution to the low-energy expansion of closed string scattering amplitudes. Modular graph forms with holomorphic subgraphs enjoy the simplifying propert
We study the structure of the non-perturbative free energy of a one-parameter class of little string theories (LSTs) of A-type in the so-called unrefined limit. These theories are engineered by $N$ M5-branes probing a transverse flat space. By analys