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Modular Graph Forms and Scattering Amplitudes in String Theory

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 نشر من قبل Jan E. Gerken
 تاريخ النشر 2020
  مجال البحث
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 تأليف Jan E. Gerken




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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 allow for a systematic evaluation of the low-energy expansion and satisfy many non-trivial algebraic and differential relations. We study these relations in detail, leading to basis decompositions for a large number of modular graph forms which greatly reduce the complexity of the expansions of the integrals appearing in the amplitude. One of the results of this thesis is a Mathematica package which automatizes these simplifications. We use these techniques to compute the leading low-energy orders of the scattering amplitude of four gluons in the heterotic string at one-loop level. Furthermore, we study a generating function which conjecturally contains the torus integrals of all perturbative closed-string theories. We write this generating function in terms of iterated integrals of holomorphic Eisenstein series and use this approach to arrive at a more rigorous characterization of the space of modular graph forms than was possible before. For tree-level string amplitudes, the single-valued map of multiple zeta values maps open-string amplitudes to closed-string amplitudes. The definition of a suitable one-loop generalization, a so-called elliptic single-valued map, is an active area of research and we provide a new perspective on this topic using our generating function of torus integrals. The original version of this thesis, as submitted in June 2020 to the Humboldt University Berlin, is available under the DOI 10.18452/21829. The present text contains minor updates compared to this version, reflecting further developments in the literature, in particular concerning the construction of an elliptic single-valued map.



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125 - Jan E. Gerken 2020
Modular graph forms (MGFs) are a class of non-holomorphic modular forms which naturally appear in the low-energy expansion of closed-string genus-one amplitudes and have generated considerable interest from pure mathematicians. MGFs satisfy numerous non-trivial algebraic- and differential relations which have been studied extensively in the literature and lead to significant simplifications. In this paper, we systematically combine these relations to obtain basis decompositions of all two- and three-point MGFs of total modular weight $w+bar{w}leq12$, starting from just two well-known identities for banana graphs. Furthermore, we study previously known relations in the integral representation of MGFs, leading to a new understanding of holomorphic subgraph reduction as Fay identities of Kronecker--Eisenstein series and opening the door towards decomposing divergent graphs. We provide a computer implementation for the manipulation of MGFs in the form of the $texttt{Mathematica}$ package $texttt{ModularGraphForms}$ which includes the basis decompositions obtained.
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