Basis decompositions and a Mathematica package for modular graph forms

Abstract

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 + w ̄ ⩽ 12 , 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 toward decomposing divergent graphs. We provide a computer implementation for the manipulation of MGFs in the form of the Mathematica package ModularGraphForms which includes the basis decompositions obtained.

Towards closed strings as single-valued open strings at genus one

“…Both eMZVs [17] and MGFs [11,62,[88][89][90] exhibit a multitude of relations over rational combinations of MZVs, all of which are automatically exposed in their iterated-Eisenstein-integral representation 13 . A computer implementation for the decomposition of a large number of eMZVs and MGFs into basis elements is available in [62,92], respectively.…”

“…21 All examples of α j 1 j 2 k 1 k 2 up to including k 1 + k 2 = 12 were checked to preserve the shuffle relations, and their explicit form can also be found in an ancillary file to the arXiv submission of this work. Note that these checks cover the more intricate cases with (k 1 , k 2 ) = (4, 6) and (k 1 , k 2 ) = (4, 8) where imaginary cusp forms occur among the MGFs [18,62].…”

“…We shall finally exemplify the appearance of cuspidal MGFs from single-valued openstring integrals whose Laurent polynomial at the order of q 0 q0 vanishes. A systematic study of imaginary cusp forms among the two-loop MGFs can be found in [54], also see [62] for examples of real cusp forms. The simplest imaginary cusp forms occur among the lattice sums (2.26) at modular weights (5, 5) whose basis can be chosen 25 to 25 The choice of basis in [18] is tailored to delay the appearance of holomorphic Eisenstein to higher Cauchy-Riemann derivatives as far as possible.…”

“…Moreover, any relation among eMZVs induces a relation among the MGFs through the lattice-sum representation of their single-valued images. Hence, the database of MGF relations [62] can be complemented by applying (5.39) to the eMZV relations on the website [92]. The lattice sums contributing to SV ω(n 1 , .…”

Poisson equation for genus two string invariants: a conjecture

“…Obtaining eigenvalue equations satisfied by these graphs is very useful in performing the integrals over moduli space, along with a knowledge of their asymptotic expansions around various degenerating nodes of the worldsheet. This has yielded various results at genus one [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and two [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] leading to an intricate underlying structure.…”