Holomorphic subgraph reduction of higher-point modular graph forms

Abstract

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 property that they may be reduced to sums of products of modular graph forms of strictly lower loop order. In the particular case of dihedral modular graph forms, a closed form expression for this holomorphic subgraph reduction was obtained previously by D'Hoker and Green. In the current work, we extend these results to trihedral modular graph forms. Doing so involves the identification of a modular covariant regularization scheme for certain conditionally convergent sums over discrete momenta, with some elements of the sum being excluded. The appropriate regularization scheme is identified for any number of exclusions, which in principle allows one to perform holomorphic subgraph reduction of higher-point modular graph forms with arbitrary holomorphic subgraphs. arXiv:1809.05122v2 [hep-th]

All-order differential equations for one-loop closed-string integrals and modular graph forms

“…where our normalization conventions differ from [9,13,[20][21][22], see Footnote 8. In the above expression we have suppressed the redundant delta function from overall momentum conservation.…”

“…In particular, the differential equations (3.25) and (3.27) of the component integrals bypass the need to perform holomorphic subgraph reduction (see Appendix B.4) to all orders in α . This is exemplified by the identities for modular graph forms C A c d B 0 0 in (3.33) and becomes particularly convenient at n ≥ 3 points, where the state-of-the-art methods for holomorphic subgraph reduction [20] are recursive and may generate huge numbers of terms in intermediate steps.…”

“…Similarly to the two-point results outlined in Section 3.4.3, also the three-point Cauchy-Riemann equations imply infinitely many relations between modular graph forms, now also including trihedral topologies. In particular, these identities allow to circumvent the convoluted trihedral holomorphic subgraph reduction [20]. In the example above, when (4.22) is simplified by means of the factorization and momentum-conservation identities spelled out in Appendices B.2 and B.3, one obtains…”

“…Higher-point generalizations of these sums were worked out in [20] and hence higher-point generalizations of (B.11) can be obtained. A lengthy, but closed, formula for trihedral graphs is available in [20].…”

“…More specifically, the right-hand side of our definition (2.53) does not include the factor of e∈E Γ ( Im τ π ) 1 2 (ae+be) from[9,13,20,22] and the factor of e∈E Γ ( Im τ π ) be from[21].…”