Non-holomorphic modular forms from zeta generators
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arXiv
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| Autores principales: | , , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912056042586112 |
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| author | Dorigoni, Daniele Doroudiani, Mehregan Drewitt, Joshua Hidding, Martijn Kleinschmidt, Axel Schlotterer, Oliver Schneps, Leila Verbeek, Bram |
| author_facet | Dorigoni, Daniele Doroudiani, Mehregan Drewitt, Joshua Hidding, Martijn Kleinschmidt, Axel Schlotterer, Oliver Schneps, Leila Verbeek, Bram |
| contents | We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL$(2,\mathbb Z)$ known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown's framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14816 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-holomorphic modular forms from zeta generators Dorigoni, Daniele Doroudiani, Mehregan Drewitt, Joshua Hidding, Martijn Kleinschmidt, Axel Schlotterer, Oliver Schneps, Leila Verbeek, Bram High Energy Physics - Theory Algebraic Geometry Number Theory We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL$(2,\mathbb Z)$ known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown's framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes. |
| title | Non-holomorphic modular forms from zeta generators |
| topic | High Energy Physics - Theory Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2403.14816 |