Elliptic modular graph forms, equivariant iterated integrals and single-valued elliptic polylogarithms
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arXiv
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| Format: | Preprint |
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2025
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| author | Schlotterer, Oliver Sohnle, Yoann Tao, Yi-Xiao |
| author_facet | Schlotterer, Oliver Sohnle, Yoann Tao, Yi-Xiao |
| contents | The low-energy expansion of genus-one string amplitudes produces infinite families of non-holomorphic modular forms after each step of integrating over a point on the torus worldsheet which are known as elliptic modular graph forms (eMGFs). We solve the differential equations of eMGFs depending on a single point $z$ and the modular parameter $τ$ via iterated integrals over holomorphic modular forms which individually transform inhomogeneously under ${\rm SL}_2(\mathbb Z)$. Suitable generating series of these iterated integrals over $τ$, their complex conjugates and single-valued multiple zeta values (svMZVs) are combined to attain equivariant transformations under ${\rm SL}_2(\mathbb Z)$ such that their components are modular forms.
Our generating series of equivariant iterated integrals for eMGFs is related to elliptic multiple polylogarithms (eMPLs) through a gauge transform of the flat Calaque-Enriquez-Etingof connection. By converting iterated $τ$-integrals to iterated integrals over points on a torus, we arrive at an explicit construction of single-valued eMPLs where all the monodromies in the points cancel. Each single-valued eMPL depending on a single point $z$ is found to be a finite combination of meromorphic eMPLs, their complex conjugates, svMZVs and equivariant iterated Eisenstein integrals. Our generating series determines the latter two admixtures via so-called zeta generators and Tsunogai derivations which act on the two generators $x$, $y$ of a free Lie algebra and where the coefficients of words in $x,y$ define the single-valued eMPLs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15883 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elliptic modular graph forms, equivariant iterated integrals and single-valued elliptic polylogarithms Schlotterer, Oliver Sohnle, Yoann Tao, Yi-Xiao High Energy Physics - Theory Algebraic Geometry Number Theory The low-energy expansion of genus-one string amplitudes produces infinite families of non-holomorphic modular forms after each step of integrating over a point on the torus worldsheet which are known as elliptic modular graph forms (eMGFs). We solve the differential equations of eMGFs depending on a single point $z$ and the modular parameter $τ$ via iterated integrals over holomorphic modular forms which individually transform inhomogeneously under ${\rm SL}_2(\mathbb Z)$. Suitable generating series of these iterated integrals over $τ$, their complex conjugates and single-valued multiple zeta values (svMZVs) are combined to attain equivariant transformations under ${\rm SL}_2(\mathbb Z)$ such that their components are modular forms. Our generating series of equivariant iterated integrals for eMGFs is related to elliptic multiple polylogarithms (eMPLs) through a gauge transform of the flat Calaque-Enriquez-Etingof connection. By converting iterated $τ$-integrals to iterated integrals over points on a torus, we arrive at an explicit construction of single-valued eMPLs where all the monodromies in the points cancel. Each single-valued eMPL depending on a single point $z$ is found to be a finite combination of meromorphic eMPLs, their complex conjugates, svMZVs and equivariant iterated Eisenstein integrals. Our generating series determines the latter two admixtures via so-called zeta generators and Tsunogai derivations which act on the two generators $x$, $y$ of a free Lie algebra and where the coefficients of words in $x,y$ define the single-valued eMPLs. |
| title | Elliptic modular graph forms, equivariant iterated integrals and single-valued elliptic polylogarithms |
| topic | High Energy Physics - Theory Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2511.15883 |