# Modular and Holomorphic Graph Functions from Superstring Amplitudes

@article{Zerbini2018ModularAH, title={Modular and Holomorphic Graph Functions from Superstring Amplitudes}, author={Federico Zerbini}, journal={Texts \& Monographs in Symbolic Computation}, year={2018} }

We compare two classes of functions arising from genus-one superstring amplitudes: modular and holomorphic graph functions. We focus on their analytic properties, we recall the known asymptotic behaviour of modular graph functions and we refine the formula for the asymptotic behaviour of holomorphic graph functions. Moreover, we give new evidence of a conjecture appeared in [4] which relates these two asymptotic expansions.

## 27 Citations

### Poincaré series for modular graph forms at depth two. Part I. Seeds and Laplace systems

- MathematicsJournal of High Energy Physics
- 2022

We derive new Poincaré-series representations for infinite families of non-holomorphic modular invariant functions that include modular graph forms as they appear in the low-energy expansion of…

### Eigenvalue equation for genus two modular graphs

- MathematicsJournal of High Energy Physics
- 2019

A bstractWe obtain a second order differential equation on moduli space satisfied by certain modular graph functions at genus two, each of which has two links. This eigenvalue equation is obtained by…

### Holomorphic subgraph reduction of higher-valence modular graph forms

- Mathematics
- 2018

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 subgraph reduction of higher-point modular graph forms

- MathematicsJournal of High Energy Physics
- 2019

A bstractModular 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…

### Poincaré series for modular graph forms at depth two. Part II. Iterated integrals of cusp forms

- MathematicsJournal of High Energy Physics
- 2022

We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré series in a companion paper. The source term…

### Poincaré series for modular graph forms at depth two. Part II. Iterated integrals of cusp forms

- MathematicsJournal of High Energy Physics
- 2022

We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré series in a companion paper. The source term…

### Integrating three-loop modular graph functions and transcendentality of string amplitudes

- MathematicsJournal of High Energy Physics
- 2022

Modular graph functions (MGFs) are SL(2, ℤ)-invariant functions on the Poincaré upper half-plane associated with Feynman graphs of a conformal scalar field on a torus. The low-energy expansion of…

### Building blocks of closed and open string amplitudes

- MathematicsProceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)
- 2022

In this text we review various relations between building blocks of closed and open string amplitudes at tree-level and genus one. We explain that KLT relations between tree-level closed and open…

### Symmetries in A-type little string theories. Part II. Eisenstein series and generating functions of multiple divisor sums

- MathematicsJournal of High Energy Physics
- 2020

We continue our study of symmetries of a class of little string theories of A-type, which are engineered by N parallel M5-branes probing a flat transverse space. Extending the analysis of the…

### Little String Instanton Partition Functions and Scalar Propagators

- Mathematics
- 2022

We discuss a class of Little String Theories (LSTs) whose low energy descriptions are supersymmetric gauge theories on the Ω-background with gauge group U ( N ) and matter in the adjoint…

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