Axel Kleinschmidt:
Titel: Single-valued periods in string theory
Abstract: I will review the occurrence of single-valued periods in string perturbation theory starting from genus zero and moving on to higher genus. The special role of certain algebraic structures and the relation between open and closed string theory will be emphasised. Time permitting, I will also touch upon the (conjectural) appearance of single-valued structures in AdS amplitudes.
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Aurélie Sangaré:
Title: On Single-Valued Periods in String Theory: From Flat Space to AdS
Abstract: Multiple zeta values (MZVs) - the periods of the moduli space of punctured genus-zero surfaces - and their single-valued sub-class (SVMZVs) appear in the low-energy expansion of tree-level string amplitudes in flat space. In this talk, we discuss the emergence of MZVs and SVMZVs in tree-level string amplitudes in AdS and its consequences for the structure of tree-level string theory in this space. First, we briefly review the relevant structure in flat space, where the low-energy expansion of closed-string tree amplitudes casts the latter as single-valued open-string tree amplitudes. In particular, we discuss the related KLT and monodromy relations of tree amplitudes. Then, we explain how imposing the low-energy transcendental structure of open- and closed-string tree-level amplitudes in AdS - which are currently still elusive - allows to establish an expression for the building blocks of these amplitudes, while simultaneously endowing them with a mathematical structure which manifestly mimics the flat-space structure introduced earlier in the talk.
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Yi-Xiao Tao: Elliptic modular graph forms, equivariant iterated integrals and single-valued elliptic polylogarithms
Abstract: 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 $\tau$ 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 $\tau$, 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 $\tau$-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.
This talk is based on a joint work 2511.15883 with Oliver Schlotterer and Yoann Sohnle.
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Marco Baccianti:
Title: Rademacher evaluations of modular integrals
Abstract: String theory provides us with UV-finite amplitudes of quantum gravity at every order in perturbation theory.
However, explicit computations become quickly very complicated, to the point that their evaluation have been possible only in the low- and high- energy expansion. Essentially no results are known at intermediate values of \alpha’.
In this talk, I will present a novel technique to evaluate one-loop amplitudes at finite \alpha’, which also implements the i \epsilon prescription in string theory.
Such technology opens a window for computations that were previously inaccessible.
Based on https://arxiv.org/abs/2501.13827, https://arxiv.org/pdf/2507.22105 and other unpublished results.
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Emiel Claasen:
Title: Single-valued periods in one-loop superstring amplitudes
Abstract: The low-energy expansion of closed-string genus-one amplitudes introduces non-holomorphic modular forms for SL(2,Z) known as modular graph forms (MGFs). By relating MGFs to single-valued iterated Eisenstein integrals, they can be interpreted as families of single-valued periods parametrized by the complex structure of the torus. The physical amplitudes are obtained by integrating these families over the moduli space of genus-one Riemann surfaces. At four points, the resulting coefficients involve single-valued multiple zeta values, logarithmic derivatives of zeta values, and the Euler–Mascheroni constant, suggesting the possible appearance of structures beyond the familiar ring of periods. At five points, we find an additional constant whose nature is presently unknown.
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Carlos Rodriguez:
Title: Hypergeometric Integrals on a Punctured Riemann Surface
Abstract: We study integrals on a punctured Riemann surface of
genus g > 1. The twisted cohomology groups associated to
these integrals were studied by Watanabe. Here we study
the corresponding twisted homology groups of these inte-
grals, and bilinear pairings among homology and cohomol-
ogy groups. This is joint work with Andrzej Pokraka and
Lecheng Ren.
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Stephan Stieberger: Periods in Low- and High-Energy String Expansions
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Simone Hu
Title: Cocycles for graph complexes via Feynman integrals
Abstract: In this talk we will introduce Kontsevich's commutative graph complexes and describe a program to detect its non-trivial (co)homology classes via associating convergent Feynman integrals to combinatorial graphs. In particular, we study closed differential forms constructed using the Pfaffian of skew-symmetric matrices and whose integrals give rise to cocycles in the odd graph complex, two of which we can show are non-trivial. More generally, these integrals also provide interesting families of periods. This is in part based on joint work with Francis Brown and Erik Panzer.
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Clement Dupont:
Title: Single-valued integration, a cohomological viewpoint
Abstract: I will explain the cohomological viewpoint on integration and single-valued integration, and how it explains several classical constructions and relations, such as double copy formulas. This is based on joint work with Francis Brown.
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Franca Lippert: Generating non-holomorphic modular forms from the canonical differential equation
Abstract: It is known how to construct modular equivariant iterated integrals of Eisenstein series of arbitrary depth for the full modular group. For congruence subgroups, the situation is different. In this talk, I will review the construction of depth one integrals for subgroups and show an example how to generate equivariant integrals of arbitrary depth - and from that non-holomorphic modular forms - from a given differential equation in canonical form.
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Ulysse Mounoud:
Title: The Motivic Coaction of One-loop Feynman Integrals
Abstract: A key insight of the motivic approach is that Feynman integrals naturally carry a coaction. What is really remarkable is that this coaction seems to admit a diagrammatic description. In this talk, I focus on convergent one-loop integrals with Euclidean kinematics, non-vanishing masses, and even spacetime dimension. Using a suitable compactification of the momentum space, I show how to derive an explicit, diagrammatic coaction formula for an integral $I_G$ associated with a graph $G$. The formula expresses $\Delta I_G$ in terms of integrals of pinches of $G$ and de Rham periods of cuts of $G$.
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Claude Duhr:
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Gaia Fontana:
Title: Parametric annihilators and differential equations for twisted integrals
Abstract: We elaborate on the method of parametric annihilators for deriving relations among integrals. Annihilators are differential operators that annihilate multi-valued integration kernels appearing in suitable integral representations of special functions and Feynman integrals. We describe a method for computing parametric annihilators based on efficient linear solvers and show how to use them to derive relations between a wide class of special functions. These include hypergeometric functions, Feynman integrals relevant to high-energy physics and duals of Feynman integrals. We finally present the public Mathematica package CALICO for computing parametric annihilators and its usage in several examples.
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Cathrin Semper:
Title: Symmetry Invariant Twisted Cohomology
Abstract: It is well known that, in the absence of symmetries, the dimension of a twisted cohomology group—and therefore the number of master integrals—is given by the Euler characteristic. However, this relation changes in the presence of symmetries. Such symmetries typically arise when differentials belonging to different cohomology classes integrate to the same value over the Feynman contour; hence the cohomology group is, a priori, not sensitive to them.
We show that in this situation the number of master integrals is given by the Euler characteristic of the orbit space arising from the action of the symmetry group. This corresponds to the dimension of the invariant subspace of the cohomology group.
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Felix Forner:
Title: Calabi-Yau Periods in QED
Abstract: It is well-known that transcendental numbers associated with Calabi-Yau varieties appear in the (g-2) of the electron in QED beyond one-loop order. In this talk, I will explore the related problem of computing the self-energies in QED at higher loop orders, involving periods of Calabi-Yau varieties of increasing dimension. In particular, I will discuss the role of the latter in the context of canonical differential equations for the self-energies up to four loops. Here, I will focus mainly on the photon self-energy at three loops and partial results for the electron self-energy at four loops.
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Pouria Mazloumi:
Title: Geometry of Feynman Integrals: A (slightly) New perspectives
Abstract: In this talk, I will introduce and explain recent developments in identifying and understanding the twisted cohomology associated with Feynman integrals. This includes an attempt to construct the mixed Hodge structure on the underlying geometry of a given Feynman integral, such as elliptic curves and Calabi–Yau varieties. I will also discuss how the periods of these geometries and how they appear naturally in this new method.