YOUNGST@RS - Loops & Geometry: Hidden Structures in Multi-loop Amplitudes

20 Apr 2026, 09:00 → 22 Apr 2026, 17:30 Europe/Berlin

Mainz Institute for Theoretical Physics, Johannes Gutenberg University

The “Loops & Geometry: Hidden Structures in Multi-loop Amplitudes” workshop will bring together mathematicians and theoretical physicists with expertise in algebraic geometry and multi-loop scattering amplitudes computation, with the goal of exploring recent developments in advanced mathematical techniques that clarify the common structure underlying  particle physics, string theory and gravitational observables . 

By fostering dialogue across communities, the workshop will provide both a survey of recent advances and a platform for presenting new ideas.

The workshop will be focused on the following three main topics: 

·             Amplitudes structure and algebraic periods

·             Intersection Theory and String Amplitudes

·             Gravitational waves and Differential equations

Three advanced lectures on Calabi-Yau periods, String Amplitudes and D-Modules, are scheduled to serve as a conceptual bridge between the expert overview and the participants’ talks.

Monday 20 April

Welcome: Carlos Tamarit

Geometric, algebraic and analytic structures in scattering amplitudes: Johannes Broedel

Abstract: Calculating an observable in physics usually boils down to answering the question: which class of special functions is suitable to accomodate the symmetries and describe the particular solution to the problem? In this talk I am asking the other way around: what does it take to define a system of special functions?
I will explore several types of differential equations with different boundary conditions, show how they (and their solutions) can be related and describe, how they are connected to the most basic Knizhnik-Zamolodchikov system. I will briefly comment on how to find functional relations for particular classes of solutions and thereby touch upon geometric approaches to modelling the spaces of functional relations.

Single-valued polylogarithms for all genera: Konstantin Baune

Polylogarithms on Riemann surfaces of different genera have been proven
useful in many QFT and string amplitude calculations. Furthermore, the notion of singlevalued
functions without monodromies on the underlying surface can be of importance for
some of these computations.
In this talk, I will construct single-valued polylogarithms on Riemann surfaces of arbitrary
genus.
I will start by reviewing Brown’s construction of single-valued polylogarithms on the
punctured Riemann sphere and then generalize this formalism to define single-valued
elliptic and higher-genus polylogarithms.
This talk is based on joint work with Johannes Broedel and Yannis Moeckli.

10:45 Break

Computational Algebraic Geometry for Feynman Integrals and Their Singularities: Claudia Fevola

The study of singularities of Feynman integrals is a classical problem in scattering amplitudes. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram.
I will illustrate several examples using the package PLD.jl. By means of numerical nonlinear algebra methods, PLD.jl makes it possible to compute components of the Landau singular locus that were previously out of reach. I will also briefly discuss the singularities of an Euler integral in the case of a hyperplane arrangement, which can be applied to the computation of singularities of cosmological correlators. This work is based on joint work with Sebastian Mizera and Simon Telen, and with Saiei Matsubara-Heo.

Feynman Integrals, Calabi-Yau Manifolds, and Abelian Curves: Pyry Kuusela

In this talk, I first give a brief overview of the connection between Feynman
integrals and Calabi-Yau geometries. Then I present a novel construction of a family of
Abelian curves whose periods are associated to Feynman integrals. To find such curves,
we first use the connection between Feynman integrals and Calabi-Yau manifolds to
express the integrals in terms of Calabi-Yau periods. Natural candidates for Abelian curves
encoding these periods are then given by intermediate Jacobians, which are well-studied
tori that can be associated to any Calabi-Yau manifolds. The classical (Griffiths and Weil)
Jacobians are either Abelian or vary holomorphically, but not both, and thus do not provide
a simple relation between Feynman integrals and periods. This motivates us to construct a
novel type of intermediate Jacobian with both of the desirable properties. The price one
has to pay is that the Jacobian is not defined everywhere in the moduli space of the
Calabi-Yau manifold. However, this restriction is very natural from the amplitudes
perspective: the moduli of the Calabi-Yau manifold are already restricted by conditions
such as that the masses appearing in the corresponding Feynman integrals are real.
This talk is based on joint work with Jockers, Kotlewski, McLeod, Pögel, Sarve, Wang, and
Weinzierl.

ε-Factorisation for Feynman Integrals in Higher Genus: Iris Bree

In this talk, we describe how a recently proposed systematic procedure for
obtaining ε-factorised differential equations for Feynman integrals can be applied beyond
the polylogarithmic setting. The method is developed independently of the underlying
geometry of the integral family. We illustrate how it works in practice in examples
associated with elliptic curves, and then show how the same strategy extends to
hyperelliptic cases of higher genus.

12:30 Lunch Break

Calabi Yau periods in string compactifications and beyond: Damian van de Heisteeg

In this review I discuss techniques for computing Calabi-Yau period integrals in compactifications of string theory. I highlight how these periods encode physical data in string compactifications and comment on possible applications to scattering amplitudes.

Numerical computation of periods: Eic Pichon-Pharabod

The period matrix of a smooth complex projective variety encodes the
isomorphism between its singular homology and its algebraic De Rham cohomology.
Numerical approximations with sufficient precision of the entries of the period matrix allow
to recover some algebraic invariants of the variety. Such approximations can be obtained
from an effective description of the homology of the variety, which itself can be obtained
from the monodromy representation associated to a generic fibration. I will describe
methods to compute periods to several hundreds digits of certified precision, and
showcase implementations and applications, in particular to computation of the Picard
rank of certain K3 surfaces appearing in Feynman integrals.

15:30 Break

Appearances of Calabi-Yau periods in classical gravitational observables: Benjamin Sauer

Classical gravitational observables of a binary black hole system play a key role
in gravitational waves physics and gravitational wave modeling. The new generation of
gravitational waves detectors demand high precision calculations of these observables. It
is remarkable that these calculations are also the first instance in which periods over
Calabi-Yau 3-folds contribute to an analytic result of a physical observable. I will discuss
the structure of the computation of the scattering angle and radiated energy up to fifth
Post-Minkowskian order (4-loop order) and in which way Calabi-Yau periods play a role in
the computation and analytic expression of these observables.

New algorithms for Feynman integral reduction and epsilon-factorised differential equations: Prof. Stefan Weinzierl

Precision calculations in quantum field theory rely very often on perturbation
theory and thus on the computation of Feynman integrals. Two of the basic algorithms are
integration-by-parts and the method of differential equations. In this talk I show how the
efficiency of these algorithms can be improved by taking geometric information of the
Feynman integrals into account. The method does not require any prior knowledge of the
geometry and merely amounts to counting poles and residues.

Tuesday 21 April

Intersection Theory and String Amplitudes: Prof. Oliver Schlotterer

Abstract: This talk will start by recalling basics of string scattering amplitudes and the twisted (co-)homologies at the heart of intersection theory. I will then review the Kawai-Lewellen-Tye (KLT) relations between closed- and open-string tree amplitudes in flat background in the light of twisted intersection numbers. Recent progress on string amplitudes in AdS spacetime can be elegantly understood from a non-commutative version of intersection theory. I will finally discuss different flavors of loop-level KLT relations, namely those of so-called Riemann Wirtinger integrals as mathematical laboratories and those of loop integrands of closed-string one-loop amplitudes.

Improving Integration-by-parts: Sid Smith

Abstract: Integration-by-parts (IBP) reduction is a central tool in the computation of scattering amplitudes, which provide the essential link between theoretical predictions and experimental observations in particle physics and gravitational-wave science. In perturbative quantum field theory, scattering amplitudes are expressed as sums of large numbers of highly non-trivial Feynman integrals. IBP identities enable the systematic reduction of these integrals to a finite and minimal basis of so-called master integrals, thereby drastically simplifying the problem. Despite its conceptual simplicity, IBP reduction remains a major computational bottleneck in modern amplitude calculations, primarily due to the enormous systems of equations that must be solved. In this talk, I will review contemporary approaches to IBP reduction, highlighting recent algorithmic and computational advances. I will conclude by presenting a new code we have developed that leverages these modern techniques in an attempt to improve the efficiency of IBP reduction.

10:45 Break

Magic relations and critical varieties: Hjalte Frellesvig

Abstract: Feynman integrals are related through linear relations, often called Integration-By-Parts relations, or IBPs for short.
Usually IBPs relate integrals in a generating sector, with integrals in its subsectors. It can, however, happen that the integrals in the generating sector drop out, and the IBP relates subsector integrals only, and that type of IBPs are known as magic relations. Magic relations are of interest, since they cause problems for cut-based approaches to IBP reduction, as well as to intersection theory in the framework of relative twisted cohomology.
The main result of this presentation, and of the paper on which it is based, is the discovery that magic relations always come together with a generating sector that has a higher dimensional critical variety.
We also outline a proof of this result, discuss how to count master integrals in the presence of higher dimensional critical varieties (using Morse-Bott theory), and discuss how magic relations interact with symmetry relations.

The spectrum of Feynman integral geometries at two loops: Florian Raphael Seefeld

In this work, we provide a complete classification of the Feynman-integral geometries at two-loop order in four-dimensional Quantum Field Theory with standard quadratic propagators. We do this by considering a finite basis of integrals in the 't Hooft-Veltman scheme and analysing the leading singularities of the integrals for generic kinematic values through the loop-by-loop Baikov representation. Aside from the Riemann sphere, we find elliptic curves, hyperelliptic curves of genus 2 and 3 as well as K3 surfaces, and in addition a smooth and non-degenerate Del Pezzo surface of degree 2, resulting in a curve of geometric genus 3. These geometries determine the space of functions relevant for Quantum Field Theories at two-loop order, including in the Standard Model.

Intersection theory and symmetry relations for Feynman integrals: Sara Maggio

Intersection theory for twisted (co-)homology has become a powerful tool in the
study of Feynman integrals. First introduced as an alternative approach to integral
decomposition, it has since developed into a versatile framework with many applications.
In this talk, I will explain how twisted (co-)homology can be used to encode symmetry
relations among Feynman integrals. I will then show how this framework leads to a formula for the number of master integrals in the presence of symmetries.

12:30 Lunch Break

An introduction to tree-level string scattering amplitudes: Chrysoula Markou

In this short introductory lecture, we will review the rudiments of the calculation
of tree-level string scattering amplitudes by means of the generating functional method.
This entails first translating external states’ vertex operators into suitable differential
operators acting on exponential factors. Calculating then a given tree-level amplitude
amounts to acting on a “generalised” Koba-Nielsen factor with its external states’
differential operators and computing the resulting worldsheet integral.

Superstring amplitudes from the field-theory limit: an n-point map at one loop: Lecheng Ren

Are perturbative superstring amplitudes for massless external states just an $ \alpha'$ dressing of super-Yang-Mills/supergravity? This is the case at tree level, where
the worldsheet correlators at $n$ points can be written in a natural worldsheet basis, such
that the kinematic coefficients are BCJ numerators of super-Yang-Mills/supergravity
amplitudes, with the non-trivial $\alpha'$ dependence carried only by the Koba-Nielsen
factor. Motivated by this construction, we present for the first time a complete worldsheet
basis of one-loop superstring correlators at $n$ points. All the kinematic coefficients
associated to non-cusp basis elements are identified with pieces of one-loop BCJ
numerators of super-Yang-Mills/supergravity. This determines the superstring correlators
up to 15 points in terms of field theory. Starting at 16 points (modular weight 12), the
worldsheet basis may include cusp forms, which vanish in the field-theory degeneration,
such that the associated coefficients cannot be fixed in this manner. Therefore, the oneloop
answer to our initial question is determined, at high multiplicity, by whether the
coefficients of cusp basis elements vanish or not. As a by-product of our construction, we
present new constraints on the field-theory limit that result from string modularity. These
are expressed as additional relations among one-loop BCJ numerators in maximal super-
Yang-Mills/supergravity starting at 6 points.

15:30 Break

One-Loop string amplitudes exactly in \alpha’: Marco Baccianti

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, and illustrate new
computational windows it opens.
Based on https://arxiv.org/abs/2501.13827, https://arxiv.org/pdf/2507.22105 and https://
arxiv.org/pdf/2601.09707.

String Amplitudes and Twisted Co(homology): Prof. Stephan Stieberger

Wednesday 22 April

From Loops to Gravitational Waves: A Journey through the Integral Landscape: Prof. Radu Roiban

The post-Minkowskian expansion of gravitational dynamics provides a rich arena in which quantum field theory and multi-loop integration techniques meet gravitational-wave physics. In this talk, I will survey some of the underpinnings of state-of-the-art calculations of both conservative dynamics and gravitational-wave emission. A central theme will be the mathematical structures and special functions that arise at various orders in Newton's constant. We will also discuss unexpected features, including nontrivial cancellations of special functions and the interplay between the amplitudes-based and multipolar-post-Minkowskian approaches to gravitational radiation.

Analytic structure of the Black Hole S-matrix: Anna M. Wolz

The analytic structure of the black hole S-matrix encodes information about the
response of a black hole to external perturbations and is required as input for S-matrix
bootstrap applications. In this talk, I will describe how to define an IR-finite S-matrix for the
scattering of a classical wave off a Schwarzschild black hole background. To understand
its analytic structure, I will first present a proof of analyticity, unitarity, and reflection
symmetry for wave scattering off generic backgrounds, based on properties of the
background potential. I will then apply this result to the Schwarzschild black hole, deriving
regions of analyticity of the S-matrix and validating them with both perturbative results from
black hole perturbation theory and examples in exactly solvable regimes. I will show that
the reflection amplitude of the S-matrix has a branch cut in the upper-half frequency plane
that is consistent with causality, analyticity, unitarity, and reflection symmetry.

10:45 Break

Magnusian Generating Function of Observables: Trevor Schoepner

Abstract: The exponential representation of the S-matrix behaves in a relatively simple way under the classical limit, allowing a classical function on phase space we call the Magnusian’’ to appear and act as a generating function of all classical observables, thus defining aclassical S-matrix’’ through the exponentiated action of the Poisson bracket. The Magnusian is naturally an in-in quantity, distinct from but related to the in-out on-shell action, and is a generalization of a notion of the eikonal that has been gaining popularity in amplitudes literature. It is naturally computed by a diagrammatic ``Magnus series’’ with well-understood iepsilons. This diagrammatic expansion naturally pre-incorporates all physical implications of unitarity, so that perturbative results at a fixed order are exactly unitary, rather than only being so up to higher order corrections. The Magnusian can be defined even outside of a scattering context, and in a general setting it provides the canonically-compatible way to time-average degrees of freedom within the Hamiltonian.

Pushing the Post-Newtonian Frontier: Six Loops and Beyond: Giacomo Brunello

The detection of gravitational waves from coalescing binary systems requires increasingly accurate theoretical description of the gravitational dynamics. In the inspiral regime, the conservative dynamics can be described within an effective field theory framework, where the two-body interactions are treated perturbatively in the post-Newtonian expansion and reformulated in terms of Feynman diagrams.
One of the main challenges in determining higher-order corrections is the appearance of multi-loop
massless two-point Feynman integrals.
In this talk, I will discuss recent developments leading to the determination of static gravitational interactions
at sixth and seventh post-Newtonian order, where six- and seven-loop Feynman integrals arise.
Multi-loop techniques for integral reduction and evaluation, including integration-by-parts identities
and numerical methods, play a central role in obtaining these results.
These developments pave the way toward the completion of the conservative dynamics at these orders and beyond.

Waveforms at Infinity and at the Horizon: Carlo Heissenberg

Abstract: The direct detection of gravitational waves has put the relativistic
two-body problem in the spotlight and stimulated progress in
perturbative approaches that provide analytic insight into its dynamics.
Two strategies that have been witnessing interesting developments to
this end are the ones based on scattering amplitudes, which apply to
binary scatterings at large impact parameter, and on black-hole
perturbation theory, which applies to extreme-mass-ratio binaries. In
this talk, I will discuss how these two methods play complementary
roles, and how they naturally feature differential equations that
determine the gravitational perturbations induced by scatterings of
black holes both far away, at null infinity, and close to the event
horizons. Such waveforms characterize in particular the amounts of
energy and angular momentum that are emitted and reabsorbed by the system.

12:30 Lunch Break

Introduction to D-modules: Kostiantyn Tolmachov

Theory of D-modules is the (algebro-)geometric theory of linear partial
differential equations. Developed in the last quarter of the 20th century, it found many
(sometimes very unexpected) applications not only in the theory of differential equations,
but also in representation theory, algebraic geometry and algebraic topology. I will give a
very brief introduction to the theory of D-modules, with the view towards their use in the
study of Gelfand-Kapranov-Zelevinsky systems.

D modules and singular expansions of Feynman integrals: Seva Chestnov

Linear differential operators play a prominent role in the study of Feynman
integrals and related special functions. In this talk, I will discuss two complementary
aspects of this framework. First, I will describe an algorithm for constructing differential
operators that annihilate Feynman integrals and for deriving the associated Pfaffian
systems of first-order linear differential equations. Second, I will explain how singular limits
of Pfaffian systems can be studied systematically through restriction theory, leading to
asymptotic expansions near singular limits. As an illustration, I will show how these ideas
apply to the post-Newtonian expansion of gravitational waveforms.

15:30 Break

D modules and integral reduction with CALICO: Gaia Fontana

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 and modern applications.

Fontana_LoopsGeometry.pdf

Hidden Structures in Feynman Amplitudes: Complexity Theory and GKZ Reductions: Prof. Thomas Grimm