## Seminar series on motives and period integrals in Quantum field theory and String theory |

Francis Brown (University of Oxford), Erik Panzer (University of Oxford), Federico Zerbini (University of Oxford), Pierre Vanhove (IPhT CEA-Saclay)

Talks (titles, abstracts and slides) for the years 2017 - 2019 -- 2020 -- 2021 -- 2022 -- 2023 -- 2024 -- next talk

This is the page for the seminar series on motives and period integrals in Quantum field theory and String theory. The seminars are taking place in an hybrid form. The Zoom Meeting ID and the password, will be sent to the participants by email before the semaine via the mailing list

If you are interested in joining this seminar series you should contact the organisers.

pdf de l'exposé

I will review the progress in computing cosmological correlation functions in momentum space. I will focus on the differences between scattering amplitudes in flat space vs cosmological correlation functions. I will then review some of the techniques from scattering amplitudes that have been developed for cosmological correlations and discuss how one can tackle these at loop order for scalar fields. Finally, I will end with some open-ended questions and work in progress on some general properties of such correlators.

pdf de l'exposé --- feuille de calcul

The period matrix of a smooth complex projective variety X encodes the isomorphism between the singular homology of X and its algebraic De Rham cohomology. Numerical approximations with sufficient precision of the entries of this matrix, called periods, allow to recover some algebraic invariants of the varieties, such as the Picard rank in the case of surfaces. I will present a method relying on the computation of an effective description of the homology for obtaining such numerical approximations of the periods of hypersurfaces.

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Generalised Eisenstein series are modular-invariant functions that satisfy a Laplace eigenvalue equation with a source-term that is bilinear in ordinary non-holomorphic Eisenstein series. These functions appear in a variety of contexts related to string theory and N=4 super Yang-Mills theory, where, in particular, they encode non-perturbative information. In this talk I will look at certain resurgent properties these functions have and discuss a construction of spectra that are relevant for physical applications. The non-trivial zeros of the Riemann zeta function also make an unexpected appearance

vidéo de l'exposé

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Finding supersymmetric flux vacua (SFV) in IIB compactifications on Calabi-Yau threefolds necessitates a constraint on the complex structure moduli. The Flux Modularity Conjecture of Kachru, Nally, and Yang states that threefolds with such moduli are weight-two modular. By this, one means that the zeta function of the manifold factorises and the coefficients in a quadratic factor are identified with the Fourier coefficients of a weight-two modular form. We provide evidence for this conjecture by computing the zeta function for a number of multiparameter Calabi-Yau threefolds, with complex structure restricted to a locus that we identify as a solution of the SFV equations. The set of manifolds that we propose to have weight-two modularity is very large, as our SFV solution method works for any threefold whose complex structure moduli spaces possesses a $Z_2$ symmetry that we describe. We discuss the geometric explanation of this modularity, and numerically verify Deligne's conjecture for our examples. Our principal example is a five-parameter geometry that has seen recent study in the context of banana Feynman graphs.

vidéo de l'exposé

To prove that $\zeta(3)$ is irrational, Apéry realized $\zeta(3)$ as the limit of the ratio of two solutions of a certain linear recurrence with polynomial coefficients. Similar recurrences and their Apéry limits have been studied in mirror symmetry. In this talk, I will sketch a method to realize vacuum Feynman integrals as Apéry limits, using a combinatorial graph invariant arXiv:2304.05299. The examples cover all 6-loop periods, which includes fourth order recurrences involving the double zeta value $\zeta(3,5)$. I will explain connections to diagonals of rational functions and point-counts over finite fields. This is work in progress with Francis Brown.

Hybrid talk in the lecture room C1 of the Mathematical Institute of the University of Oxford and on zoom

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The periods of the mirror quintic contain important information such as the numbers of rational curves of a given degree on the quintic. In this talk, we express these periods as integrals of periods of K3 surfaces and use this to prove that mixed periods of the conifold fiber are given by integrals of modular forms. As an application, we get identities for the growth of the number of rational curves. The strategy can be generalized to other Calabi-Yau motives and this is work in progress with Vasily Golyshev and Albrecht Klemm.

I will present a construction of the two-loop amplitude of four graviton supermultiplets in $AdS_5 \times S^5$. Starting from an ansatz for a preamplitude, the full amplitude is generated by the action of a specific Casimir operator (related to the hidden 10-dimensional conformal symmetry). In passing, I will describe a class of four-dimensional "zigzag" integrals which capture the leading logarithmic discontinuity to all loop orders, and also revisit the prescription of the bulk-point limit of AdS amplitudes. From the two-loop result, one can then extract the two-loop anomalous dimensions of twist-four double-trace operators of generic spin, which includes dependence on (alternating) harmonic sums up to weight three. Based on [2204.01829] with James Drummond.

The Coon amplitude is a one-parameter family of 4-point tree-level amplitudes that describes an infinite exchange of massive resonances and exhibits duality symmetry. It is essentially the only generalisation of the famous Veneziano amplitude known to exist, and reduces to the Veneziano amplitude in some limit of the parameter space.

In this talk, I will review the properties of dual model (Veneziano, Coon, in particular) amplitudes, and present its unitarity properties and low-energy expansion, which can be written in terms of q-deformed zeta-values. If time allows, I'll describe the challenges : extension to N-point amplitudes, and finding other classes of similar objects.