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

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

The low-energy expansion of closed-string scattering amplitudes at genus one introduces infinite families of non-holomorphic modular forms called modular graph forms. Their differential and number-theoretic properties motivated Brown's alternative construction of non-holomorphic modular forms in the recent mathematics literature from so-called equivariant iterated Eisenstein integrals. I will spell out the dictionary between the two formalisms and discuss examples beyond the depth-one case of non-holomorphic Eisenstein series. These examples are strong validations of Brown's conjecture that equivariant iterated Eisenstein integrals contain modular graph forms. Based on analogies with single-valued polylogarithms at genus zero, certain elements of Brown's construction will be made fully explicit to all orders.

Hybrid talk in the lecture room L6 of the Mathematical Institute of the University of Oxford and on zoom

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We consider the four-point correlator of the stress-energy tensor in ${\cal N}=4$ SYM, to leading order in inverse powers of the central charge, but including all order corrections in $1/\lambda$. This corresponds to the AdS version of the Virasoro-Shapiro amplitude to all orders in the small $\alpha'$/low energy expansion. Using dispersion relations in Mellin space, we derive an infinite set of sum rules. These sum rules strongly constrain the form of the amplitude and determine all Wilson coefficients in the low energy expansion in terms of the CFT data for heavy string operators. The assumption that the Wilson coefficients are in the ring of single-valued multiple zeta values, as expected for closed string amplitudes, is surprisingly powerful and leads to a unique solution to the sum rules for the $1/\sqrt{\lambda}$ correction. The corresponding OPE data fully agrees with and extends the results from integrability. The Wilson coefficients to order $1/\sqrt{\lambda}$ can be summed into an expression whose structure of poles and residues generalises that of the Virasoro-Shapiro amplitude in flat space.

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We argue that Yangian-invariant l-loop fishnet integrals in 2 dimensions are closely related to a family of Calabi-Yau l-folds. This allows us to reduce the computation of these integrals to the computation of the Calabi-Yau periods. The periods are solutions of the Picard-Fuchs differential equations, which in turn are determined by the Yangian symmetry. Finally, we show that the values of these fishnet integrals admit a natural interpretation as the quantum volume of the Calabi-Yau.

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We study the analytic structure of late-time cosmological correlations functions in rigid de Sitter space time. It was recently pointed out that the cosmological correlators can be mapped to Euclidean anti de Sitter correlators, therefore reducing the computation to correlator in EAdS. For the case of conformally coupled scalars we show that the evaluation of the AdS Witten diagrams can be reintepreted as flat-space Feynman integrals. This simplifies drastically the perturbative computations, and allow to analytical evaluations of the correlator in perturbation. From these analytical evaluations, we extract loop corrections to the anomalous dimensions, and get some insights into the analytic structure of conformal correlators in rigid AdS and dS.

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I will present Poincaré series representations for infinite families of modular forms including depth-two non-holomorphic modular graph forms as they appear in the low-energy expansion of closed-string scattering amplitudes at genus one. These Poincaré series are constructed from iterated integrals over holomorphic Eisenstein series and their complex conjugates, decorated by suitable combinations of zeta values. The space of modular forms thus obtained is an extension of the world of modular graph forms, and it does also contain iterated integrals of holomorphic cusp forms and their L-values.

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Mirror symmetry predicts the "classical" algebraic and transcendental invariants of degenerate Calabi-Yau threefold to match with the symplectic "quantum" invariants of the mirror manifold. An instance of this correspondence arises in the context of open-string mirror symmetry, in which the Abel-Jacobi map of families of algebraic cycles corresponds to instanton-corrected invariants of Lagrangian submanifolds of the mirror manifold. Using a technique introduced by Griffiths, Green and Kerr, we calculate certain values of the Abel-Jacobi map for specific algebraic cycles in a suitable degeneration limit of the Calabi-Yau threefold, recovering previously obtained numerical results. We interpret the calculated invariants in the context of open-string mirror symmetry. This talk is based on work in progress with Dave Morrison and Johannes Walcher.

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Does our world respect causality at all energy scales? We explore constraints on gravitational dynamics which stem from this assumption. Parameterizing long-distance effects of possible new heavy particles using effective field theory (EFT), we investigate causality of 2 to 2 scattering processes. Due to its growth with energy, the gravitational force turns out to be particularly constrained. I will present two-sided bounds which show that a wide class of modifications to four-dimensional Einstein's gravity, require either the existence of light higher-spin states, or violation of causality as we understand it.

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This talk will present the exact expression for an integrated correlator of four BPS operators in ${\cal N}=4$ supersymmetric Yang-Mills with any classical gauge group $G_N$. I will show that this can be expressed as a two-dimensional lattice sum, which is a function of $N$ and of the complex Yang--Mills coupling, $\tau=\theta/2\pi + i 4\pi/g_{_{YM}}^2,$. This expression is manifestly invariant under $SL(2,Z)$ duality in the case of simply-laced gauge groups ($SU(N)$ and $SO(2N)$), and $\Gamma_0(2)$ in the case of non-simply laced gauge groups ($SO(2N+1)$ and $USp(2N)$). Furthermore, it satisfies a striking `Laplace difference' equation that relates the correlator for gauge group $G_N$ to those for gauge groups $G_{N-1}$ and $G_{N+1}$. For any $G_N$ the correlator can be expressed as an infinite rational sum of non-holomorphic Eisenstein series with integer indices, and can be expanded to arbitrary order in perturbation theory, together with an infinite series of instanton terms. This reproduces and extends perturbative and non-perturbative ${\cal N}=4$ SYM results. In the $N\to \infty$ limit the expansion in powers of $1/N$ has coefficients that are rational sums of non-holomorphic Eisenstein series with half-integer indices and is holographically related to the low energy expansion of the type IIB superstring four-graviton amplitude in an orientifold of $AdS_5\times S^5$ if $G_N=SU(N)$, or in $AdS_5\times S^5/Z_2$ if $G_N=SO(2N)$, $SO(2N+1)$ or $USp(2N)$.

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Iterated integrals encode important information about fundamental groups, and are a rich source of periods of mixed motives. Such integrals are often encountered in the computation of Feynman amplitudes. Motivated by string theory computations, I will focus on the case of Riemann surfaces, reviewing the known genus-zero and genus-one case, which give rise to classical and elliptic polylogarithms, and reporting on recent progress at higher genus in collaboration with Benjamin Enriquez. More specifically, we have constructed explicit flat connections, conjugated to higher-genus analogues of the KZ-connection, which give rise to spaces of multi-valued functions that may be viewed as higher-genus analogues of polylogarithms.