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

Federico Zerbini (IPhT CEA-Saclay & IRMA, Strasbourg), Pierre Vanhove (IPhT CEA-Saclay & CERN)

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 compactifications of type II string theory on a Calabi-Yau 3-fold where there is an extremal black hole in the remaining four dimensions. As noted by Moore (hep-th/9807087), such compactifications are often interesting from an arithmetic geometry point of view. I will explain how one might use techniques from arithmetic geometry (namely, the zeta-function of a variety) to find certain "rank two attractors" where the Hodge structure of the underlying CY variety splits. I will also discuss some interesting consequences of such a splitting. This is the result of work produced in collaboration with Philip Candelas, Xenia de la Ossa and Duco van Straten (1912.06146). The periods of the holomorphic 3-form of the underlying CY 3-fold will play a central role in our study and these same periods also appear as sunset/banana Feynman integrals.

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We provide a summary of concepts from Calabi-Yau motives relevant to the computation of multi-loop Feynman integrals. From this we derive several consequences for multi-loop integrals in general, and we illustrate them on the example of multi-loop banana integrals. For example, we show how Griffiths transversality, known from the theory of variation of mixed Hodge structures, leads quite generically to a set of quadratic relations among maximal cut integrals associated to Calabi-Yau motives. These quadratic relations then naturally lead to a compact expression for $l$-loop banana integrals in $D=2$ dimensions in terms of an integral over a period of a Calabi-Yau $(l-1)$-fold. This new integral representation generalizes in a natural way the known representations for $l\le 3$ involving logarithms with square root arguments and iterated integrals of Eisenstein series. We show how the results can be extended to dimensional regularization. We present a method to obtain the differential equations for banana integrals with an arbitrary number of loops in dimensional regularization without the need to solve integration-by-parts relations. We also present a compact formula for the leading asymptotics of banana integrals with an arbitrary number of loops in the large momentum limit that generalizes the $\widehat{\Gamma}$-class formalism to dimensional regularization and provides a convenient boundary condition to solve the differential equations for the banana integrals. As an application, we present for the first time numerical results for equal-mass banana integrals with up to four loops and up to second order in the dimensional regulator.

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A deep idea, going back to Grothendieck, is that there should be a conceptual framework for studying periods, provided by the category of motives, which should be the category of representations of a pro-algebraic group called the motivic Galois group. This group should act on the ring of periods, preserving all algebraic relations, and thus generalize the classical Galois theory for algebraic numbers to a large class of transcendental numbers. Computations done in the context of dimensionally regularised Feynman integrals by Abreu, Britto, Duhr, Gardi, and Matthew, as well as the work of Brown and Dupont on Lauricella hypergeometric functions, suggest that more should be true, namely, that the action of the motivic Galois group on infinitely many different periods arising as coefficients of a certain type of series expansion, the prototype of which is the Taylor expansion of a Mellin transform, can be captured by finite formulas. In this talk we will explore this phenomenon with a particular emphasis on examples arising as dimensionally regularized Feynman integrals.

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Vanhove's differential operators are holonomic annihilators of off-shell Bessel moments, an important class of Feynman diagrams. We show that Vanhove's operators enjoy nice properties, such as Fuchsian condition and parity. These nice analytic properties enable us to construct algebraic relations among on-shell and off-shell Bessel moments, thereby covering the Broadhurst-Mellit determinant formulae and the Broadhurst-Roberts quadratic relations for Feynman diagrams.

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We will review the progress in formulating intersection theory for cohomologies with local coefficients associated to Feynman integrals in dimensional regularization. After explaining the physical motivation for this construction, we will go through the methods used to compute the intersection pairings in practical applications.

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A number of classical relations between periods, such as Legendre's relation for periods of elliptic curves, arise from the compatibility of the intersection pairing on de Rham and Betti cohomology with the period isomorphism. In the first part of the talk, I will give a friendly introduction to this phenomenon, with emphasis on how to compute intersection matrices. In the second part, I will explain how to interpret the moments of the Bessel functions as periods and derive quadratic relations between them that were found empirically by Broadhurst and Roberts. The talk is based on joint work with Claude Sabbah and Jeng-Daw Yu [arXiv:2006.02702].

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Closed string amplitudes at genus one can be described in terms of modular graph forms. These real-analytic functions with definite transformation properties under SL(2,Z) can be represented either as lattice sums over world-sheet torus momenta or as configuration space integrals. It is convenient to package them into a generating series involving Kronecker-Eisenstein series and a Koba-Nielsen factor. The generating series can be shown to satisfy a differential equation that contains a (conjectural) matrix representation of Tsunogai's derivation algebra and whose formal solution involves iterated Eisenstein integrals as studied by Brown. The closed string differential equation can be viewed as a single-valued version of a corresponding open string equation and, together with the input of the genus zero single-valued map in the form of initial conditions, this suggests a candidate genus one single-valued map.

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The philosophy of motives suggests the existence of a Galois theory for periods, which can be explicitly determined for certain families of periods such as hyperlogarithms. Recently, Abreu, Britto, Duhr, Gardi, and Matthew computed the Galois theory (a.k.a. the motivic coaction) of the coefficients in the epsilon-expansion of certain Feynman integrals in dimensional regularization, and observed that it could be packaged into succinct formulas at the level of power series. I will explain a proof of this phenomenon on the toy example of Lauricella hypergeometric functions, and suggest a geometric framework in which more general hypergeometric-type integrals are equipped with a Galois theory. This is based on the paper Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions written with Francis Brown, and ongoing work with Francis Brown, Javier Fresán, and Matija Tapušković.

vidéo de l'exposé | Notes on f-hyperlogs | Notes on f-functions

Multiple zeta values (and their cousins) are best handled in the motivic f-alphabet. The f-alphabet is a free shuffle algebra with deconcatenation as coaction. Because the f-alphabet is free, the conversion into the f-alphabet effectively serves as a reduction to a Q-basis. Thanks to a conversion algorithm (the decomposition algorithm) by F. Brown basis-reductions of MZVs of weights >30 are possible.

Here, we suggest to extend the f-alphabet to functions. We consider the case of (possibly generalized, single-valued) hyperlogarithms. We present the conjectured structure of such f-hyperlogs. In particular, we propose that integration of f-hyperlogs is facilitated by a commutative hexagon linking integration with monodromy.

Because hyperlogarithms are well understood in the classical setup (where they are a free shuffle algebra, too) these f-hyperlogarthms have the flavor of a toy model as a first attempt to understand general (elliptic, ...) f-functions. Beyond that, a benefit of f-hyperlogarithms can be their structural simplicity (for computer implementations) which may lead to speed gains (for code generation as well as in computing time). The drawback of using f-hyperlogarithms is that expressions become lengthier because they carry explicit information on monodromy which is somewhat concealed in the classical setup.

Note that this talk has the character of a proposal (rather than being a presentation of fully developed and established results).