• Periods in Number Theory, Algebraic Geometry and Physics
• Workshop: Amplitudes and Periods

• Schedule
• Participants

We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the ρ-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-N space either. The solution of the homogeneous equations is possible in terms of convergent close integer power series as 2F1 Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using q-product and series representations implied by Jacobi's ϑ_i functions and Dedekind's η-function. The corresponding representations can be traced back to polynomials out of Lambert-Eisenstein series, having representations also as elliptic polylogarithms, a q-factorial 1/η^k(τ), logarithms and polylogarithms of q and their q-integrals. Due to the specific form of the physical variable x(q) for different processes, different representations do usually appear. Numerical results are also presented. The talk would cover the way from large systems of ordinary differential equation systems, which partly have irreducible 2nd order systems beyond the well understood 1st order systems to 2F1-solutions. In the present cases we even obtain elliptic solutions and can express them in terms of modular functions, not only in modular forms. The topic has quite a series of yet open ends raising e.g. the following questions to the mathematics colleagues:

i) How to get to short (or even shortest) bases for the modular forms occurring ? [as we usually run into bigger representations first].
ii) Which systems are known to solve only by modular forms?
iii) Modularity for 2F1 solutions, which are not elliptic.
iv) Abel integral solutions? [Perhaps for 3x3 systems onwards] - what does one know there etc.
v) General aspects out of algebraic geometry, which could serve as construction principles? [It might be we did not pay enough attention to this yet and our structures could greatly simplify in doing so.]

Recorded Talk  Slides

The integrals mentioned in the title are Feynman integrals, which are of particular interest in recent developments, as they can not be expressed in terms of multiple polylogarithms. Physicists have been trying to find appropriate classes of functions for the computation of such integrals and in the last few years, elliptic generalizations of polylogarithms have shown to be useful. In my talk I would discuss properties of these Feynman integrals by use of periods of corresponding elliptic curves and of the Picard-Lefschetz-theorem. The talk is based on joint work with Schweitzer and Weinzierl: arXiv:1705.08952 [hep-ph] and an introduction would include aspects of previous work with the same authors and Adams (e.g. arXiv:1607.01571 [hep-ph]). 

Recorded talk

Physicists have recently come to appreciate the utility of Eichler integrals of quotients of the Dedekind eta function. In this survey, I focus on some of the mathematical features that are involved, including modular transformations of the higher normal functions investigated by Bloch, Kerr and Vanhove, critical and non-critical values of L-series, quasi-periods in the sense of Brown and Hain, and quadratic relations between periods encoded by Betti and de Rham matrices. No familiarity with the physical context will be assumed.

Slides

We relate one-loop scattering amplitudes of massless open- and closed-string states at the level of their low-energy expansion. The modular graph functions resulting from integration over closed-string punctures are observed to follow from symmetrized open-string integrals through a tentative generalization of the single-valued projection known from genus zero.

Further information (preprint on arXiv)

Recorded talk

Modular graph functions are non-holomorphic modular forms arising from string perturbation theory in genus 1 studied by Green, Russo, Vanhove, Zagier and most recently Zerbini. They are conjectured to satisfy a long list of properties. I will explain how to take real and imaginary parts of iterated integrals of Eisenstein series regularised at tangential base points to construct a large new class of non-holomorphic modular forms which satisfy all the desired properties and more. If time permits, I will explain how these modular forms are associated to mixed (Tate) motives, and use them to define a new class of `non-abelian' L-functions associated to universal mixed elliptic motives.

Recorded Talk 

The first nontrivial amplitudes in N=4 Super-Yang-Mills Theory are for the scattering of six or seven gluons. These amplitudes appear to be constructible to all orders in perturbation theory from multiple polylogarithms, where the weight (number of integrations) is twice the (loop) order in perturbation theory. I will describe the space of "Steinmann hexagon functions" of three variables relevant for the six-gluon case, including the restricted set of multiple zeta values to which they evaluate at a special point. I will also provide evidence, through weight 12, that the function space is stable under the co-action associated with a cosmic Galois group.

Recorded Talk    Lecture notes

I will show how to use a construction due to Brown to define a certain coaction on (a variant of) elliptic polylogarithms (eMPLs). This coaction can be used to study relations among these classes of functions. We apply it in particular to study eMPLs evaluated at torsion points of the elliptic curve, and we show that they can always be expressed as iterated integrals of Eisenstein series for the congruence subgroup Gamma(N), where N is determined by the torsion points.

Recorded Talk

Joint work with Michael Groechenig.

If we dropped ‘Cohomologically’ from the title, this would be a complete (positive) answer to Simpson’s conjecture. We prove the conjecture under the (perhaps ?) stronger assumption that the local systems are cohomologically rigid. Our proof can’t be extended to the case of rigid connections, and I shall explain why. Initially, in the projective case, our proof consisted in going to the de Rham side, showing that the restriction of the connection on the p-adic varieties is a Frobenius isocrystal, descend mod p, considering the ℓ-adic companions (the existence of which has been proven recently by Abe and myself) and going back to the complex numbers by showing that the induced local systems stemming from the companions are still cohomologically rigid. This last step requires the study of weights and the L-functions of the companions. The new proof is purely Betti-ℓ-adic. We also  prove that the  overconvergent isocrystals with a Frobenius structure coming from cohomologically rigid connectionshave a full package of companions. There are also mild consequences for the p-curvature conjecture.

I will discuss recent results linking the Hopf algebras of Goncharov for multiple zetas, the Hopf algebra of Connes and Kreimer for renormalisation and their Hopf algebra of graphs. These are several levels and sublteties in this construction which I will discuss using examples. A surprising realisation is that the first Hopf algebra is simplicial and is closely related to a Hopf algebra appearing in the context of double loop spaces. One very concise form is that these Hopf algebras arise as deconcatenation Hopf algebras for certain categories called Feynman categories. This general structure has already provided links between fields, such as decorations, factorisations, and an understanding of the cubical complexes underlying Cutkosky rules and those of moduli spaces, which we will very briefly mention at the end.

Recorded Talk 

String theory generates differential equations for automorphic functions that are inhomogeneous and violate the Z-finiteness condition. I will discuss old and new instances of this and how one can formally solve these equations, highlighting open problems in connection with representation theory. Based on unpublished work with O. Ahlen.

I will try to explain the notion of a normal and anomalous threshold for a Feynman graph G. Such a graph G is evaluated by renormalized Feynman rules. The result depends on various parameters -masses and external momenta- and just the question where to find threshold divisors leads to the study of interesting graph complexes.

Recorded Talk   Slides

Maxim Kontsevich gave a universal formula for the quantization of Poisson brackets (the star product), as a formal power series in differential operators. The terms in this series are indexed by directed graphs, each of which is associated to a certain integral over the moduli space of punctured disks. In a forthcoming paper with Peter Banks and Brent Pym, we use Francis Brown's approach to the periods of the moduli space of genus zero curves, together with the single-valued integration method of Oliver Schnetz, to give a general algorithm for the computation of any of these integrals in terms of multiple zeta values. This allows us to calculate several terms in the expansion of the star product for the first time and provides evidence for several open conjectures.

Recorded Talk Lecture notes 

Lecture notes Recorded Talk

We introduce a new class of hyperlogs which generalizes iterated integrals (it is closely related to cell zeta values by F. Brown). These graphical hyperlogs admit a graphical representation. We present a coaction formula for these graphical hyperlogs in terms of subquotient graphs. In particular, the space of graphical hyperlogs is closed under the coaction.

Recorded Talk   Lecture notes

Racah coefficients, or 6j-symbols for tensor products of infinite-dimensional unitary principal series representations of the group SL(2,C) are considered. They were constructed earlier by Ismagilov, but we derive them by a different method (and get a slightly different result) using the Feynman diagrams technique. The resulting 6j-symbols are expressed either as a triple integral over complex plane or as an infinite bilateral sum of integrals of the Mellin-Barnes type. This is a joint work with S.E. Derkachov (arXiv:1711.07073 [math-ph]).

Recorded Talk

A class of (N-3)-dimensional real iterated integrals on (RP1-{0,1,∞})N-3 (integrating to Aomoto-Gelfand hypergeometric functions and GKZ structures) will be related to (N-3)-dimensional complex integrals on (CP1-{0,1,∞})N-3 via the single-valued map. Both integrals give rise to periods of the moduli space M0,N of genus zero curves with N ordered marked points and the single-valued map projects onto a subspace.

From the physics point of view the first class of integrals describes tree-level open string scattering amplitudes, while the second class accounts for tree-level closed string scattering amplitudes. Hence, the single-valued projection relates open and closed string scattering amplitudes.

A graphical insight into this relation is obtained by looking at the underlying beta-functions of two-dimensional sigma models describing world-sheets of open and closed (heterotic) strings. We analyze the sigma-model Feynman diagrams generating identical effective action terms in both theories and verify the single-valued map at the level of individual Feynman diagrams.

Recorded Talk   Slides 

Feynman integrals are easily solved if their system of differential equations is in epsilon-form. In this talk I will show that an epsilon-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. This is achieved by a non-algebraic change of basis for the master integrals. In the case of the sunrise and the kite integral one obtains in this way iterated integrals of modular forms and the epsilon-expansion of the Feynman integral is at each order homogeneous in the number of iterated integrations. For these integrals I will also discuss the implications of different choices of periods of the elliptic curve spanning the same lattice of periodicity.

Recorded Talk

We study the Feynman integral for the sunset graph defined as the scalar two-point self-energy at two-loop order. The Feynman integral is evaluated for all inequal internal masses in two space-time dimensions. Two calculations are given for the Feynman integral; one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of elliptic curves. Using an Hodge theoretic (B-model) approach, we show that the integral is given by a sum of elliptic dilogarithms evaluated at the divisors determined by the punctures. Secondly we associate to the sunset elliptic curve a local non-compact Calabi-Yau 3-fold, obtained as a limit of elliptically fibered compact Calabi-Yau 3-folds. By considering the limiting mixed Hodge structure of the Batyrev dual A-model, we arrive at an expression for the sunset Feynman integral in terms of the local Gromov-Witten prepotential of the del Pezzo surface of degree 6. This expression is obtained by proving a strong form of local mirror symmetry which identifies this prepotential with the second regulator period of the motivic cohomology class. The talk is based on work done in collaboration with Spencer Bloch and Matt Kerr.

Recorded Talk       Slides