Amplitudes, Motives and Beyond

26 May 2015, 08:00 → 12 Jun 2015, 18:00 Europe/Berlin

02.430 (Mainz Institute for Theoretical Physics<br>Johannes Gutenberg University)

The interplay and similarities between physics and mathematics found in the study of scattering amplitudes on the physics side and in the theory of motives on mathematics side is a fascinating and rapid developing subject of current research. The MITP scientific program “Amplitudes, Motives and beyond” will bring together the world leading experts in this intersection from the fields of high energy physics, mathematics and string theory. Ultimately, the proposed program should ignite further progress in this direction.

Dates of participants Participantslist_(Indico)_Amplitudes,_Motives_and_Beyond2015.pdf

Executive Summary executive_summary.pdf

Tuesday 26 May

1 Welcome

Speaker: Stefan Weinzierl

Slides mitp2_workshop.pdf

2 Multiple polylogarithms in cyclotomic fields and subfields

Single-scale Feynman diagrams evaluate to periods that may (but need not) be multiple polylogarithms with arguments in algebraic number fields, such as Nth roots of unity, with N=1,2,6 being prominent. In this introductory talk I shall give examples of such evaluations and give evidence for some new conjectures, in real subfields of cyclotomic fields, that have emerged from recent discussion with Pierre Deligne. I hope to pursue the latter as the workshop progresses.

Speaker: David Broadhurst

Slides mitp_broadhurst.pdf

3 Elliptic multiple zeta values and superstring one-loop amplitudes, part I

In the first half of this talk, we introduce elliptic multiple zeta values (eMZVs) as iterated integrals on a genus-one curve and illustrate their natural appearance in one-loop scattering amplitudes of the open superstring. The underlying elliptic iterated integrals are shown to require an infinite alphabet of differential forms subject to a rich network of shuffle- and (so-called) Fay-relations. Based on these identities, any worldsheet integral in the low-energy expansion of one-loop open superstring amplitudes can be expressed in terms of eMZVs. We conclude with an overview of eMZVs indecomposable under Fay and shuffle relations.

Speaker: Oliver Schlotterer

Slides mitp_schlotterer.pdf

4 Elliptic multiple zeta values and superstring one-loop amplitudes, part II

The second half of the talk opens up a new perspective on eMZVs based on their dependence on the modular parameter of the elliptic curve. A differential equation gives rise to an alternative description of eMZVs in terms of iterated integrals over Eisenstein series. The resulting counting of indecomposable eMZVs ties in with Fay and shuffle relations and leads to selection rules among the occurring iterated Eisenstein integrals. When formulated in terms of non-commutative variables, these selection rules reproduce all known commutator relations from an algebra of special derivations on a free Lie algebra.

Speaker: Johannes Broedel

Slides mitp_broedel.pdf

Wednesday 27 May

5 Yangian invariant scattering amplitudes as unitary matrix integrals

I show how to derive the deformed Yangian invariant tree level amplitudes from the quantum inverse scattering method. The construction naturally leads to deformed Graßmannian contour integrals. The novelty of our construction is that the contours are fixed from the beginning. In the split helicity case the contours correspond to the manifolds of the unitary groups, and the contour integration is just invariant Haar integration. The resulting integrals are analytic functions of the deformation parameters as long as the scattering data is generic, i.e. stays away from collinear configurations. This is joint work with Nils Kanning and Yumi Ko.

Speaker: Matthias Staudacher

6 The Cluster Bootstrap for Scattering Amplitudes

The singularities of scattering amplitudes in planar N=4 super Yang-Mills theory are conjecturally described by a cluster algebra structure. I will present a number of tests of this hypothesis and show that it can be used to determine scattering amplitudes from very little information.

Speaker: James Drummond

Thursday 28 May

7 Periods and Superstring Amplitudes

We present (some) connections and implications of superstring amplitudes from and to number theory. These relations include motivic multiple zeta values, single-valued multiple zeta values, Drinfeld, and Deligne associators. More concretely, we will show that tree-level superstring amplitudes provide a beautiful link between generalized multiple Gaussian hypergeometric functions and the decomposition of motivic multiple zeta values. Furthermore, we establish relations between complex integrals on CP^1 minus 3 points as single-valued projection of iterated real integrals on RP^1 minus 3 points. From the the physical point of view this relation expresses closed string amplitudes as projections of open string amplitudes: a relation, which goes far beyond, what is known from the notorious Kawai-Lewellen-Tye (KLT) relations.

Speaker: Stephan Stieberger

Slides mitp_stieberger_corr.pdf

Friday 29 May

8 Zhegalkin zebra motives

A Zhegalkin polynomial in n variables is a polynomial function on {F_2}^n; here F_2={0,1} is the field with 2 elements. The zebra with frequency vector v is the F_2 - valued function Z on the plane R^2 defined by the formula Z(x) = floor(2v\cdot x) mod 2. By inserting an n-tuple of zebras into an n-variable Zhegalkin polynomial one obtains an F_2 - valued function F on R^2, which we call a Zhegalkin zebra function. The graph of such a function gives a partion of the plane into black (i.e. F=1) and white (i.e. F=0) areas. This easily produces many beautiful pictures. We will focus on examples for which the picture is a doubly periodic tiling of the plane by black and white convex polygons. These tilings have a very rich geometric structure (including so-called brane tilings or dimer models, Poisson structures, Calabi-Yau manifolds). We focus in particular on examples which reproduce the quantum-periods of the DelPezzo surfaces, and may therefore be considered as mirrors of DelPezzo surfaces.

Speaker: Jan Stienstra

9 Calculating Higgs production at three loops in QCD

I report on our recent calculation of the Higgs cross section at three loops in QCD. I will illustrate how techniques from the study of amplitudes in N=4 can be employed to aid calculations in non-supersymmetric QCD.

Speaker: Falko Dulat

Monday 1 June

10 Computing Feynman integrals via Fuchsian differential equations

I will review recent progress in the evaluation of Feynman integrals via differential equations and point out a number of interesting open problems. For further reading see the lecture notes arXiv:1412.2296 [hep-ph].

Speaker: Johannes Henn

11 Amplituhedric open problems

In this talk I will present some of the questions which remain open in the (deformed) Grassmannian/Amplituhedron formulation of scattering amplitudes in N=4 super Yang-Mills.

Speaker: Livia Ferro

12 On-shell methods for cross-sections and anomalous dimensions in N=4 SYM

In this talk we use on-shell unitarity methods to study anomalous dimensions of non-protected operators in N=4 SYM from their form factors up to two loop orders. We also construct IR-finite cross-section type quantity for such operators in N=4 SYM.

Speaker: Dhritiman Nandan

Slides mitp_nandan.pdf

Tuesday 2 June

13 Cluster Algebra Structures in Scattering Amplitudes

I will review cluster algebra structure of scattering amplitudes in planar N=4 Yang-Mills, and apply it to compute several amplitudes.

Speaker: Anastasia Volovich

Wednesday 3 June

14 Kloosterman sums, modular forms and sunrise diagrams

The on-shell equal-mass sunrise diagram with N-2 loops in two space-time dimensions is an integral of a product of N Bessel functions. Using Schwinger parameters one may try to guess modular forms that prevent reduction to polylogarithms for N>4. I was able to do so for N=5,6 and 8, obtaining evaluations of integrals of N Bessel functions in terms of L-series of modular forms with weight N-2. The case N=7 proved harder, but was conquered last week, thanks to work by Ronald Evans, on Kloosterman sums, and an inspired suggestion for the functional equation of the corresponding L-series, from Anton Mellit.

Speaker: David Broadhurst

Slides mitp_broadhurst2.pdf

15 The Galois coaction on phi4 periods

The periods of primitive divergent phi4 graphs are renormalization scheme independent contributions to the phi4 beta function. They are mathematical periods in the sense of Kontsevich and Zagier. By general principles there exists a Galois coaction on these numbers. Recently it has become possible to calculate more than 300 distinct periods of graphs up to 11 loops. The analysis of these data leads to two (possibly false) conjectures on the coaction structure of phi4 periods.

Speaker: Oliver Schnetz

Friday 5 June

16 Integral invariants, unitarity, and the correlahedron

Speaker: Burkhard Eden

17 Elliptic multiple zeta values and the elliptic KZB equation

After recalling the connection between classical multiple zeta values and the Knizhnik-Zamolodchikov equation, I will review work of Enriquez, which uses the elliptic Knizhnik-Zamolodchikov-Bernard (KZB) equation to construct elliptic analogues of multiple zeta values (eMZVs). The construction given here is also closely related to work of Brown and Levin on multiple elliptic polylogarithms. I will then present some results about the structure of the algebra of eMZVs with an emphasis on linear relations between elliptic double zeta values.

Speaker: Nils Matthes

18 B-formula: sums over graphs, Macdonald polynomials and character varieties

Character varieties parametrize local systems on punctured Riemann surfaces with prescribed monodromies around punctures. Hausel, Letellier and Rodriguez-Villegas studied character varieties from a motivic point of view and guessed a formula which should compute the mixed Hodge structures of all character varieties. The formula is quite complicated: it has infinite products, infinitely many variables, and Macdonald polynomials. However in the case when all but two monodromies have ramification index 1, I discovered that their formula can be reduced to an elementary expression with a sum over graphs of certain type, which I call the B-formula. This case can be seen as an analogue of the famous Hurwitz number problem (ELSV formula), and I hope it to have some connection to physics.

Speaker: Anton Mellit

Monday 8 June

19 E_n minuscule graphs and motives

First I will briefly describe similarities between some topics of the last week and the study of mirror symmetry for Fano manifolds. Then I will give an example of an interesting series of 5 graphs, obtained from each other by inductive construction, and their associated Laurent polynomials, periods and motives. These series might be interesting in other contexts as well.

Speaker: Sergey Galkin

20 Around motivic structure of quantum cohomology

Speaker: Yuri Manin

Slides mitp_manin.pdf

Tuesday 9 June

21 A quasi-finite basis for Feynman integrals

In this talk, I describe a new method for the decomposition of multi-loop Feynman integrals into quasi-finite Feynman integrals. These are defined in shifted dimensions with higher powers of the propagators, make explicit both infrared and ultraviolet divergences, and allow for an immediate and trivial expansion in the parameter of dimensional regularization. The approach avoids the introduction of spurious structures and thereby leaves integrals particularly accessible to direct analytical integration techniques. Alternatively, the resulting convergent Feynman parameter integrals may be evaluated numerically. The method is based on integration by parts reductions and allows for automation in a straight- forward fashion. Further reading: arXiv:1411.7392.

Speaker: Andreas von Manteuffel

22 Cross-Order Integral Relations from Maximal Cuts

We explain how to reconstruct cross-order integral relations from maximal cuts. The underlying phenomenon is that integrals across different loop orders can have support on the same generalized unitarity cuts and can share global poles. The technique is demonstrated for the two-loop ABDK relation with up to five external legs. Our analysis suggests that maximal and near-maximal cuts can be used to infer the existence of integral identities more generally.

Speaker: Mads Sogaard

Wednesday 10 June

23 Ambitwistors and the scattering equations

Cachazo, He and Yuan have produced remarkable formulae that express tree-level scattering amplitudes as sums over certain solutions to the scattering equations defined on the Riemann sphere. The lecture will explain how these formulae arise from a holomorphic string theory in ambitwistor space and will go on to explain some further developments such as soft limits and/or loop amplitudes.

Speaker: Lionel Mason

Slides mitp_mason.pdf

24 String amplitudes, simple and multiple zeta values, partI

Speaker: Don Zagier

25 String amplitudes, simple and multiple zeta values, part II

Speaker: Don Zagier

Thursday 11 June

26 Integral reduction from elliptic and hyperelliptic curves

I show that several classes of two-loop Feynman integrals can be explicitly reduced to master integrals, by the analysis on elliptic and hyperelliptic curves. The key is that IBP relations correspond to exact meromorphic 1-forms on algebraic curves.

Speaker: Yang Zhang

Friday 12 June

27 Hexagon OPE Resummation and 2-dimensional Harmonic Polylogarithms

In the Wilson loop OPE approach, scattering amplitudes in N=4 super Yang-Mills theory are given by an infinite sum over all excitations of an integrable flux tube.  As a first step towards the resummation of the entire series, we compute the contribution of all single gluon bound states in the weak coupling expansion of the MHV 6-point (or hexagon) amplitude, with the help of nested sum technology. The result is expressed in terms of 2-dimensional harmonic polylogarithms, and very interestingly it also yields the full amplitude in multi-Regge kinematics, under the assumption that the natural function space for the latter are single-valued harmonic polylogarithms.

Speaker: Georgios Papathanasiou