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Amplitudes, Motives and Beyond
from
Tuesday, May 26, 2015 (8:00 AM)
to
Friday, June 12, 2015 (6:00 PM)
Monday, May 25, 2015
Tuesday, May 26, 2015
10:55 AM
Welcome

Stefan Weinzierl
Welcome
Stefan Weinzierl
10:55 AM  11:00 AM
Room: 02.430
11:00 AM
Multiple polylogarithms in cyclotomic fields and subfields

David Broadhurst
Multiple polylogarithms in cyclotomic fields and subfields
David Broadhurst
11:00 AM  11:45 AM
Room: 02.430
Singlescale 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.
4:00 PM
Elliptic multiple zeta values and superstring oneloop amplitudes, part I

Oliver Schlotterer
Elliptic multiple zeta values and superstring oneloop amplitudes, part I
Oliver Schlotterer
4:00 PM  4:40 PM
Room: 02.430
In the first half of this talk, we introduce elliptic multiple zeta values (eMZVs) as iterated integrals on a genusone curve and illustrate their natural appearance in oneloop 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 (socalled) Fayrelations. Based on these identities, any worldsheet integral in the lowenergy expansion of oneloop open superstring amplitudes can be expressed in terms of eMZVs. We conclude with an overview of eMZVs indecomposable under Fay and shuffle relations.
4:45 PM
Elliptic multiple zeta values and superstring oneloop amplitudes, part II

Johannes Broedel
Elliptic multiple zeta values and superstring oneloop amplitudes, part II
Johannes Broedel
4:45 PM  5:25 PM
Room: 02.430
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 noncommutative variables, these selection rules reproduce all known commutator relations from an algebra of special derivations on a free Lie algebra.
Wednesday, May 27, 2015
11:00 AM
Yangian invariant scattering amplitudes as unitary matrix integrals

Matthias Staudacher
Yangian invariant scattering amplitudes as unitary matrix integrals
Matthias Staudacher
11:00 AM  11:45 AM
Room: 02.430
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.
4:00 PM
The Cluster Bootstrap for Scattering Amplitudes

James Drummond
The Cluster Bootstrap for Scattering Amplitudes
James Drummond
4:00 PM  4:40 PM
Room: 02.430
The singularities of scattering amplitudes in planar N=4 super YangMills 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.
Thursday, May 28, 2015
11:00 AM
Periods and Superstring Amplitudes

Stephan Stieberger
Periods and Superstring Amplitudes
Stephan Stieberger
11:00 AM  11:45 AM
Room: 02.430
We present (some) connections and implications of superstring amplitudes from and to number theory. These relations include motivic multiple zeta values, singlevalued multiple zeta values, Drinfeld, and Deligne associators. More concretely, we will show that treelevel 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 singlevalued 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 KawaiLewellenTye (KLT) relations.
Friday, May 29, 2015
11:00 AM
Zhegalkin zebra motives

Jan Stienstra
Zhegalkin zebra motives
Jan Stienstra
11:00 AM  11:45 AM
Room: 02.430
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 ntuple of zebras into an nvariable 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 socalled brane tilings or dimer models, Poisson structures, CalabiYau manifolds). We focus in particular on examples which reproduce the quantumperiods of the DelPezzo surfaces, and may therefore be considered as mirrors of DelPezzo surfaces.
4:00 PM
Calculating Higgs production at three loops in QCD

Falko Dulat
Calculating Higgs production at three loops in QCD
Falko Dulat
4:00 PM  4:40 PM
Room: 02.430
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 nonsupersymmetric QCD.
Saturday, May 30, 2015
Sunday, May 31, 2015
Monday, June 1, 2015
11:00 AM
Computing Feynman integrals via Fuchsian differential equations

Johannes Henn
Computing Feynman integrals via Fuchsian differential equations
Johannes Henn
11:00 AM  11:45 AM
Room: 02.430
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 [hepph].
4:00 PM
Amplituhedric open problems

Livia Ferro
Amplituhedric open problems
Livia Ferro
4:00 PM  4:40 PM
Room: 02.430
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 YangMills.
4:40 PM
Onshell methods for crosssections and anomalous dimensions in N=4 SYM

Dhritiman Nandan
Onshell methods for crosssections and anomalous dimensions in N=4 SYM
Dhritiman Nandan
4:40 PM  5:20 PM
Room: 02.430
In this talk we use onshell unitarity methods to study anomalous dimensions of nonprotected operators in N=4 SYM from their form factors up to two loop orders. We also construct IRfinite crosssection type quantity for such operators in N=4 SYM.
Tuesday, June 2, 2015
11:00 AM
Cluster Algebra Structures in Scattering Amplitudes

Anastasia Volovich
Cluster Algebra Structures in Scattering Amplitudes
Anastasia Volovich
11:00 AM  11:45 AM
Room: 02.430
I will review cluster algebra structure of scattering amplitudes in planar N=4 YangMills, and apply it to compute several amplitudes.
Wednesday, June 3, 2015
11:00 AM
Kloosterman sums, modular forms and sunrise diagrams

David Broadhurst
Kloosterman sums, modular forms and sunrise diagrams
David Broadhurst
11:00 AM  11:45 AM
Room: 02.430
The onshell equalmass sunrise diagram with N2 loops in two spacetime 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 Lseries of modular forms with weight N2. 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 Lseries, from Anton Mellit.
4:00 PM
The Galois coaction on phi4 periods

Oliver Schnetz
The Galois coaction on phi4 periods
Oliver Schnetz
4:00 PM  4:40 PM
Room: 02.430
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.
Thursday, June 4, 2015
Friday, June 5, 2015
11:00 AM
Integral invariants, unitarity, and the correlahedron

Burkhard Eden
Integral invariants, unitarity, and the correlahedron
Burkhard Eden
11:00 AM  11:45 AM
Room: 02.430
3:00 PM
Elliptic multiple zeta values and the elliptic KZB equation

Nils Matthes
Elliptic multiple zeta values and the elliptic KZB equation
Nils Matthes
3:00 PM  3:40 PM
Room: 02.430
After recalling the connection between classical multiple zeta values and the KnizhnikZamolodchikov equation, I will review work of Enriquez, which uses the elliptic KnizhnikZamolodchikovBernard (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.
3:40 PM
Bformula: sums over graphs, Macdonald polynomials and character varieties

Anton Mellit
Bformula: sums over graphs, Macdonald polynomials and character varieties
Anton Mellit
3:40 PM  4:20 PM
Room: 02.430
Character varieties parametrize local systems on punctured Riemann surfaces with prescribed monodromies around punctures. Hausel, Letellier and RodriguezVillegas 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 Bformula. 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.
Saturday, June 6, 2015
Sunday, June 7, 2015
Monday, June 8, 2015
11:00 AM
E_n minuscule graphs and motives

Sergey Galkin
E_n minuscule graphs and motives
Sergey Galkin
11:00 AM  11:45 AM
Room: 02.430
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.
4:00 PM
Around motivic structure of quantum cohomology

Yuri Manin
Around motivic structure of quantum cohomology
Yuri Manin
4:00 PM  4:40 PM
Room: 02.430
Tuesday, June 9, 2015
11:00 AM
A quasifinite basis for Feynman integrals

Andreas von Manteuffel
A quasifinite basis for Feynman integrals
Andreas von Manteuffel
11:00 AM  11:45 AM
Room: 02.430
In this talk, I describe a new method for the decomposition of multiloop Feynman integrals into quasifinite 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.
4:00 PM
CrossOrder Integral Relations from Maximal Cuts

Mads Sogaard
CrossOrder Integral Relations from Maximal Cuts
Mads Sogaard
4:00 PM  4:40 PM
Room: 02.430
We explain how to reconstruct crossorder 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 twoloop ABDK relation with up to five external legs. Our analysis suggests that maximal and nearmaximal cuts can be used to infer the existence of integral identities more generally.
Wednesday, June 10, 2015
11:00 AM
Ambitwistors and the scattering equations

Lionel Mason
Ambitwistors and the scattering equations
Lionel Mason
11:00 AM  11:45 AM
Room: 02.430
Cachazo, He and Yuan have produced remarkable formulae that express treelevel 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.
3:00 PM
String amplitudes, simple and multiple zeta values, partI

Don Zagier
String amplitudes, simple and multiple zeta values, partI
Don Zagier
3:00 PM  3:40 PM
Room: 02.430
4:00 PM
String amplitudes, simple and multiple zeta values, part II

Don Zagier
String amplitudes, simple and multiple zeta values, part II
Don Zagier
4:00 PM  4:40 PM
Room: 02.430
Thursday, June 11, 2015
11:00 AM
Integral reduction from elliptic and hyperelliptic curves

Yang Zhang
Integral reduction from elliptic and hyperelliptic curves
Yang Zhang
11:00 AM  11:45 AM
Room: 02.430
I show that several classes of twoloop 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 1forms on algebraic curves.
Friday, June 12, 2015
11:00 AM
Hexagon OPE Resummation and 2dimensional Harmonic Polylogarithms

Georgios Papathanasiou
Hexagon OPE Resummation and 2dimensional Harmonic Polylogarithms
Georgios Papathanasiou
11:00 AM  11:45 AM
Room: 02.430
In the Wilson loop OPE approach, scattering amplitudes in N=4 super YangMills 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 6point (or hexagon) amplitude, with the help of nested sum technology. The result is expressed in terms of 2dimensional harmonic polylogarithms, and very interestingly it also yields the full amplitude in multiRegge kinematics, under the assumption that the natural function space for the latter are singlevalued harmonic polylogarithms.