Polylogarithms as a Bridge between Number Theory and Particle Physics

I will review a recent progress in computing correlation functions in N=4 SYM at weak coupling and present a new approach to constructing four-point correlation functions of half-BPS operators which does not rely on Feynman diagrams and makes use of the recently discovered symmetry of the integrand under permutations of external and integration points. Combined with the conjectured amplitude/correlation function duality, this symmetry alone allows us to construct the four-point correlation function in terms of a set of conformal four point integrals.

In the first part of this lecture series we will begin with the basic definition of the multiple polylogarithms and their different representations. We then turn to the variation of mixed Hodge structures associated with them and derive their single-valued versions as an important application. We will mostly use classical polylogs and the double logs as concrete examples. In the second part of the series, we will turn to the multiple harmonic sums and multiple zeta values which are special values of multiple polylogarithms and are intimately connected to Feynman integrals. The algebraic structure/relations will be our main focus and pointers to possible generalizations to special values of multiple polylogarithms at roots of unity will be provided.

I will review a recent progress in computing correlation functions in N=4 SYM at weak coupling and present a new approach to constructing four-point correlation functions of half-BPS operators which does not rely on Feynman diagrams and makes use of the recently discovered symmetry of the integrand under permutations of external and integration points. Combined with the conjectured amplitude/correlation function duality, this symmetry alone allows us to construct the four-point correlation function in terms of a set of conformal four point integrals.

I'll provide a set of introductory lectures outlining the recent and profound advances in our understanding of quantum field theory and the connections between its analytic structure and the geometry of Grassmannian polytopes. I will review the recently discovered tools which allow us to rewrite the Feynman expansion much more efficiently, and describe how the terms in these recursively-generated formulae are classified by simple combinatorics, and can be understood geometrically in terms of the positroid stratification of the Grassmannian.

I will review a recent progress in computing correlation functions in N=4 SYM at weak coupling and present a new approach to constructing four-point correlation functions of half-BPS operators which does not rely on Feynman diagrams and makes use of the recently discovered symmetry of the integrand under permutations of external and integration points. Combined with the conjectured amplitude/correlation function duality, this symmetry alone allows us to construct the four-point correlation function in terms of a set of conformal four point integrals.

I'll provide a set of introductory lectures outlining the recent and profound advances in our understanding of quantum field theory and the connections between its analytic structure and the geometry of Grassmannian polytopes. I will review the recently discovered tools which allow us to rewrite the Feynman expansion much more efficiently, and describe how the terms in these recursively-generated formulae are classified by simple combinatorics, and can be understood geometrically in terms of the positroid stratification of the Grassmannian.

In the first part of this lecture series we will begin with the basic definition of the multiple polylogarithms and their different representations. We then turn to the variation of mixed Hodge structures associated with them and derive their single-valued versions as an important application. We will mostly use classical polylogs and the double logs as concrete examples. In the second part of the series, we will turn to the multiple harmonic sums and multiple zeta values which are special values of multiple polylogarithms and are intimately connected to Feynman integrals. The algebraic structure/relations will be our main focus and pointers to possible generalizations to special values of multiple polylogarithms at roots of unity will be provided.

I'll provide a set of introductory lectures outlining the recent and profound advances in our understanding of quantum field theory and the connections between its analytic structure and the geometry of Grassmannian polytopes. I will review the recently discovered tools which allow us to rewrite the Feynman expansion much more efficiently, and describe how the terms in these recursively-generated formulae are classified by simple combinatorics, and can be understood geometrically in terms of the positroid stratification of the Grassmannian.

Approximate Schedule:
Depart from Grey College 9.00 a.m. (packed lunches available).
Arrive Alnwick Castle and Gardens 10.30 a.m.
Lunch 12.00 p.m.
Depart for Durham 4.00 p.m.
Arrive Grey College 5.30 p.m.

There is growing evidence, from both massless and massive Feynman diagrams, that modular forms first arise in quantum field theory when polylogarithms no longer suffice. In this talk, I aim to (1) introduce a variety of modular forms, (2) link enumerations of modular forms and multiple zeta values, (3) identify modular forms that obstruct evaluations to polylogs, (4) give a modular form that controls massive and massless diagrams, (5) use L-functions of modular forms to evaluate Feynman diagrams. En route, I shall report significant progress, made in recent months by Erik Panzer, Francis Brown and Oliver Schnetz, in many massless cases, and by Spencer Bloch and Pierre Vanhove, in a two-scale massive case.

The planar N=4 S-matrix and polygonal Wilson loop can be both reformulated as a holomorphic Wilson loop in twistor space. In joint work with Lipstein, we show that at MHV this formulation naturally leads to an all-loop integrand in d log form. We show how this can be evaluated to give poly-logs for some basic examples.

Hexagon functions are a class of iterated integrals, depending on three variables (dual conformal cross ratios) which have the correct branch cut structure and other properties to describe the scattering of six gluons in planar N=4 super-Yang-Mills theory. We classify all hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. As an example, the three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematics limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multi-Regge factorization directly at the function level, and thereby to fix uniquely a set of Riemann-Zeta-valued constants that could not be fixed at the level of the symbol. The near-collinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with a factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to -7.

We discuss the structure of amplitudes in parametric space, for scalar and gauge field theories, with or without a vanishing $\beta$-function.

Superstring disk amplitudes are described by generalized hypergeometric functions, whose low-energy expansion leads to multiple polylogarithms and multiple zeta values. In fact, the low-energy expansion is encoded by the decomposition of motivic multi zeta values and takes a particular intriguing form after mapping onto a non-commutative Hopf algebra. Furthermore, the low-energy expansion of superstring amplitudes gives rise to graded Lie algebra structures related to Grothendiecks Galois theory. Finally, after applying multiple inverse Mellin transformations string amplitudes give rise to a distributional setup.

Single-valued multiple zeta values are special values of single-valued multiple polylogarithms which are generalizations of the Bloch-Wigner dilogarithm. Using the Ihara action the motivic counterparts of single-valued multiple-zeta-values can be characterized in terms of the Galois coaction on motivic multiple-zeta-values. The K5 conjecture gives a simple combinatorial description of the family of graphs in phi^4 theory whose contributions to the beta function are single-valued multiple zeta values.

The nontriviality of a nullhomologous algebraic cycle on a smooth projective variety is measured by its image under an Abel-Jacobi (AJ) map. Given a family of cycles on such varieties, it is natural to pair the resulting family of AJ classes with the class of a holomorphic form, which yields a multivalued holomorphic function (over a Zariski open set in parameter space, with logarithmic singularities on the complement) called a "truncated normal function". These have recently appeared in type IIB string theory where they are mirror to the generating functions for open Gromov-Witten invariants (a phenomenon discovered by Morrison and Walcher). In the same way, given a family of higher cycles in the sense of Bloch, we can define a higher (truncated) normal function (HNF). These functions (as above, multivalued/holomorphic with log singularities) were studied in my joint 2011 CNTP paper with C. Doran in connection with local mirror symmetry and asymptotic growth of Gromov-Witten invariants. Two other places where they arise are in the Apery irrationality proofs and the evaluation of certain integrals arising from Feynman graphs. In this talk I will focus on the latter, including examples arising in the work of Bloch-Esnault-Kreimer and (especially) more recent work of Bloch-Vanhove. The main point is that by recognizing an integral as a HNF, one knows with very little (or at least significantly reduced) work the inhomogeneous Picard-Fuchs equation it satisfies, as well as its monodromy and "special values" at degenerate fibers. I'll devote some time at the beginning to explaining what the higher cycles are and what a "higher AJ" map is.

This talk will survey some perturbative and non-perturbative properties of string scattering amplitudes and how they are interrelated by duality symmetries.

14:30 Cathedral Guided Tour
16:00 Lindisfarne Gospels - ticket price 6.50

In the first part of the talk I consider the approach of integrating over Feynman parameters such that the result is given by multiple polylogarithms in several variables. I discuss aspects of a sufficient criterion for a given Feynman integral to be computable by this approach. In the second part I focus on the two-loop sunrise integral with arbitrary masses, which fails the criterion and which, to the best of our knowledge, cannot be expressed by multiple polylogarithms. I discuss a new result for this integral, involving elliptic integrals.

The $c_2$ invariant is the coefficient of $q^2$ in the number of rational points over $\mathbb{F}_q$ of the hypersurface associated to a Feynman graph. This invariant respects many relations between the periods of Feynman graphs. I will explain how to define $c_2$ for different representations of a graph and show that they all coincide.

Mid 90's, Broadhurst and Kreimer observed that multiple zeta values persist to appear in Feynman integral computations. Following this observation, Kontsevich proposed a conceptual explanation, that is, the loci of divergence in these integrals must be very particular type of object in algebraic geometry; mixed Tate motives. In 2000, Belkale and Brosnan disproved this conjecture. In this talk, I will describe a way to `correct' Kontsevich's proposal and show that the regularized Feynman integrals in position space setting as well as their ambiguities are given in terms of periods of suitable configuration spaces, which are mixed Tate. Therefore, the Feynman integrals for massless scalar QFTs are indeed Q[(2 π i)-1]-linear combinations of multiple zeta values due a theorem by F. Brown. Moreover, the regularized Feynman amplitude for a massive scalar Euclidean field have an asymptotic expansion given by a formal series where the terms are combinations of multiple zeta values with coefficients that are polynomials in Q[√(π/2),(2 π i)-1,m-1]. This is a joint work with Matilde Marcolli.

I will describe a relation between the alpha' expansion of open superstring amplitudes at tree-level and Drinfel'd's associator derived from the Knizhnik-Zamolodchikov equation.

I discuss the recent computation of the soft triple-real corrections to Higgs production. The result is expressed in terms of zeta values up to weight six and exhibits the feature that all coefficients in front of the zeta values are integers.

In this talk we take the first steps towards a new framework for computing two-loop amplitudes, based on unitarity rather than Feynman diagrams. In this approach, the two-loop amplitude is first expanded in a basis of integrals. The expansion coefficients are then determined by applying generalized unitarity cuts. We find explicit formulas for the integral coefficients as products of tree-level amplitudes integrated over specific multidimensional complex contours, thus allowing the construction of the two-loop amplitude from appropriately defined tree amplitudes. The validity of this method extends to all four-dimensional gauge theories, in particular QCD. Finally, we comment on the non-expressibility in terms of polylogarithms of a particular two-loop amplitude in N=4 SYM theory.

Scattering amplitudes in super-Yang Mills theory beautifully tie together two subjects of intense interest to mathematicians: polylogarithm functions and cluster algebras. I will show, by drawing on the specific example of two-loop MHV amplitudes, that the cluster structure of the kinematic configuration space Conf_n(P^3) underlies the structure of n-particle amplitudes.