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Register a new userOctagons I: Combinatorics and Non-Planar Resummations
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We explain how the t Hooft expansion of correlators of half-BPS operators can be resummed in a large-charge limit in N=4 super Yang-Mills theory. The full correlator in the limit is given by a non-trivial function of two variables: One variable is the charge of the BPS operators divided by the square root of the number Nc of colors; the other variable is the octagon that contains all the t Hooft coupling and spacetime dependence. At each genus g in the large Nc expansion, this function is a polynomial of degree 2g+2 in the octagon. We find several dual matrix model representations of the correlators in the large-charge limit. Amusingly, the number of colors in these matrix models is formally taken to zero in the relevant limit.
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The octagon function is the fundamental building block yielding correlation functions of four large BPS operators in N=4 super Yang-Mills theory at any value of the t Hooft coupling and at any genus order. Here we compute the octagon at strong coupling, and discuss various interesting limits and implications, both at the planar and non-planar level.
Inspired by the topological sign-flip definition of the Amplituhedron, we introduce similar, but distinct, positive geometries relevant for one-loop scattering amplitudes in planar $mathcal{N}=4$ super Yang-Mills theory. The simplest geometries are those with the maximal number of sign flips, and turn out to be associated with chiral octagons previously studied in the context of infrared (IR) finite, pure and dual conformal invariant local integrals. Our result bridges two different themes of the modern amplitudes program: positive geometry and Feynman integrals.
We give a geometric realization of the polyhedra governed by the structure of associative algebras with co-inner products, or more precisely, governed by directed planar trees. Our explicit realization of these polyhedra, which include the associahedra in a special case, shows in particular that these polyhedra are homeomorphic to balls. We also calculate the number of vertices of the lowest generalized associahedra, giving appropriate generalizations of the Catalan numbers.
Multi-loop scattering amplitudes/null polygonal Wilson loops in ${mathcal N}=4$ super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an $1+1$ dimensional subspace of Minkowski spacetime (or boundary of the $rm AdS_3$ subspace). Since the edges of a $2n$-gon with even and odd labels go along two different null directions, the kinematics is reduced to two copies of $G(2,n)/T sim A_{n{-}3}$. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping $A_2$ functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters $v, 1+v, w, 1+w$ of $A_1 times A_1$, there are two letters $v-w, 1- v w$ mixing the two sectors but they never appear together in the same term; these are the reduced version of four-mass-box algebraic letters. Evidence supporting our conjecture includes all known octagon amplitudes as well as new computations of multi-loop integrals in reduced kinematics. By leveraging this alphabet and conditions on first and last entries, we initiate a bootstrap program in reduced kinematics: within the remarkably simple space of overlapping $A_2$ functions, we easily obtain octagon amplitudes up to two-loop NMHV and three-loop MHV. We also briefly comment on the generalization to $2n$-gons in terms of $A_2$ functions and beyond.
We consider the Bremsstrahlung function associated to a 1/6-BPS Wilson loop in ABJM theory, with a cusp in the couplings to scalar fields. We non-trivially extend its recent four-loop computation at weak coupling to include non-planar corrections. We have recently proposed a conjecture relating this object to supersymmetric circular Wilson loops with multiple windings, which can be computed via localization. We find agreement between this proposal and the perturbative computation of the Bremsstrahlung function, including color sub-leading corrections. This supports the conjecture and hints at its validity beyond the planar approximation.
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