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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.
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We compute three-point functions of single trace operators in planar N=4 SYM. We consider the limit where one of the operators is much smaller than the other two. We find a precise match between weak and strong coupling in the Frolov-Tseytlin classical limit for a very general class of classical solutions. To achieve this match we clarify the issue of back-reaction and identify precisely which three-point functions are captured by a classical computation.
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.
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.
This paper continues our studies in arXiV:1608.06452 [hep-th] of ${cal N}=1$ gauge theories in the strongly coupled regimes. We also consider here the ${cal N}=1$ SQCD-like theories with $SU(N_c)$ colors (and their Seibergs dual), with $N_F$ flavors of light quarks and $N_F^2$ additional colorless flavored scalars $Phi^j_i$, but now with $N_F$ in the range $N_F>3N_c$. The mass spectra of these direct and dual theories in various vacua are calculated within the dynamical scenario introduced by the author in [8]. It assumes that quarks in such ${cal N}=1$ SQCD-like theories without elementary colored adjoint scalars can be in two {it standard} phases only. These are either the HQ (heavy quark) phase where they are confined or the Higgs phase. Recall that this scenario satisfies all those tests which were used as checks of the Seiberg hypothesis about the equivalence of the direct and dual theories. Calculated mass spectra of the direct $SU(N_c)$ theory are compared to those of its Seibergs dual $SU(N_F-N_c)$ variant and appeared to be parametrically different.
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.
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