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We establish an all-loop conformal Yangian symmetry for the full set of planar amplitudes in the recently proposed integrable bi-scalar field theory in four dimensions. This chiral theory is a particular double scaling limit of γ-twisted weakly coupled N = 4 SYM theory. Each amplitude with a certain order of scalar particles is given by a single fishnet Feynman graph of disc topology cut out of a regular square lattice. The Yangian can be realized by the action of a product of Lax operators with a specific sequence of inhomogeneity parameters on the boundary of the disc. Based on this observation, the Yangian generators of level one for generic bi-scalar amplitudes are explicitly constructed. Finally, we comment on the relation to the dual conformal symmetry of these scattering amplitudes.This paper is dedicated to the memory of L.D.Faddeev.

Various classes of fishnet Feynman graphs are shown to feature a Yangian symmetry over the conformal algebra. We explicitly discuss scalar graphs in three, four and six spacetime dimensions as well as the inclusion of fermions in four dimensions. The Yangian symmetry results in novel differential equations for these families of largely unsolved Feynman integrals. Notably, the considered fishnet graphs in three and four dimensions dominate the correlation functions and scattering amplitudes in specific double scaling limits of planar, γ-twisted N = 4 super Yang-Mills or ABJM theory. Consequently, the study of fishnet graphs allows us to get deep insights into the integrability of the planar AdS/CFT correspondence.

We explore the idea to bootstrap Feynman integrals using integrability. In particular, we put the recently discovered Yangian symmetry of conformal Feynman integrals to work. As a prototypical example we demonstrate that the D-dimensional box integral with generic propagator powers is completely fixed by its symmetries to be a particular linear combination of Appell hypergeometric functions. In this context the Bloch-Wigner function arises as a special Yangian invariant in 4D. The bootstrap procedure for the box integral is naturally structured in algorithmic form. We then discuss the Yangian constraints for the six-point double box integral as well as for the related hexagon. For the latter we argue that the constraints are solved by a set of generalized Lauricella functions and we comment on complications in identifying the integral as a certain linear combination of these. Finally, we elaborate on the close relation to the Mellin-Barnes technique and argue that it generates Yangian invariants as sums of residues. *

We show that appropriately supersymmetrized smooth Maldacena-Wilson loop operators in N = 4 super Yang-Mills theory are invariant under a Yangian symmetry Y [psu(2, 2|4)] built upon the manifest superconformal symmetry algebra of the theory. The existence of this hidden symmetry is demonstrated at the one-loop order in the weak coupling limit as well as at leading order in the strong coupling limit employing the classical integrability of the dual AdS 5 × S 5 string description. The hidden symmetry generators consist of a canonical non-local second order variational derivative piece acting on the superpath, along with a novel local path dependent contribution. We match the functional form of these Yangian symmetry generators at weak and strong coupling and find evidence for an interpolating function. Our findings represent the smooth counterpart to the Yangian invariance of scattering superamplitudes dual to light-like polygonal super Wilson loops in the N = 4 super Yang-Mills theory.

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We perform a detailed study of the Yangian symmetry of smooth supersymmetric Maldacena-Wilson loops in planar N = 4 super Yang-Mills theory. This hidden symmetry extends the global superconformal symmetry present for these observables. A gauge-covariant action of the Yangian generators on the Wilson line is established that generalizes previous constructions built upon path variations. Employing these generators the Yangian symmetry is proven for general paths in non-chiral N = 4 superspace at the first perturbative order. The bi-local piece of the level-one generators requires the use of a regulator due to divergences in the coincidence limit. We perform regularization by point splitting in detail, thereby constructing additional local and boundary contributions as regularization for all level-one Yangian generators. Moreover, the Yangian algebra at level one is checked and compatibility with local kappa-symmetry is established. Finally, the consistency of the Yangian symmetry is shown to depend on two properties: the vanishing of the dual Coxeter number of the underlying superconformal algebra and the existence of a novel superspace "G-identity" for the gauge field theory. This tightly constrains the conformal gauge theories to which integrability can possibly apply.

A D3-D5 intersection gives rise to a defect CFT, wherein the rank of the gauge group jumps by k units across a domain wall. The one-point functions of local operators in this setup map to overlaps between on-shell Bethe states in the underlying spin chain and a boundary state representing the D5 brane. Focussing on the k = 1 case, we extend the construction to gluonic and fermionic sectors, which was prohibitively difficult for k > 1. As a byproduct, we test an all-loop proposal for the one-point functions in the su(2) sector at the half-wrapping order of perturbation theory.