Lectures on Yangian symmetry

Abstract

In these introductory lectures we discuss the topic of Yangian symmetry from various perspectives. Forming the classical counterpart of the Yangian and an extension of ordinary Noether symmetries, first the concept of nonlocal charges in classical, two-dimensional field theory is reviewed. We then define the Yangian algebra following Drinfel'd's original motivation to construct solutions to the quantum Yang-Baxter equation. Different realizations of the Yangian and its mathematical role as a Hopf algebra and quantum group are discussed. We demonstrate how the Yangian algebra is implemented in quantum, two-dimensional field theories and how its generators are renormalized. Implications of Yangian symmetry on the twodimensional scattering matrix are investigated. We furthermore consider the important case of discrete Yangian symmetry realized on integrable spin chains. Finally we give a brief introduction to Yangian symmetry in planar, four-dimensional super Yang-Mills theory and indicate its impact on the dilatation operator and tree-level scattering amplitudes. These lectures are illustrated by several examples, in particular the two-dimensional chiral Gross-Neveu model, the Heisenberg spin chain and N = 4 superconformal Yang-Mills theory in four dimensions. This review arose from lectures given at the Young Researchers Integrability School at Durham University (UK).

Yangian bootstrap for conformal Feynman integrals

“…In fact, this can also be observed directly by inspecting the recurrence relations (45,46). Note that if we move x or y slightly away from zero, the sum in equation (48) will extend over all of Z and diverge.…”

Classifying integrable spin-1/2 chains with nearest neighbour interactions

“…This operator can be written in terms of densities that have interaction range 3. The explicit expression for all higher operators becomes more cumbersome, but they can be elegantly described in terms of the so-called boost operator B [2,3]. and is only defined on open spin chains of infinite length.…”

“…. can be recursively generated by means of the so-called Boost operator B[Q 2 ] defined by [2,3] B[Q 2 ] := ∞ n=−∞ nH n,n+1 .…”

Wilson loops and integrability

“…A. For an introduction to Yangian symmmetry, the reader is invited to consult references [76,77,78,79]. Below, we will extract Yangian symmetry generators for the Maldacena-Wilson loop at strong coupling from the finding that these charges vanish, which follows directly from the triviality of the monodromy.…”