Yangians and Kac-moody loop algebras in superconformal gauge theory

We discuss the Yangian symmetry recently found for superconformal Yang Mills theory, and contrast it with the Kac-Moody loop algebra that can be defined in various integrable models. 1. Yangian Symmetry and Kac-Moody Algebras We describe an infinite-dimensional non-abelian symmetry algebra that is found in planar superconformal Yang-Mills theory. The generators have a basis J A n where J A 0 = J, J A 1 = Q, and J A n , n = 0, 1, 2, . We have given the first two generators special names, since only those occur in the defining relations for the algebra. The higher charges will arise from commutators of the Q’s. The algebra, called a Yangian Y (G), is an associative Hopf algebra [1]-[5] that satisfies: [J, J ] = f C J C , [J, Q ] = f C Q C , (1.1) and the Serre relations [Q, [Q , J ]] + [Q , [Q , J]] + [Q , [Q, J ]] = 1 24 ffffKLM{JD, JE , JF } , (1.2) [[Q, Q ], [J , Q]] + [[Q , Q], [J, Q]] , = 1 24 (ffffLMNf CD K +ffffLMNf AB K ){JG, JE , JF } , (1.3) for J taking values in the Lie algebra of an arbitrary semi-simple Lie group G. ∗Speaker. †Work partially supported by DOE grant DE-FG02-03ER-41262. j h w 2 0 0 3 27th Johns Hopkins Workshop on Current Problems in Particle Theory: Symmetries and Mysteries of M Theory L. Dolan Here {J, J , J } is the totally symmetrized product, {J, J , J } = JJJ + JJJ + JJJ +JJJ + JJJ + JJJ . (1.4) For SU(2) the relation (1.2) is trivial. For other cases such as SU(N) with N ≥ 3, the relation (1.2) implies the following one (1.3). In [6] we give the construction of the generators for the free superconformal Yang-Mills theory, and conjecture that they will exist to all orders of perturbation theory. We then check that the free generators commute with the planar one-loop anomalous dimension operator, i.e. the spin chain Hamiltonian [7]-[9]. Our construction is motivated by the existence of non-local symmetries in the AdS/CFT dual worldsheet theory of the type IIB superstring on AdS5 × S5, as conjectured [10] and found in [11]. Non-local symmetries in two-dimensional models and their possible role in four-dimensional gauge theory has a long history. In 1980, Polykov discussed “rings of glue” [12],[13], where the untraced Wilson loop of ordinary Yang-Mills theory, ψ[ξμ(s)] = Pe ∮ AμT adξμ , satisfies the on-shell functional equation δ δξμ (ψ−1 δ δξμ ψ) ∼ DF a μνT ξ̇ ∼ 0 , (1.5) with 0 ≤ μ ≤ 3. He used the similarity of (1.5) to the equations of motions of the twodimensional principal chiral model (PCM) ∂μ(g −1∂μg) = 0 , (1.6) where g is a matrix valued field talking values in a group G, gij(x, t) ∈ G, and now 0 ≤ μ ≤ 1. In 1978, Lusher and Pohlmeyer [14] had shown that the conserved current of a PCM Lagrangian L = 12Tr∂μg∂g, given by jμ(x, t) = g−1∂μg = jA μ (x, t)TA, which takes values in the Lie algebra of G, was also a flat connection ∂μjν − ∂νjμ + [jμ, jν ] = 0 . (1.7) From this they could construct an infinite number of non-local symmetries iteratively, since in two dimensions the conservation of the ordinary global current, ∂jμ = 0, implies jμ = μν∂ χ, and this in turn leads to the conservation of the non-local current j μ ≡ ∂μχ+ g−1∂μg χ . (1.8) The conserved charges associated with these currents are J = ∫∞ −∞ dxj 0A(x, t) Q = f BC ∫∞ −∞ dx ∫∞ x dyj 0B(x, t)j0c(y, t)− 2 ∫∞ −∞ dxj A 1 (x, t) . (1.9) A set of infinitesimal field transformations that leave (1.6) invariant g(x, t)→ g(x, t) + δ n g(x, t), (1.10)