Planar Script N = 4 gauge theory and the Hubbard model

Abstract

Recently it was established that a certain integrable long-range spin chain describes the dilatation operator of N = 4 gauge theory in the su(2) sector to at least three-loop order, while exhibiting BMN scaling to all orders in perturbation theory. Here we identify this spin chain as an approximation to an integrable short-ranged model of strongly correlated electrons: The Hubbard model.Recently it was discovered that the planar one-loop dilatation operator of supersymmetric N = 4 gauge theory is completely integrable [1,2]. This means that its spectrum may be exactly determined in the form of a set of non-linear Bethe equations. Evidence was found that this integrability is preserved beyond the one-loop approximation, and it was conjectured that the dilatation operator might be integrable to all orders in perturbation theory [3]. Given the usually benign, analytic nature of planar perturbation theory, one may then even hope for the theory's complete large N integrability at all values of the Yang-Mills coupling constant.Deriving the dilatation operator from the field theory, and subsequently demonstrating its integrability, is not easy. The three-loop planar dilatation operator in the maximally compact su(2|3) sector was found by Beisert, up to two unknown constants, by algebraic means in [4]. These constants could later be unequivocally fixed from the results of a solid field theory calculation of Eden, Jarczak and Sokatchev [5]. This basically completely determines the planar dilatation operator in this large sector up to three loops. Its restriction to su(2) agrees with the original conjecture of [3]. Three-loop integrability in su (2) was then demonstrated in [6] by embedding the dilatation operator into an integrable long-range spin chain due to Inozemtsev, and a three-loop Bethe ansatz was derived.The Inozemtsev spin chain exhibited a four-loop breakdown of BMN scaling [8]. This scaling behavior seemed, and still seems, to be a desirable, albeit unproven, property of perturbative gauge theory. Mainly for that reason an alternative long-range spin chain, differing from the Inozemtsev model at and beyond four loops, was conjectured to exist in [7]. Its construction principles were an extension of the ones already laid out in [3]:(1) Structural consistency with general features of Yang-Mills perturbation theory, (2) perturbative integrability and (3) qualitative BMN scaling. The model's Hamiltonian is only known up to five loops, and increases exponentially in complexity with the loop order. In striking contrast, a very compact Bethe ansatz may be conjectured for the model and shown to diagonalize the Hamiltonian to the known, fifth, order. The conjecture reads e ip k L = M j=1 j =k

Wrapping interactions at strong coupling: The giant magnon

“…The curious numerical prefactor includues e = exp(1), the base of natural logarithm. This result is even more interesting when one compares it to finite size corrections for the magnon computed within the Hubbard model approach [36] which gives [36,35] …”

The AdS5× S5superstring in light-cone gauge and its Bethe equations

“…In appendix A we introduce the Dirac matrices for SO(5) γ s , s = 1, 2, 3, 4 and γ 5 ≡ , which we all take to be Hermitian. These matrices obey the relations 17) and, therefore, span the orthogonal complement to the Lie algebra so(5). The same matrices can be used to build the set of Dirac matrices for so(4, 1); one takes {i , γ a } with a = 1, 2, 3, 4.…”

“…This conjecture has been by now extended to the full P SU (2, 2|4) case in the gauge theory [16] and is believed to hold in an asymptotic sense, where the classical scaling dimension of the operator in question determines the loop order to which a prediction is made by the Bethe equations. Moreover, in the prominent minimal compact, bosonic subsector of SU (2) the conjectured gauge theory Bethe equations [15] were recently shown to arise microscopically from the well-known Hubbard model at half filling [17]. Whether this surprising connection is indeed fully realized beyond three loops remains to be seen; for this a four-loop computation on the gauge theory side would have to be performed.…”

Constructing the AdS/CFT dressing factor

“…In [30,31] the classical algebraic curve for the string moving in S 5 ⊂ AdS 5 ×S 5 was obtained as the classical limit of the quantum nested Bethe ansatz equations coming from the Zamolodchikov's bootstrap procedure [32] where in addition to the magnons described by the roots u j we have the rapidities θ α of the relativistic particles with O(N) isotopic degree of freedom. In [33] it was observed that the BDS equations [6] mentioned in the introduction (1) could be obtained from the Hubbard model where the electron has a spin which can create spin waves described by the roots u j , but also has momentum p α . In both cases the introduction of the extra level simplifies the structure of the Bethe equations considerably.…”