Towards the exact dilatation operator of N=4 super Yang-Mills theory

TL;DR

This work advances an exact, all-orders understanding of the dilatation operator $D$ in planar ${\cal N}=4$ SYM by organizing it into independent spin-interaction blocks $D^{(n)}$ and determining the linear-in-$Q$ part $D^{(1)}$ from the regular BMN continuum limit and BMN spectrum. The authors obtain closed-form, all-orders-in-$\lambda$ expressions for the coefficients via hypergeometric functions ${}_2F_1$, predicting $D^{(1)}$-driven anomalous dimensions that grow as $\sqrt{\lambda}$ for long operators and hinting at integrability of the resulting long-range spin chain. Finite-length corrections and the Konishi operator check are discussed, indicating that higher-order $D^{(n)}$ terms may be necessary to recover the expected short-operator strong-coupling behavior. The work links gauge-theory dilatation operators to long-range spin chains and semiclassical string dynamics in AdS$_5\times$S$^5$, with potential implications for integrability and exact spectra in ${\cal N}=4$ SYM.

Abstract

We investigate the structure of the dilatation operator D of planar N=4 SYM in the sector of single trace operators built out of two chiral combinations of the 6 scalars. Previous results at low orders in `t Hooft coupling λsuggest that D has the form of an SU(2) spin chain Hamiltonian with long range multiple spin interactions. Instead of the usual perturbative expansion in powers of λ, we split D into parts D^(n) according to the number n of independent pairwise interactions between spins at different sites. We determine the coefficients of spin-spin interaction terms in D^(1) by imposing the condition of regularity of a BMN-type scaling limit. For long spin chains, these coefficients turn out to be expressible in terms of hypergeometric functions of λ, which have regular expansions at both small and large values of λ. This suggest that anomalous dimensions of ``long'' operators in the two-scalar sector should generically scale as square root of λat large coupling, i.e. in the same way as energies of semiclassical states in dual AdS5 x S5 string theory. We speculate that D^(1) may be a Hamiltonian of a new integrable spin chain.