Matching Higher Conserved Charges for Strings and Spins

TL;DR

The paper extends the known one-loop gauge/string energy matching in AdS/CFT to the entire infinite set of local commuting charges, constructing a gauge-theory resolvent from Bethe root densities and an exact string-theory generating function from Backlund transformations. It demonstrates explicit correspondences for folded and circular two-spin solutions, including BMN scaling via linear redefinitions and a Gauss-Landen transformation connecting the gauge and string descriptions. A full exact generating function on the string side is derived and shown to reproduce the gauge charges after appropriate normalization, providing a deep link between the integrable structures of the spin chain and the classical string sigma model. Overall, the work argues for a universal, operator-level matching of integrable hierarchies across the planar AdS/CFT correspondence, at least at the one-loop level for the full tower of charges.

Abstract

We demonstrate that the recently found agreement between one-loop scaling dimensions of large dimension operators in N=4 gauge theory and energies of spinning strings on AdS_5 x S^5 extends to the eigenvalues of an infinite number of hidden higher commuting charges. This dynamical agreement is of a mathematically highly intricate and non-trivial nature. In particular, on the gauge side the generating function for the commuting charges is obtained by integrable quantum spin chain techniques from the thermodynamic density distribution function of Bethe roots. On the string side the generating function, containing information to arbitrary loop order, is constructed by solving exactly the Backlund equations of the integrable classical string sigma model. Our finding should be an important step towards matching the integrable structures on the string and gauge side of the AdS/CFT correspondence.