Decoupling limits of N=4 super Yang-Mills on R x S^3
Troels Harmark, Kristjan R. Kristjansson, Marta Orselli
TL;DR
The paper develops a comprehensive framework for decoupling limits of ${ m N}=4$ SYM on ${ m R} imes S^3$ by turning on five chemical potentials for the ${ m SO}(4)$ and ${ m SU}(4)$ charges and approaching a near-critical, zero-temperature point. In these limits the grand canonical partition function reduces to a decoupled theory with an effective Hamiltonian ${D_0 + ilde{ ext{λ}} D_2}$ acting on a truncated Hilbert space ${ m H}$ of states with ${D_0=J}$; in the planar limit each decoupled sector maps to an integrable spin chain, enabling exact Bethe-ansatz spectra and thermodynamics. They classify 14 decoupling limits (12 non-trivial, 2 trivial) corresponding to subgroups of ${PSU(2,2|4)}$, of which 9 contain scalars and exhibit string-like thermodynamics at large ${ ilde{ ext{λ}}}$, while 3 scalar-free theories remain more challenging to analyze. The authors also formulate microcanonical decoupling limits and present a decoupling limit for pure Yang–Mills, showing integrable spin-chain structures in planar YM as well as discussions of the difficulties in finding a string dual for certain sectors. Together, these results provide a controlled setting to study weakly coupled gauge theories with string-like excitations and connect gauge theory dynamics to integrable spin chains and, in favorable regimes, to semiclassical string dynamics via AdS/CFT.
Abstract
We find new decoupling limits of N=4 super Yang-Mills (SYM) on R x S^3 with gauge group SU(N). These decoupling limits lead to decoupled theories that are much simpler than the full N=4 SYM but still contain many of its interesting features. The decoupling limits correspond to being in a near-critical region, near a point with zero temperature and critical chemical potentials. The new decoupling limits are found by generalizing the limits of hep-th/0605234 to include not only the chemical potentials for the SU(4) R-symmetry of N=4 SYM but also the chemical potentials corresponding to the SO(4) symmetry. In the decoupled theories it is possible to take a strong coupling limit in a controllable manner since the full effective Hamiltonian is known. For planar N=4 SYM on R x S^3 all the decoupled theories correspond to fully integrable spin chains. We study the thermodynamics of the decoupled theories and find the Hagedorn temperature for small and large values of the effective coupling. We find an alternative formulation of the decoupling limits in the microcanonical ensemble. This leads to a characterization of certain regimes of weakly coupled N=4 SYM in which there are string-like states. Finally, we find a similar decoupling limit for pure Yang-Mills theory, which for the planar limit leads to a fully integrable decoupled theory.