Spin Matrix theory: a quantum mechanical model of the AdS/CFT correspondence

Abstract

We introduce a new quantum mechanical theory called Spin Matrix theory (SMT). The theory is interacting with a single coupling constant g and is based on a Hilbert space of harmonic oscillators with a spin index taking values in a Lie (super)algebra representation as well as matrix indices for the adjoint representation of U(N ). We show that SMT describes N = 4 super-Yang-Mills theory (SYM) near zero-temperature critical points in the grand canonical phase diagram. Equivalently, SMT arises from non-relativistic limits of N = 4 SYM. Even though SMT is a non-relativistic quantum mechanical theory it contains a variety of phases mimicking the AdS/CFT correspondence. Moreover, the g → ∞ limit of SMT can be mapped to the supersymmetric sector of string theory on AdS 5 × S 5 . We study SU(2) SMT in detail. At large N and low temperatures it is a theory of spin chains that for small g resembles planar gauge theory and for large g a nonrelativistic string theory. When raising the temperature a partial deconfinement transition occurs due to finite-N effects. For sufficiently high temperatures the partially deconfined phase has a classical regime. We find a matrix model description of this regime at any coupling g. Setting g = 0 it is a theory of N 2 + 1 harmonic oscillators while for large g it becomes 2N harmonic oscillators.

Relating non-relativistic string theories

“…Furthermore, non-relativistic string theory appears to be related to limits of the AdS/CFT correspondence, as shown in particular in [12,16] by taking a further non-relativistic limit on the worldsheet. In this setting, the connection with nonrelativistic geometry elucidates the dual space-time formulation of Spin Matrix Theory (SMT) [43,44]. SMT is a quantum mechanical theory obtained by considering near-BPS limits of AdS 5 /CFT 4 , which reduces to tractable sectors in which quantum gravity effects are potentially easier to compute.…”

Nonrelativistic Corners of N=4 Supersymmetric Yang-Mills Theory

“…The spectrum of two-point functions is described by the dilatation operator D [9][10][11]. One can then follow the Spin Matrix theory limit procedure of [8] with D as starting point, and take the near-BPS limit in which only the one-loop contribution to D survives, and the Hilbert space reduces to the SU(1, 1) subsector. We show in this letter that this matches perfectly with H q .…”

Nonrelativistic pulsating strings

“…This serves as an important as well as interesting result in the context of nonrelativistic AdS/CFT correspondence which eventually indicates that there is a finite possibility that some of the conserved charges associated to certain specific sectors within N = 4 SYM might survive the SMT limit of [18] thereby preserving the underlying integrable structure in the near BPS bound. It remains to be an open question whether Kovacic's algorithm plays an useful tool in order to check integrability of NR strings in the presence of background fields.…”