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Spin Matrix Theory: A quantum mechanical model of the AdS/CFT correspondence

Troels Harmark, Marta Orselli

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 infinite g limit of SMT can be mapped to the supersymmetric sector of string theory on AdS_5 x 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 non-relativistic 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.

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 infinite g limit of SMT can be mapped to the supersymmetric sector of string theory on AdS_5 x 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 non-relativistic 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.

Paper Structure

This paper contains 20 sections, 83 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction and summary
  2. Spin Matrix theory
  3. Definition of Spin Matrix theory
  4. Spin chains from Spin Matrix theory
  5. Spin Matrix theory from $\mathcal{N}=4$ SYM near critical points
  6. Partition function of $\mathcal{N}=4$ SYM
  7. Zero-temperature critical points in the grand canonical ensemble
  8. Low energy and non-relativistic limits (microcanonical ensemble)
  9. The $g \rightarrow \infty$ limit, supersymmetry and the AdS/CFT correspondence
  10. $SU(2)$ Spin Matrix theory at low temperature (nearly planar)
  11. $SU(2)$ Spin Matrix theory
  12. Planar limit
  13. Large $N$ and low temperature: Nearly-planar regime
  14. $SU(2)$ Spin Matrix theory at high temperature (non-planar)
  15. Free $SU(q)$ Spin Matrix theory at high temperature
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Phase diagram of $SU(2)$ Spin Matrix theory as function of the temperature $T$ and the coupling $g$. The stipled line marks the temperature $T_c$ where a partial deconfinement transition occurs. At zero coupling this meets the Hagedorn temperature $T_H$.
  • Figure 2: Illustration of our general philosophy for Spin Matrix theory. The diagram represents the regimes of $\mathcal{N}=4$ SYM. Towards the right one approaches the planar limit regime, depicted in blue. Towards the bottom one approaches the Spin Matrix theory regime depicted in red. The black area depicts the regime in which one finds black holes and D-branes.
  • Figure 3: Above $L_q(x)$ is plotted as function of $x$ for $q=2,3,4,5$. These plots are found numerically starting from $x=1/q$ and ending at $x=(1/q)^{1/11}$. The linear curve $(b_q)^{\frac{1}{q-1}} ( 1-x )$ has been plotted as estimate for the behavior of $L_q(x)$ for $x \rightarrow 1$.