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Plane-wave Matrix Theory from N=4 Super Yang-Mills on RxS^3

Nakwoo Kim, Thomas Klose, Jan Plefka

TL;DR

This work shows that the plane-wave matrix theory can be obtained from a consistent truncation of D=4 N=4 SYM on R×S^3 via Kaluza-Klein reduction. It demonstrates a precise one-loop equivalence between the plane-wave matrix model’s scalar sector and the 1-loop dilatation operator of N=4 SYM, while revealing a breakdown at two loops, as illustrated by Konishi-like states. The BMN limit analysis indicates that higher-loop discrepancies persist, suggesting that integrating out non-singlet KK modes or additional interactions are required for a full correspondence. These results illuminate the curved-background origin of the MM and hint at underlying integrable structures, with implications for holography and beyond.

Abstract

Recently a mass deformation of the maximally supersymmetric Yang-Mills quantum mechanics has been constructed from the supermembrane action in eleven dimensional plane-wave backgrounds. However, the origin of this plane-wave matrix theory in terms of a compactification of a higher dimensional Super Yang-Mills model has remained obscure. In this paper we study the Kaluza-Klein reduction of D=4, N=4 Super Yang-Mills theory on a round three-sphere, and demonstrate that the plane-wave matrix theory arises through a consistent truncation to the lowest lying modes. We further explore the relation between the dilatation operator of the conformal field theory and the hamiltonian of the quantum mechanics through perturbative calculations up to two-loop order. In particular we find that the one-loop anomalous dimensions of pure scalar operators are completely captured by the plane-wave matrix theory. At two-loop level this property ceases to exist.

Plane-wave Matrix Theory from N=4 Super Yang-Mills on RxS^3

TL;DR

This work shows that the plane-wave matrix theory can be obtained from a consistent truncation of D=4 N=4 SYM on R×S^3 via Kaluza-Klein reduction. It demonstrates a precise one-loop equivalence between the plane-wave matrix model’s scalar sector and the 1-loop dilatation operator of N=4 SYM, while revealing a breakdown at two loops, as illustrated by Konishi-like states. The BMN limit analysis indicates that higher-loop discrepancies persist, suggesting that integrating out non-singlet KK modes or additional interactions are required for a full correspondence. These results illuminate the curved-background origin of the MM and hint at underlying integrable structures, with implications for holography and beyond.

Abstract

Recently a mass deformation of the maximally supersymmetric Yang-Mills quantum mechanics has been constructed from the supermembrane action in eleven dimensional plane-wave backgrounds. However, the origin of this plane-wave matrix theory in terms of a compactification of a higher dimensional Super Yang-Mills model has remained obscure. In this paper we study the Kaluza-Klein reduction of D=4, N=4 Super Yang-Mills theory on a round three-sphere, and demonstrate that the plane-wave matrix theory arises through a consistent truncation to the lowest lying modes. We further explore the relation between the dilatation operator of the conformal field theory and the hamiltonian of the quantum mechanics through perturbative calculations up to two-loop order. In particular we find that the one-loop anomalous dimensions of pure scalar operators are completely captured by the plane-wave matrix theory. At two-loop level this property ceases to exist.

Paper Structure

This paper contains 12 sections, 83 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. $\mathcal{N}=4$ Super Yang-Mills on $\mathbb{R}\times S^3$
  3. Harmonic expansion on $S^3$
  4. Derivation of Plane-wave matrix theory
  5. Perturbation Theory
  6. The Berenstein-Maldacena-Nastase limit
  7. Discussion
  8. Dimensional split of Clifford algebras
  9. Killing spinors and Killing vectors
  10. Reduction of equation of motion
  11. A useful identity for gauge invariant states
  12. Effective 2-loop vertex

Figures (1)

  • Figure 1: Kaluza-Klein particle spectrum of $\mathcal{N}=4$ Super Yang-Mills on $\mathbb{R}\times S^3$. The states are labeled by representations of $\mathrm{SU}(2)_L \otimes \mathrm{SU}(2)_R \otimes \mathrm{SU}(4)$. One climbs upward to the left by acting with the supercharge $Q_L=(\mathbf{2},\mathbf{1},\mathbf{\bar{4}})$ and upward to the right with $Q_R=(\mathbf{1},\mathbf{2},\mathbf{4})$. The encircled states comprise the consistent truncation to the plane-wave matrix theory.