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On the Integrability of large N Plane-Wave Matrix Theory

Thomas Klose, Jan Plefka

TL;DR

The paper investigates whether the integrable structure observed at planar, one-loop level in N=4 SYM persists at higher (three) loops within a reduced, large-N plane-wave matrix theory in the SU(2) scalar subsector. Using degenerate perturbation theory and a careful mapping to the SYM dilatation operator, the authors demonstrate three-loop integrability in the planar limit, contingent on a renormalized mass-coupling relation, and show the resulting spectra exhibit parity-degenerate pairs due to a hidden commuting charge. They provide explicit planar dilatation operators and parity-degenerate state constructions, and verify integrability remains robust under certain nonsupersymmetric deformations, though not universally across all bosonic deformations. The work thus supports the conjectured three-loop integrability of planar N=4 SYM and clarifies the role of mass renormalization and deformation stability in the corresponding large-N quantum mechanics.

Abstract

We show the three-loop integrability of large N plane-wave matrix theory in a subsector of states comprised of two complex light scalar fields. This is done by diagonalizing the theory's Hamiltonian in perturbation theory and taking the large N limit. At one-loop level the result is known to be equal to the Heisenberg spin-1/2 chain, which is a well-known integrable system. Here, integrability implies the existence of hidden conserved charges and results in a degeneracy of parity pairs in the spectrum. In order to confirm integrability at higher loops, we show that this degeneracy is not lifted and that (corrected) conserved charges exist. Plane-wave matrix theory is intricately connected to N=4 Super Yang-Mills, as it arises as a consistent reduction of the gauge theory on a three-sphere. We find that after appropriately renormalizing the mass parameter of the plane-wave matrix theory the effective Hamiltonian is identical to the dilatation operator of N=4 Super Yang-Mills theory in the considered subsector. Our results therefore represent a strong support for the conjectured three-loop integrability of planar N=4 SYM and are in disagreement with a recent dual string theory finding. Finally, we study the stability of the large N integrability against nonsupersymmetric deformations of the model.

On the Integrability of large N Plane-Wave Matrix Theory

TL;DR

The paper investigates whether the integrable structure observed at planar, one-loop level in N=4 SYM persists at higher (three) loops within a reduced, large-N plane-wave matrix theory in the SU(2) scalar subsector. Using degenerate perturbation theory and a careful mapping to the SYM dilatation operator, the authors demonstrate three-loop integrability in the planar limit, contingent on a renormalized mass-coupling relation, and show the resulting spectra exhibit parity-degenerate pairs due to a hidden commuting charge. They provide explicit planar dilatation operators and parity-degenerate state constructions, and verify integrability remains robust under certain nonsupersymmetric deformations, though not universally across all bosonic deformations. The work thus supports the conjectured three-loop integrability of planar N=4 SYM and clarifies the role of mass renormalization and deformation stability in the corresponding large-N quantum mechanics.

Abstract

We show the three-loop integrability of large N plane-wave matrix theory in a subsector of states comprised of two complex light scalar fields. This is done by diagonalizing the theory's Hamiltonian in perturbation theory and taking the large N limit. At one-loop level the result is known to be equal to the Heisenberg spin-1/2 chain, which is a well-known integrable system. Here, integrability implies the existence of hidden conserved charges and results in a degeneracy of parity pairs in the spectrum. In order to confirm integrability at higher loops, we show that this degeneracy is not lifted and that (corrected) conserved charges exist. Plane-wave matrix theory is intricately connected to N=4 Super Yang-Mills, as it arises as a consistent reduction of the gauge theory on a three-sphere. We find that after appropriately renormalizing the mass parameter of the plane-wave matrix theory the effective Hamiltonian is identical to the dilatation operator of N=4 Super Yang-Mills theory in the considered subsector. Our results therefore represent a strong support for the conjectured three-loop integrability of planar N=4 SYM and are in disagreement with a recent dual string theory finding. Finally, we study the stability of the large N integrability against nonsupersymmetric deformations of the model.

Paper Structure

This paper contains 7 sections, 47 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and Conclusions
  2. Plane-wave Matrix Theory
  3. Perturbation Theory
  4. Energy Operator in $\mathrm{SU}(2)$ subsector
  5. Planar limit
  6. Integrability
  7. Nonsupersymmetric Deformations

Figures (1)

  • Figure 1: The complete spectrum of unprotected states in the $\mathrm{SU}(2)$ subsector up to bare dimension 10 at three-loop order for $\Lambda=0.125$. States of positive (negative) parity are depicted as dark (light) lines. The states are grouped according to the Dynkin labels of their representation. Note that whenever states of positive and negative parity exist within a representation, then they are degenerate. The explicit results for the energy shifts are spelled out in the apppendix.