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All loop BMN state energies from matrices

David Berenstein, Diego H. Correa, Samuel E. Vazquez

TL;DR

This paper develops a quantum-corrected matrix quantum mechanics for the s-wave scalars of ${\cal N}=4$ SYM on $S^3$ in the commuting-matrix sector, where ground-state eigenvalues form a 5-sphere $S^5$ in a six-dimensional phase space, identified with the $S^5$ of $AdS_5\times S^5$. Off-diagonal modes, interpreted as string bits, are treated perturbatively atop the commuting background, allowing a Born-Oppenheimer analysis that yields the BMN energy spectrum to all orders in the 't Hooft coupling in the large-$J$ limit; the resulting energies coincide with the BMN results and connect to the Beisert-Dippel-Staudacher magnon energies. The work also discusses tensions with the all-loop Bethe Ansatz and highlights the limits of the free-string-bit approximation, particularly at finite $J$, where the number of impurities may not be conserved. Overall, the approach provides a geometrical, eigenvalue-density framework for capturing strong-coupling string dynamics in $AdS_5\times S^5$ and offers insights into integrability and its possible limitations at strong coupling.

Abstract

We study a quantum corrected SO(6) invariant matrix quantum mechanics obtained from the s-wave modes of the scalars of N = 4 SYM on S^3. For commuting matrices, this model is believed to describe the 1/8 BPS states of the full SYM theory. In the large N limit the ground state corresponds to a distribution of eigenvalues on a S^5 which we identify with the sphere on the dual geometry AdS_5x S^5. We then consider non-BPS excitations by studying matrix perturbations where the off-diagonal modes are treated perturbatively. To a first approximation, these modes can be described by a free theory of "string bits" whose energies depend on the diagonal degrees of freedom. We then consider a state with two string bits and large angular momentum J on the sphere. In the large J limit we use a simple saddle point approximation to show that the energy of these states coincides precisely with the BMN spectrum to all orders in the 't Hooft coupling. We also find some new problems with the all loop Bethe Ansatz conjecture of the N=4 SYM planar spin chain model.

All loop BMN state energies from matrices

TL;DR

This paper develops a quantum-corrected matrix quantum mechanics for the s-wave scalars of SYM on in the commuting-matrix sector, where ground-state eigenvalues form a 5-sphere in a six-dimensional phase space, identified with the of . Off-diagonal modes, interpreted as string bits, are treated perturbatively atop the commuting background, allowing a Born-Oppenheimer analysis that yields the BMN energy spectrum to all orders in the 't Hooft coupling in the large- limit; the resulting energies coincide with the BMN results and connect to the Beisert-Dippel-Staudacher magnon energies. The work also discusses tensions with the all-loop Bethe Ansatz and highlights the limits of the free-string-bit approximation, particularly at finite , where the number of impurities may not be conserved. Overall, the approach provides a geometrical, eigenvalue-density framework for capturing strong-coupling string dynamics in and offers insights into integrability and its possible limitations at strong coupling.

Abstract

We study a quantum corrected SO(6) invariant matrix quantum mechanics obtained from the s-wave modes of the scalars of N = 4 SYM on S^3. For commuting matrices, this model is believed to describe the 1/8 BPS states of the full SYM theory. In the large N limit the ground state corresponds to a distribution of eigenvalues on a S^5 which we identify with the sphere on the dual geometry AdS_5x S^5. We then consider non-BPS excitations by studying matrix perturbations where the off-diagonal modes are treated perturbatively. To a first approximation, these modes can be described by a free theory of "string bits" whose energies depend on the diagonal degrees of freedom. We then consider a state with two string bits and large angular momentum J on the sphere. In the large J limit we use a simple saddle point approximation to show that the energy of these states coincides precisely with the BMN spectrum to all orders in the 't Hooft coupling. We also find some new problems with the all loop Bethe Ansatz conjecture of the N=4 SYM planar spin chain model.

Paper Structure

This paper contains 4 sections, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Gauged matrix quantum mechanics of commuting matrices
  3. A saddle point approximation to BMN state energies
  4. Discussion

Figures (2)

  • Figure 1: Diagram of forces for the statistical problem
  • Figure 2: Geometry of the string bit between two eigenvalues, string bit shown in red. The angle between the eigenvalues is $\theta = k/2J$