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Quantum Mechanical Sectors in Thermal N=4 Super Yang-Mills on RxS^3

Troels Harmark, Marta Orselli

TL;DR

This work shows that thermal $\mathcal{N}=4$ SYM on $\mathbb{R}\times S^3$ with $SU(4)$ R-symmetry chemical potentials possesses three quantum-mechanical sectors that emerge near zero temperature and near-critical chemical potentials, namely the half-BPS, $SU(2)$, and $SU(2|3)$ sectors. In the planar limit, these sectors map to spin-chain systems, with the $SU(2)$ sector corresponding to the ferromagnetic $XXX_{1/2}$ Heisenberg chain, and an exact decoupling limit exists in which the full theory reduces to the corresponding quantum-mechanical Hamiltonian $H=D_0+\tilde{\lambda}D_2$ in the relevant sector, with $\tilde{\lambda}=\lambda/(1-\Omega)$. The paper derives both the free and one-loop partition functions, determines the (modified) Hagedorn temperatures for near-critical regions, and shows how the decouplings survive interactions, yielding explicit spin-chain descriptions and ground-state structures. These results illuminate the thermodynamics near the Hagedorn transition and establish a concrete bridge between four-dimensional gauge dynamics and integrable spin-chain physics, with potential links to pp-wave backgrounds and Hawking-Page transitions.

Abstract

We study the thermodynamics of U(N) N=4 Super Yang-Mills (SYM) on RxS^3 with non-zero chemical potentials for the SU(4) R-symmetry. We find that when we are near a point with zero temperature and critical chemical potential, N=4 SYM on RxS^3 reduces to a quantum mechanical theory. We identify three such critical regions giving rise to three different quantum mechanical theories. Two of them have a Hilbert space given by the SU(2) and SU(2|3) sectors of N=4 SYM of recent interest in the study of integrability, while the third one is the half-BPS sector dual to bubbling AdS geometries. In the planar limit the three quantum mechanical theories can be seen as spin chains. In particular, we identify a near-critical region in which N=4 SYM on RxS^3 essentially reduces to the ferromagnetic XXX_{1/2} Heisenberg spin chain. We find furthermore a limit in which this relation becomes exact.

Quantum Mechanical Sectors in Thermal N=4 Super Yang-Mills on RxS^3

TL;DR

This work shows that thermal SYM on with R-symmetry chemical potentials possesses three quantum-mechanical sectors that emerge near zero temperature and near-critical chemical potentials, namely the half-BPS, , and sectors. In the planar limit, these sectors map to spin-chain systems, with the sector corresponding to the ferromagnetic Heisenberg chain, and an exact decoupling limit exists in which the full theory reduces to the corresponding quantum-mechanical Hamiltonian in the relevant sector, with . The paper derives both the free and one-loop partition functions, determines the (modified) Hagedorn temperatures for near-critical regions, and shows how the decouplings survive interactions, yielding explicit spin-chain descriptions and ground-state structures. These results illuminate the thermodynamics near the Hagedorn transition and establish a concrete bridge between four-dimensional gauge dynamics and integrable spin-chain physics, with potential links to pp-wave backgrounds and Hawking-Page transitions.

Abstract

We study the thermodynamics of U(N) N=4 Super Yang-Mills (SYM) on RxS^3 with non-zero chemical potentials for the SU(4) R-symmetry. We find that when we are near a point with zero temperature and critical chemical potential, N=4 SYM on RxS^3 reduces to a quantum mechanical theory. We identify three such critical regions giving rise to three different quantum mechanical theories. Two of them have a Hilbert space given by the SU(2) and SU(2|3) sectors of N=4 SYM of recent interest in the study of integrability, while the third one is the half-BPS sector dual to bubbling AdS geometries. In the planar limit the three quantum mechanical theories can be seen as spin chains. In particular, we identify a near-critical region in which N=4 SYM on RxS^3 essentially reduces to the ferromagnetic XXX_{1/2} Heisenberg spin chain. We find furthermore a limit in which this relation becomes exact.

Paper Structure

This paper contains 14 sections, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Free thermal $\mathcal{N}=4$ SYM on $\mathbb{R} \times S^3$
  3. Calculation of the partition function
  4. Hagedorn temperature for non-zero chemical potentials
  5. Decoupling for near-critical chemical potentials
  6. Above the Hagedorn temperature
  7. Quantum mechanical sectors for near-critical chemical potential
  8. Low temperature limit and the Heisenberg spin chain
  9. Decoupling limit to exact quantum mechanical Hamiltonian
  10. One-loop partition function
  11. Discussion and conclusions
  12. Oscillator representation of $\mathcal{N}=4$ SYM
  13. The $XXX_{1/2}$ Heisenberg spin chain
  14. Computations for one-loop partition function

Figures (2)

  • Figure 1: The Hagedorn temperature $T_H$ as function of $\Omega$ in the case $(\Omega_1,\Omega_2,\Omega_3)=(\Omega,0,0)$.
  • Figure 2: The Hagedorn temperature $T_H$ as function of $\Omega$ in the two cases $(\Omega_1,\Omega_2,\Omega_3)=(\Omega,\Omega,0)$, displayed on the left, and $(\Omega_1,\Omega_2,\Omega_3)=(\Omega,\Omega,\Omega)$, displayed on the right.