Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bose-Fermi Degeneracies in Large $N$ Adjoint QCD

Gokce Basar, Aleksey Cherman, David A. McGady

TL;DR

This work analyzes the large-$N$ limit of $SU(N)$ adjoint QCD with $N_f$ massless adjoint Majorana fermions on $S^3 \times S^1$, in the weakly coupled, small-$S^3$ regime. By computing the $( -1)^F$-twisted partition function and its single-particle content, the authors uncover strong boson–fermion spectral cancellations that suppress Hagedorn instabilities for any spatial circle size $L$, while preserving confinement and enabling large-$N$ volume independence. The study relates these cancellations to misaligned supersymmetry and provides evidence for emergent fermionic symmetries at large $N$, particularly for $N_f=1$, with implications for the structure of the spectrum and the role of curvature. The results establish a coherent picture in which adjoint QCD stays confining and $L$-independent at large $N$ and motivate further exploration of emergent symmetries and potential string-dual interpretations at finite coupling and for $N_f>1$.

Abstract

We analyze the large $N$ limit of adjoint QCD, an $SU(N)$ gauge theory with $N_f$ flavors of massless adjoint Majorana fermions, compactified on $S^3 \times S^1$. We focus on the weakly-coupled confining small-$S^3$ regime. If the fermions are given periodic boundary conditions on $S^1$, we show that there are large cancellations between bosonic and fermionic contributions to the twisted partition function. These cancellations follow a pattern previously seen in the context of misaligned supersymmetry, and lead to the absence of Hagedorn instabilities for any $S^1$ size $L$, even though the bosonic and fermionic densities of states both have Hagedorn growth. Adjoint QCD stays in the confining phase for any $L \sim N^0$, explaining how it is able to enjoy large $N$ volume independence for any $L$. The large $N$ boson-fermion cancellations take place in a setting where adjoint QCD is manifestly non-supersymmetric at any finite $N$, and are consistent with the recent conjecture that adjoint QCD has emergent fermionic symmetries in the large $N$ limit.

Bose-Fermi Degeneracies in Large $N$ Adjoint QCD

TL;DR

This work analyzes the large- limit of adjoint QCD with massless adjoint Majorana fermions on , in the weakly coupled, small- regime. By computing the -twisted partition function and its single-particle content, the authors uncover strong boson–fermion spectral cancellations that suppress Hagedorn instabilities for any spatial circle size , while preserving confinement and enabling large- volume independence. The study relates these cancellations to misaligned supersymmetry and provides evidence for emergent fermionic symmetries at large , particularly for , with implications for the structure of the spectrum and the role of curvature. The results establish a coherent picture in which adjoint QCD stays confining and -independent at large and motivate further exploration of emergent symmetries and potential string-dual interpretations at finite coupling and for .

Abstract

We analyze the large limit of adjoint QCD, an gauge theory with flavors of massless adjoint Majorana fermions, compactified on . We focus on the weakly-coupled confining small- regime. If the fermions are given periodic boundary conditions on , we show that there are large cancellations between bosonic and fermionic contributions to the twisted partition function. These cancellations follow a pattern previously seen in the context of misaligned supersymmetry, and lead to the absence of Hagedorn instabilities for any size , even though the bosonic and fermionic densities of states both have Hagedorn growth. Adjoint QCD stays in the confining phase for any , explaining how it is able to enjoy large volume independence for any . The large boson-fermion cancellations take place in a setting where adjoint QCD is manifestly non-supersymmetric at any finite , and are consistent with the recent conjecture that adjoint QCD has emergent fermionic symmetries in the large limit.

Paper Structure

This paper contains 22 sections, 79 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Properties of large $N$ adjoint QCD
  3. Hagedorn instability
  4. Large $N$ volume independence
  5. The tension
  6. Utility of $S^3 \times S^1$ compactifications
  7. Large $N$ partition functions on $S^3 \times S^1$
  8. Single particle partition functions
  9. Twisted and thermal partition functions of adjoint QCD
  10. Instabilities and their disappearance
  11. Thermal compactification and the Hagedorn instability
  12. Spatial compactification and the disappearance of the Hagedorn instability
  13. Twisted Casimir energy in adjoint QCD
  14. Relation to misaligned supersymmetry
  15. Emergent fermionic symmetries in adjoint QCD on $S^3 \times S^1$
  16. ...and 7 more sections

Figures (5)

  • Figure 1: (Color Online.) This plot summarizes much of the paper. The red dots are singularities of the thermal (top row) and twisted (bottom row) partition functions of adjoint QCD as a function of complex temperature $Q = e^{-L/2R}$ for $N_f=1$ (left column) and $N_f=2$ (right column). The absence of singularities on the positive real axis (except at $Q=1$, corresponding to $L=0$) is tied to the absence of Hagedorn instabilities in the twisted partition function. The evident $Q \to -Q$ symmetry relating the singularity structure of the twisted and thermal partition follows from \ref{['eq:ZNfThermal']} and \ref{['eq:ZNfTwisted']}. For visual clarity we only show singularities arising from the first $30$ terms in \ref{['eq:ZNfThermal']} and the first $45$ terms in \ref{['eq:ZNfTwisted']}.
  • Figure 2: Logarithms of the coefficients of $Q^n$ of the series expansion of the twisted partition function $\tilde{Z}(Q)$, with $+/-$ signs for bosons/fermions. The coefficients of even/odd powers of $Q$ are boson/fermion degeneracy factors. We draw lines between successive data points as a visual aid to make the oscillations easier to follow. The linearity of the envelope function means that the bosonic and fermionic densities of states both have Hagedorn growth, while the symmetry of the envelope function around zero is responsible for the elimination of Hagedorn instabilities in the twisted partition function.
  • Figure 3: Logarithms of the coefficients of $Q^n$ of the series expansion of the thermal partition function $Z(Q)$ for $N_F=2$. The bosonic and fermionic state degeneracy factors have identical asymptotic scaling with $n$.
  • Figure 4: Plot of the $N_f=2$ partition function on $S^3_R \times S^1_L$ when $R \Lambda \ll 1$, which illustrates the lack of invariance under $L\rightarrow {c\over L}$ for any $c >0$. The fact that the confined-phase twisted partition function is well-defined and continuous for any $L \sim N^0$ is a consequence of massive cancellations between bosons and fermions.
  • Figure 5: Behavior of $L^2 C(L)$ at small $L$ for $N_f=2$ (as an example) as a function of a cutoff $M$ on the upper end of the sum in \ref{['eq:ZSTrepeated']}.