Phases of $N=\infty$ QCD-like gauge theories on $S^3\times S^1$ and nonperturbative orbifold-orientifold equivalences

We study the phase diagrams of $N=\infty$ vectorlike, asymptotically free gauge theories as a function of volume, on $S^3\times S^1$. The theories of interest are the ones with fermions in two index representations [adjoint, (anti)symmetric, and bifundamental abbreviated as QCD(adj), QCD(AS/S), and QCD(BF)], and are interrelated via orbifold or orientifold projections. The phase diagrams reveal interesting phenomena such as disentangled realizations of chiral and center symmetry, confinement without chiral symmetry breaking, zero temperature chiral transitions, and in some cases, exotic phases which spontaneously break the discrete symmetries such as $C$, $P$, $T$ as well as $CPT$. In a regime where the theories are perturbative, the deconfinement temperature in SYM, and QCD(AS/S/BF) coincide. The thermal phase diagrams of thermal orbifold QCD(BF), orientifold QCD(AS/S), and $\mathcal{N}=1$ SYM coincide, provided charge conjugation symmetry for QCD(AS/S) and ${\mathbb{Z}}_2$ interchange symmetry of the QCD(BF) are not broken in the phase continuously connected to the ${\mathbb{R}}^4$ limit. When the $S^1$ circle is endowed with periodic boundary conditions, the (nonthermal) phase diagrams of orbifold and orientifold QCD are still the same, however, both theories possess chirally symmetric phases which are absent in $\mathcal{N}=1$ SYM. The match and mismatch of the phase diagrams depending on the spin structure of fermions along the $S^1$ circle is naturally explained in terms of the necessary and sufficient symmetry realization conditions which determine the validity of the nonperturbative orbifold-orientifold equivalence.

Figure 1

Schematic phase diagrams of $U(N)$ $\mathcal{N}=1$ SYM as a function of the radii $R_{S^1}(\beta )$ and $R_{S^3}$. The left figure is the nonthermal case where fermions obey periodic boundary conditions along $S^1$, and the right one is the thermal case corresponding to antiperiodic boundary conditions for fermions. Thermal $\mathcal{N}=1$ super-Yang-Mills has two confining low temperature phases in which temporal center symmetry is unbroken. The discrete chiral symmetry is spontaneously broken at sufficiently large radius $R_{S^3}>R_{\chi }$ while small radius $R_{S^3}<R_{\chi }$ shows confinement without chiral symmetry breaking. The theory also possesses a deconfined high temperature phase with unbroken chiral symmetry. The nonthermal compactification respects spatial center symmetry throughout the whole phase diagram. Regardless of the size of the $S^1$ circle, for sufficiently small radius, $R_{S^3}<R_{\chi }$, the theory is in a chirally symmetric phase. For sufficiently large radius, $R_{S^3}>R_{\chi }$, the discrete chiral symmetry is believed to be spontaneously broken. These sketches depict the simplest scenario in which only a single phase transition separates various phases; more complicated scenarios with distinct deconfinement, chiral restoration transitions are also possible.

Figure 2

The distribution of the eigenvalues of the spatial (and temporal) Wilson lines for $U(N{)}_1\times U(N{)}_2$ QCD(BF) theory. The red (light) and blue (dark) points label eigenvalues of the Wilson line $U_1$ and $U_2$, respectively. At small spatial $S^1$ circle, both spatial center symmetry ${\mathcal{G}}_s$ and $\mathcal{I}$ interchange symmetry of orbifold are broken. (Wilson lines $U_1$ and $U_2$ are antiparallel.) At small temporal $S^1$, the temporal center symmetry ${\mathcal{G}}_t$ is broken, but not $\mathcal{I}$. (Wilson lines are parallel.) At large radius of $S^1$ (either thermal or spatial), the center symmetry is restored, as well as $\mathcal{I}$ in the case of spatial $S^1$. Exactly analogously, one may regard the same picture for orientifold $U(N)$ QCD(AS/S). In this case, the role of the $\mathcal{I}$ symmetry of orbifold is taken by charge conjugation symmetry $\mathcal{C}$, and the role of the Wilson line $U_2$ is played by the mirror image, $U^{*}$. Therefore, the red (blue) points label eigenvalues of the Wilson line $U$ (its complex conjugate $U^{*}$). At small spatial $S^1$ circle, both ${\mathcal{G}}_s$ and $\mathcal{C}$ of QCD(AS/S) are broken (antiparallel $U$ and $U^{*}$). At small temporal $S^1$, ${\mathcal{G}}_t$ is broken, but not $\mathcal{C}$ (parallel $U$ and $U^{*}$). At large radius of $S^1$, all symmetries mentioned above are restored. (The blue and red points are split to guide the eye.) The orientifold QCD(AS/S) will be discussed in detail in the next section. We should also point out that for high temperature SYM, the eigenvalues clump just like QCD(AS/S/BF). At low temperature, they delocalize and distribute uniformly. On the other hand, for spatial $S^1$ circle, the eigenvalues are uniformly distributed regardless of the size of the $S^1$.

Figure 3

Schematic phase diagram of $U(N)$ QCD(AS/S/BF) with periodic boundary conditions for fermions (left) and with thermal (antiperiodic) boundary conditions (right) as a function of radii $R_{S^1}(\beta )$ and $R_{S^3}$. Thermal QCD(AS/S/BF) has two confining low temperature phases in which temporal center symmetry ${\mathcal{G}}_t$ is unbroken. These two phases are distinguished by the discrete chiral symmetry realization. The theory is chirally symmetric at sufficiently small radius $R_{S^3}<R_{\chi }$ and asymmetric at large radius $R_{S^3}>R_{\chi }$. These theories also possess a deconfined high temperature phase with broken ${\mathcal{G}}_t$ and unbroken chiral symmetry. For the nonthermal compactification where $S^1$ is a spatial circle, the symmetry realizations are essentially the same modulo the replacement of temporal spatial symmetry with the spatial one ${\mathcal{G}}_s$. As explained in the text, the underlying reason for unbroken chiral symmetry in the spatial symmetry broken phase is a one-loop quantum effect, unlike the thermal compactification where it is due do tree-level mass of fermions. These sketches depict the simplest consistent scenarios. In the thermal case, the $\mathcal{I}$ and $\mathcal{C}$ symmetries necessary for the validity of large $N$ equivalence are unbroken, explaining the matching of phase diagram to $\mathcal{N}=1$ SYM. In the nonthermal case, the phase which breaks ${\mathcal{G}}_s$ also breaks $\mathcal{I}$ and $\mathcal{C}$, explaining the mismatch to $\mathcal{N}=1$ SYM within that phase.

Figure 4

The $N$ dependence of the $\mathcal{C}$-breaking in two natural generalization of the $SU(3)$ QCD on ${\mathbb{R}}^3$ times small spatial $S^1$. For $SU(3)$ QCD with few fermions, the two minima are located at $e^{\pm i(2\pi /3)}$. For $SU(2N+1)$ QCD with fundamental fermions, the two minima gradually approach each other with increasing $N$. The $N=\infty$ limit has a unique vacuum with unbroken $\mathcal{C}$. For $SU(2N+1)$ QCD(AS), the two minima approaches to antipodal points $\pm e^{i\pi /2}$ and $N=\infty$ theory breaks $\mathcal{C}$ spontaneously, just like $SU(3)$.

Figure 5

Multiflavor generalization of the phase diagram given in Figs. 1, 3. The left figure depicts how the addition of multiple adjoint fermion flavors alters QCD(adj) with periodic boundary conditions by gradually pushing the chiral transition radius to larger values. The right figure may be thought of as corresponding to multiflavor QCD(AS/S/BF/adj) with thermal boundary conditions or QCD(AS/S/BF) with periodic boundary conditions. As seen in the figure, the net effect of adding flavors is to increase the slope of the phase transition line (associated with the appropriate center symmetry realization) in the perturbative regime and to push the nonperturbative strong confinement length to higher values (or to lower energy scales). If the theory were genuinely conformal, then there would not be such a scale and the confinement deconfinement phase boundary would not bend down around any particular scale. The phase boundaries for $n_f=2$, 3, 4 should be between $n_f=1$ and $n_f=5$ in increasing order. (The $n_f\ge 6$ theories are not asymptotically free and are not examined in this paper.)

Figure 6

The vacuum structure of the $U(N)$ QCD-like gauge theories for $1\le n_f\le 5$. The islands represents coset spaces associated with continuous symmetry breaking. There are $h=N-2$, $N+2$, $N$, $N$ such islands (corresponding to QCD(AS/S/BF/adj), respectively) distinguished by the phase of the determinant of the condensate, which is a ${\mathbb{Z}}_h$ valued object. Since these theories show both discrete and continuous chiral symmetry breaking, there exists both Goldstone bosons (for $n_f>1$) and domain walls $n_f\ge 1$. Notice that the coset spaces shrink to points in the case $n_f=1$, and the Goldstone bosons disappear.