Phases of a two-dimensional large-$N$ gauge theory on a torus

We consider two-dimensional large $N$ gauge theory with $D$ adjoint scalars on a torus, which is obtained from a $D+2$-dimensional pure Yang-Mills theory on $T^{D+2}$ with $D$ small radii. The two-dimensional model has various phases characterized by the holonomy of the gauge field around noncontractible cycles of the 2-torus. We determine the phase boundaries and derive the order of the phase transitions using a method developed in an earlier work (hep-th/0910.4526), which is nonperturbative in the ’t Hooft coupling and uses a $1/D$ expansion. We embed our phase diagram in the more extensive phase structure of the $D+2$-dimensional Yang-Mills theory and match with the picture of a cascade of phase transitions found earlier in lattice calculations [4]. We also propose a dual gravity system based on a Scherk-Schwarz compactification of a D2 brane wrapped on a 3-torus and find a phase structure which is similar to the phase diagram found in the gauge theory calculation.

Figure 1

A simple Feynman diagram contributing to (10).

Figure 2

Configurations of eigenvalue density $\rho (\theta )$ in the unitary matrix model. The left plot is the uniform distribution [corresponding to ${\rho }_1=0$ in (28)], the middle one is the nonuniform distribution ($|E/(2\xi {\rho }_1)|\ge 1$), and the right one is the gapped distribution ($|E/(2\xi {\rho }_1)|\le 1$).

Figure 3

Free energy (26) vs $\xi$ in the four phases. The gapped and nonuniform solutions here are numerically evaluated. Since $\xi$ is a monotonically increasing function of temperature [see (24)], the uniform distribution (Phase I) is stable at low temperatures and the gapped distribution (Phase III) is stable at higher temperature. A first-order phase transition between these two phases happens at ${\xi }_2$.

Figure 4

Phase structure of the 2d gauge theory at weak coupling (defined below). There are essentially four phases characterized by nonzero values of various Wilson lines. The inner region, with both Wilson lines nonzero, includes two additional phases in which the eigenvalue distribution is gapless but nonuniform. The orders of the phase transitions (1st, 2nd, 3rd) are indicated. Our analysis does not apply to the region enclosed by the dotted lines. Possible connections between the phase boundaries across this region are suggested in the inset (where boundaries of the intermediate phases are omitted for simplicity). A similar diagram is proposed in 3 for the model (3) with large mass for the adjoint scalars (see Sec. 6b for details). As we will see in Fig. 5, the gravity analysis conforms to the second pattern. We will see in Sec. 6a that the second pattern is also supported by lattice studies.

Figure 5

Conjectured phase structure of the gauge theory from the gravity analysis. We used a large $L_2$ and the (P,P,AP) spin structure of the fermions on the three-dimensional torus (with AP boundary condition on the Scherk-Schwarz (SS) circle). $\mathrm{Tr}W$ is the Polyakov loop operator along the SS circle (d16). The gravity analysis is reliable only in the region above the dotted line. The transitions in this diagram are predicted to be first order phase transitions. (See Appendix ).

Figure 6

Cascade of phase transitions for pure Yang-Mills theory on a $T^4$ (adapted from 4; see the points (a)-(e) above for more details). Our results in this paper, for the theory (3) with $D=2$, describe the indicated phases $2_c$, $3_c$ and $4_c$. The $0_c\to 1_c$ transition is found to be first order from lattice studies in 56; the $2_c\to 3_c$ transition is found to be first order from our analysis; the $3_c\to 4_c$ transition for an asymmetric torus is found in our analysis to be a double ($2\mathrm{nd}\mathrm{\text{order}}+3\mathrm{rd}\mathrm{\text{order}}$) transition for an appropriate parameter regime (see Fig. 4), although, it can be a single first-order transition at other regimes [6, 60].

Figure 7

Phase structure of the D2 brane on $T^3$ with (AP,AP,AP) boundary condition for large $L_2$. The gravity analysis is valid above the dotted line.

Figure 8

Phase structure of the D2 brane on $T^3$ with (AP,P,AP) boundary condition for large $L_2$. The gravity analysis is reliable only in the region above the dotted line.