Duality and Higher Temperature Phases of Large $N$ Chern-Simons Matter Theories on $S^2 \times S^1$

TL;DR

This work analyzes the thermal phase structure of large $N$ Chern-Simons theories coupled to fundamental matter on $S^2\times S^1$, showing the partition function reduces to a capped unitary matrix model and solving the saddle-point equations across no-gap, lower-gap, upper-gap, and two-gap phases. It provides exact density functions and free energies for the regular fermion, critical boson, and ${\cal N}=2$ SUSY CS matter theories, and demonstrates a complete level-rank duality network: RF$\leftrightarrow$CB duality, and SUSY self-duality, including analytic proofs in the two-gap (GWW-type) sector. Central to the analysis is the cut-functional approach with a $h(u)$-based Riemann-Hilbert problem that yields precise phase-transition curves and a quadruple critical point where all four phases coexist. The results illuminate the holographic interpretation in higher-spin gravity and provide a robust framework for validating dualities in CS-matter systems at finite temperature and large $N$.

Abstract

It has been recently demonstrated that the thermal partition function of any large $N$ Chern-Simons gauge theories on $S^2$, coupled to fundamental matter, reduces to a capped unitary matrix model. The matrix models corresponding to several specific matter Chern-Simons theories at temperature $T$ were determined in arXiv:1301.6169. The large $N$ saddle point equations for these theories were determined in the same paper, and were solved in the low temperature phase. In this paper we find exact solutions for these saddle point equations in three other phases of these theories and thereby explicitly determine the free energy of the corresponding theories at all values of $T^2/N$. As anticipated on general grounds in arXiv:1301.6169, our results are in perfect agreement with conjectured level rank type bosonization dualities between pairs of such theories.

Figures (11)

• Figure 1: The eigenvalue distribution $\rho(\alpha)$ in the no gap phase Fig.\ref{['nocut']}, in the lower gap phase Fig.\ref{['gwwfig']}, in the upper gap phase Fig.\ref{['lbig2']} at $\lambda = 0.51$, and in the two gap phase Fig.\ref{['twocut']}. These are the graphs calculated in the GWW toy model. We can see that the eigenvalue density in the upper gap phase Fig.\ref{['lbig2']} is saturated from above as $\rho(\alpha) = \frac{1}{2 \times 0.51 \pi}$. We can also see that the two gap density function Fig.\ref{['twocut']} has both a lower gap and an upper gap.
• Figure 2: Phase diagram of the regular fermion theory. Here $(\lambda_c,\zeta_c) = (0.596967, 2.86454)$ is the quadruple phase transition point where the four phases (no gap, lower gap, upper gap, two gap phases) coexist.
• Figure 3: These are the plots of phase transition points in the regular fermion theory. Fig. \ref{['fig:low-two-RF']} shows the plots of the phase transition points from the lower gap to the two gap phase, and Fig. \ref{['fig:up-two-RF']} shows the ones between the upper gap and the two gap phase.
• Figure 4: Phase diagram of the critical boson theory. Here $(\lambda_c,\zeta_c) = (0.403033, 4.24292)$ is the quadruple phase transition point where the four phases (no gap, lower gap, upper gap, two gap phases) coexist.
• Figure 5: These are the plots of phase transition point in the critical boson theory. Fig. \ref{['fig:low-two-CB']} shows the plots of the phase transition points from the lower gap to the two gap phase, and Fig. \ref{['fig:up-two-CB']} shows the ones between the upper gap and the two gap phase.