Phases of large $N$ vector Chern-Simons theories on $S^2 \times S^1$

Abstract

We study the thermal partition function of level $k$ U(N) Chern-Simons theories on $S^2$ interacting with matter in the fundamental representation. We work in the 't Hooft limit, $N,k\to\infty$, with $λ= N/k$ and $\frac{T^2 V_{2}}{N}$ held fixed where $T$ is the temperature and $V_{2}$ the volume of the sphere. An effective action proposed in arXiv:1211.4843 relates the partition function to the expectation value of a `potential' function of the $S^1$ holonomy in pure Chern-Simons theory; in several examples we compute the holonomy potential as a function of $λ$. We use level rank duality of pure Chern-Simons theory to demonstrate the equality of thermal partition functions of previously conjectured dual pairs of theories as a function of the temperature. We reduce the partition function to a matrix integral over holonomies. The summation over flux sectors quantizes the eigenvalues of this matrix in units of ${2π\over k}$ and the eigenvalue density of the holonomy matrix is bounded from above by $\frac{1}{2 πλ}$. The corresponding matrix integrals generically undergo two phase transitions as a function of temperature. For several Chern-Simons matter theories we are able to exactly solve the relevant matrix models in the low temperature phase, and determine the phase transition temperature as a function of $λ$. At low temperatures our partition function smoothly matches onto the $N$ and $λ$ independent free energy of a gas of non renormalized multi trace operators. We also find an exact solution to a simple toy matrix model; the large $N$ Gross-Witten-Wadia matrix integral subject to an upper bound on eigenvalue density.

Figures (11)

• Figure 1: We plot $\frac{T}{\sqrt{N}}$ as a function of $\lambda$ for ${\cal N}=2$ supersymmetric case for $\lambda<\frac{1}{2}$ in Fig.(a), $\lambda >\frac{1}{2}$ in Fig.(b). In Fig.(a), $T$ denotes the phase transition temperature from no gap phase to lower gap phase. In Fig.(b), $T$ denotes the phase transition temperature from no gap phase to upper gap phase. We demonstrate \ref{['buglg']} by Fig.(c).
• Figure 2: We plot phase transition temperature $\frac{T}{\sqrt{N}}$ as a function of $\lambda$ for critical bosonic theory for $\lambda<\lambda_{c}^{\text{Cri.B.}}(=0.403033)$ in Fig.(a) and for $\lambda >\lambda_{c}^{\text{Cri.B.}}(=0.403033)$ in Fig.(b). In Fig.(a), $T$ denotes the phase transition temperature from no gap phase to lower gap phase. In Fig.(b), $T$ denotes the phase transition temperature from no gap phase to upper gap phase.
• Figure 3: We plot phase transition temperature $\frac{T}{\sqrt{N}}$ as a function of $\lambda$ for regular fermionic theory for $\lambda<\lambda_{c}^{\text{Reg.F.}}(=0.596967)$ in Fig.(a) and for $\lambda >\lambda_{c}^{\text{Reg.F.}}(=0.596967)$ in Fig.(b). In Fig.(a), $T$ denotes the phase transition temperature from no gap phase to lower gap phase. In Fig.(b), $T$ denotes the phase transition temperature from no gap phase to upper gap phase.
• Figure 4: We plot the phase transition temperature $\frac{T}{\sqrt{N}}$ as a function of $\lambda$ for regular bosonic theory for $\lambda<\lambda_{c}^{\text{Reg.B.}}(=0.360884)$ in Fig.(a) and for $\lambda >\lambda_{c}^{\text{Reg.B.}}(=0.360884)$ in Fig.(b). In Fig.(a), $T$ denotes the phase transition temperature from no gap phase to lower gap phase. In Fig.(b), $T$ denotes the phase transition temperature from no gap phase to upper gap phase.
• Figure 5: We plot the phase transition temperature $\frac{T}{\sqrt{N}}$ as a function of $\lambda$ for critical fermionic for $\lambda<\lambda_{c}^{\text{Cri.F.}}(=0.639116)$ in Fig.(a) and for $\lambda >\lambda_{c}^{\text{Cri.F.}}(=0.639116)$ in Fig.(b). In Fig.(a), $T$ denotes the phase transition temperature from no gap phase to lower gap phase. In Fig.(b), $T$ denotes the phase transition temperature from no gap phase to upper gap phase.