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

• Published in: arXiv.org
• Main Authors: Jain, Sachin, Minwalla, Shiraz, Sharma, Tarun, Takimi, Tomohisa, Wadia, Spenta R, Yokoyama, Shuichi
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 25.07.2013
• Subjects: Density; Eigenvalues; Field theory (physics); Free energy; Integrals; Operators (mathematics); Partitions; Partitions (mathematics); Phase transitions; Quantum theory; Transition temperature; Upper bounds
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1301.6169

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 \(\lambda = N/k\) 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 \(\lambda\). 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 \({2\pi \over k}\) and the eigenvalue density of the holonomy matrix is bounded from above by \(\frac{1}{2 \pi \lambda}\). 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 \(\lambda\). At low temperatures our partition function smoothly matches onto the \(N\) and \(\lambda\) 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.