%0 Generic %1 Jain2013Phases %A Jain, Sachin %A Minwalla, Shiraz %A Sharma, Tarun %A Takimi, Tomohisa %A Wadia, Spenta R. %A Yokoyama, Shuichi %D 2013 %K chern-simons, higher-spin, hs4, phase-transition %T Phases of large \$N\$ vector Chern-Simons theories on \$S^2 S^1\$ %U http://arxiv.org/abs/1301.6169 %X 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\toınfty\$, with \$= N/k\$ and \$T^2 V\_2N\$ held fixed where \$T\$ is the temperature and \$V\_2\$ the volume of the sphere. An effective action proposed in <a href="/abs/1211.4843">arXiv:1211.4843</a> 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 \$łambda\$. 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 \$2k\$ and the eigenvalue density of the holonomy matrix is bounded from above by \$12 łambda\$. 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 \$łambda\$. At low temperatures our partition function smoothly matches onto the \$N\$ and \$łambda\$ 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.

@misc{Jain2013Phases, 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 \$\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 <a href="/abs/1211.4843">arXiv:1211.4843</a> 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.}, added-at = {2019-02-26T10:37:35.000+0100}, archiveprefix = {arXiv}, author = {Jain, Sachin and Minwalla, Shiraz and Sharma, Tarun and Takimi, Tomohisa and Wadia, Spenta R. and Yokoyama, Shuichi}, biburl = {https://www.bibsonomy.org/bibtex/212d26224a1e18af87fce9e0dbcc64442/acastro}, citeulike-article-id = {11967556}, citeulike-linkout-0 = {http://arxiv.org/abs/1301.6169}, citeulike-linkout-1 = {http://arxiv.org/pdf/1301.6169}, day = 25, eprint = {1301.6169}, interhash = {f790bfe0888cdcc0366d35d7f8eca977}, intrahash = {12d26224a1e18af87fce9e0dbcc64442}, keywords = {chern-simons, higher-spin, hs4, phase-transition}, month = jan, posted-at = {2013-01-29 06:18:11}, priority = {2}, timestamp = {2019-02-26T10:37:35.000+0100}, title = {{Phases of large \$N\$ vector Chern-Simons theories on \$S^2 \times S^1\$}}, url = {http://arxiv.org/abs/1301.6169}, year = 2013 }