ABJM Theory with mass and FI deformations and Quantum Phase Transitions

The phase structure of ABJM theory with mass $m$ deformation and non-vanishing Fayet-Iliopoulos (FI) parameter, $\zeta$, is studied through the use of localisation on ${\mathbb S}^3$. The partition function of the theory then reduces to a matrix integral, which, in the large $N$ limit and at large sphere radius, is exactly computed by a saddle-point approximation. When the couplings are analytically continued to real values, the phase diagram of the model becomes immensely rich, with an infinite series of third-order phase transitions at vanishing FI-parameter. As the FI term is introduced, new effects appear. For any given $0 < \zeta < m/2$, the number of phases is finite and for $\zeta\geq m/2$ the theory does not have any phase transitions at all. Finally, we argue that ABJM theory with physical couplings does not undergo phase transitions and investigate the case of $U(2)\times U(2)$ gauge group in detail by an explicit calculation of the partition function.