Large-N limit of non-local 2D generalized Yang-Mills theories on non-orientable surface

The large-group behavior of the non-local two dimensional generalized Yang-Mills theories (nlgYM$_2$'s) on arbitrary closed non-orientable surfaces is investigated. It is shown that all order of $\phi^{2k}$ model of these theories have thired order phase transition only on projective plane (RP$^2$). Also the phase structure of $\phi^2 + \frac{\gamma}{4}\phi^4$ model of nlgYM$_2$ is studied and it is found that for $\gamma >0$, this model has third order phase transition only on RP$^2$ and for $\gamma<0$ it has third order phase transition on any closed non-orientable surfaces except RP$^2$ and Klein bottel.