Phase transition of generalized two dimensional Yang-Mills U(N) on the sphere for $G(z)=z^4+\lambda z^3$ and Maxwell construction

The large-N behavior of the quartic-cubic generalized two dimensional Yang-Mills U(N) on the sphere is investigated for finite cubic couplings. First, it is shown that there are two phase transitions one of which is third order and the other one is second order. Second, $gY{M}_2$ and Maxwell construction are compared and a relationship between two-dimensional space-time, that is purely mathematical, and four-dimensional space-time is obtained.