3 problems

Let $U\in\operatorname{U}(N)$ have non-degenerate spectrum, and consider the effective action $S(\gamma,U,Q)$ and its leading-order saddle-point expansion. Among its critical subma…

Let ${\cal M}_S={\cal M}_{(p,r)}$ be one of the critical submanifolds other than the isolated points $\{\pm\Sigma_3\}$, and let ${\cal M}_{\bar S}$ be its particle-hole dual. Write…

Let $U$ be a unitary matrix and let $\gamma\neq 1$, with genericity meaning that the matrix $\operatorname{diag}(\gamma U,U)$ is not degenerate. For integers $p$ and $r$, let…