(which depends on $\lambda$) vanishes,

In each vacuum $X$ has a quadratic superpotential, i.e. it is massive and can be integrated out. The gauge group is broken by the $X$ expectation value: $SU(r_l)$ is the gauge group of the $l$’th decoupled SQCD theory one finds in the infrared. The existence of a large number of vacua  (labeled by partitions of $N_c$ into $k$ or less integers) after resolving the singularity  is the key fact leading to simple but non-trivial quantitative checks of duality.

For generic $\{s_i\}$ (and fixed $\{r_i\}$) the classical infrared behavior of the theory, a decoupled set of SQCD theories with gauge groups , is insensitive to the precise values of $\{s_i\}$. Quantum mechanically , this is not the case; each of the low energy SQCD theories has a scale $\Lambda_i$, $i=1,\cdots, k$. These $k$ scales are functions of the scale $\Lambda$ of the underlying theory with the adjoint superfield, and the $k-1$ couplings $s_1, \cdots, s_{k-1}$. The functions $\Lambda_i(\Lambda, s_i)$ will be computed below (in section [*4*]{}).

A similar discussion holds if the $\{s_l\}$ are fine tuned such that some of the $c_i$ in  coincide: where $\sum_i n_i=k;\;\;m\le k;\;\; n_i\ge1$. Vacua are labeled by sequences $(r_1,\cdots, r_m)$, 