  $O(n) \times O(m)$ symmetric Landau-Ginzburg models in $d=3$ dimension
possess a rich structure of the renormalization group and its understanding
offers a theoretical prediction of the phase diagram in frustrated spin models
with non-collinear order. Depending on $n$ and $m$, they may show
chiral/anti-chiral/Heisenberg/Gaussian fixed points within the same
universality class. We approach all the fixed points in the conformal bootstrap
program by examining the bound on the conformal dimensions for scalar operators
as well as non-conserved current operators with consistency crosschecks. For
large $n/m$, we show strong evidence for the existence of four fixed points by
comparing the operator spectrum obtained from the conformal bootstrap program
with that from the large $n/m$ analysis. We propose a novel non-perturbative
approach to the determination of the conformal window in these models based on
the conformal bootstrap program. From our numerical results, we predict that
for $m=3$, $n=7\sim 8$ is the edge of the conformal window for the anti-chiral
fixed points.
