  The resolvent operator plays a central role in matrix models. For instance,
with utilizing the loop equation, all of the perturbative amplitudes including
correlators, the free-energy and those of instanton corrections can be obtained
from the spectral curve of the resolvent operator. However, at the level of
non-perturbative completion, the resolvent operator is generally not sufficient
to recover all the information from the loop equations. Therefore it is
necessary to find a sufficient set of operators which provide the missing
non-perturbative information. In this paper, we study generalized Wronskians of
the Baker-Akhiezer systems as a manifestation of these new degrees of freedom.
In particular, we derive their isomonodromy systems and then extend several
spectral dualities to these systems. In addition, we discuss how these
Wronskian operators are naturally aligned on the Kac table. Since they are
consistent with the Seiberg-Shih relation, we propose that these new degrees of
freedom can be identified as FZZT-Cardy branes in Liouville theory. This means
that FZZT-Cardy branes are the bound states of elemental FZZT branes (i.e. the
twisted fermions) rather than the bound states of principal FZZT-brane (i.e.
the resolvent operator).
