  It has been understood that correlation functions of multi-trace operators in
${\cal N}=4$ SYM can be neatly computed using the group algebra of symmetric
groups or walled Brauer algebras. On the other hand such algebras have been
known to construct 2D topological field theories (TFTs). After reviewing the
construction of 2D TFTs based on symmetric groups, we construct 2D TFTs based
on walled Brauer algebras. In the construction, the introduction of a dual
basis manifests a similarity between the two theories. We next construct a
class of 2D field theories whose physical operators have the same symmetry as
multi-trace operators constructed from some matrices. Such field theories
correspond to non-commutative Frobenius algebras. A matrix structure arises as
a consequence of the noncommutativity. Correlation functions of the Gaussian
complex multi-matrix models can be translated into correlation functions of the
two-dimensional field theories.
