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We consider $\cN=1, \ D=4$ superconformal $SU(N)^{p \times q}$
Yang-Mills theories dual to \orb\ orbifolds. We construct the
dilatation operator of this superconformal gauge theory at
one-loop, planar level. We demonstrate that a specific sector of
this dilatation operator can be thought of as the transfer matrix
for a three-dimensional statistical mechanical system, which in
turn is equivalent to a $2+1$-dimensional string theory where the
spatial slices are discretized on a triangular lattice. This is a
generalization of the $SO(6)$ spin chain picture of $\cN=4$ super
Yang-Mills theory. We comment on the integrability of this $\cN=1$
gauge theory and hence the corresponding three-dimensional
statistical mechanical system, its connection to three-dimensional
lattice gauge theories, extensions to six-dimensional string
theories, AdS/CFT type dualities and finally their construction
via orbifolds and brane-box models. In the process we discover a
new class of almost-BPS BMN type operators with large engineering
dimensions but controllably small anomalous corrections.
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