Quiver matrix mechanics for IIB string theory II: generic dual tori, fractional matrix membrane and SL(2,Z) duality

With the deconstruction technique, the geometric information of a torus can be encoded in a sequence of orbifolds. By studying the matrix theory on these orbifolds as quiver mechanics, we present a formulation that (de)constructs the torus of generic shape on which matrix theory is “compactified”. The continuum limit of the quiver mechanics gives rise to a (1 + 2)-dimensional SYM. A hidden (fourth) dimension, that was introduced before in the matrix theory literature to argue for the electric-magnetic duality, can be easily identified in our formalism. We construct membrane wrapping states rigorously in terms of Dunford calculus in the context of matrix regularization. Unwanted degeneracy in the spectrum of the wrapping states is eliminated by using SL(2,Z) symmetry and the relations to the FD-string bound states. The dual IIB circle emerges in the continuum limit, constituting a critical evidence for IIB/M duality.