Quiver Matrix Mechanics for IIB String Theory (II): Generic Dual Tori, Fractional Matrix Membrane and $SL(2,\integer)$ Duality

TL;DR

This paper develops a quiver mechanics approach to encode generic torus geometries in Matrix Theory, revealing a hidden dimension, membrane states, and supporting IIB/M duality through $SL(2, ext{Z})$ symmetry.

Contribution

It introduces a novel formulation using orbifolds and quiver mechanics to construct generic-shaped tori and identifies a hidden dimension and membrane states in Matrix Theory.

Findings

Continuum limit yields a (1+2)-D SYM theory.

Membrane wrapping states are rigorously constructed.

$SL(2, ext{Z})$ symmetry eliminates degeneracy and supports IIB/M duality.

Abstract

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 {\em 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 $SL(2,\integer)$ 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.