Aspects of ALE Matrix Models and Twisted Matrix Strings

TL;DR

The paper advances the ADE ALE matrix-model program as a nonperturbative description of M-Theory on ALE spaces by: (i) establishing massless vector multiplets in blow-up resolutions via a hyperkähler quotient quiver construction; (ii) proposing a quiver-based description of wrapped membranes with calculable bound-state energies and Coulomb interactions; (iii) analyzing orbifold realizations and twisted matrix-string theories, including a Reid-based classification that yields several new gauge groups (e.g., $Sp(n)$, $G_2$, $SO(2n-1)$, $F_4$); and (iv) arguing for the correct infrared physics (no $v^2$ terms, one-loop $F^4$ corrections) and outlining avenues for further exploration of bound states and large-$N$ dynamics in these ALE matrix models.

Abstract

We examine several aspects of the formulation of M(atrix)-Theory on ALE spaces. We argue for the existence of massless vector multiplets in the resolved $A_{n-1}$ spaces, as required by enhanced gauge symmetry in M-Theory, and that these states might have the correct gravitational interactions. We propose a matrix model which describes M-Theory on an ALE space in the presence of wrapped membranes. We also consider orbifold descriptions of matrix string theories, as well as more exotic orbifolds of these models, and present a classification of twisted matrix string theories according to Reid's exact sequences of surface quotient singularities.

Figures (7)

• Figure 1: The $A_{n-1}$ quiver diagram.
• Figure 2: The $\mathbb{Z}_2$ twist on the $A_{2n-1}$ model that generates a $C_n$ model.
• Figure 3: The $\mathbb{Z}_2$ action on the $D_{2n}$ diagram induced by the reflection of Figure \ref{['fig:a2n-1-cn']}.
• Figure 4: The labeling of roots and Dynkin labels (marks) for $A_{2n- 1}$, $D_{n+2}$, and $C_n$.
• Figure 5: The $\mathbb{Z}_3$-twisted $\Gamma(D_4)$ and $\Gamma(E_6)$ models with $G_2$ gauge group.