Non-Commutative Moduli Spaces, Dielectric Tori and T-duality

TL;DR

The paper addresses the moduli space of vacua for certain deformations of $N=4$ super Yang-Mills theories using a non-commutative geometric framework. It encodes points as irreducible representations of the $q$-commutator algebra $[\phi_i,\phi_j]_q=0$ and identifies the center coordinates $x=\phi_1^n$, $y=\phi_2^n$, $z=\phi_3^n$, $w=\phi_1\phi_2\phi_3$ obeying $(-w)^n+xyz=0$, with two mirror dual string realizations connected by T-duality on a torus. D3-branes can bubble into D5-branes with topology $\mathbb{R}^4\times S^2$ in weak RR/NSNS backgrounds, and massless inter-brane modes at special separations are demonstrated via open-string analyses, consistent with a field-theory picture of brane bubbling and moduli-space structure. Degenerations of toroidal branes correspond to brane fractionation and the smoothing of singularities into hyperboloids like $xy=t_\zeta^n$, while a dual AdS/CFT description and a worldsheet derivation tie together non-commutative moduli, torus degenerations, and brane dynamics. Overall, the work integrates non-commutative geometry, brane polarization, and string dualities to illuminate the moduli spaces of deformed $N=4$ theories.

Abstract

We review and extend recent work on the application of the non-commutative geometric framework to an interpretation of the moduli space of vacua of certain deformations of N=4 super Yang-Mills theories. We present a simple worldsheet calculation that reproduces the field theory results and sheds some light on the dynamics of the D-brane bubbles. Different regions of moduli space are associated with D5-branes of various topologies; singularities in the moduli space are associated with topology change. T-duality on toroidal topologies maps between mirror string realizations of the field theory.

Figures (6)

• Figure 1: Moduli space of $q$-deformed theory.
• Figure 2: Quiver diagram for reducible representations.
• Figure 3:
• Figure 4: Moduli space of $q$-deformed theory, with rank three mass.
• Figure 5: Disc diagram for fermion EOM.