On the geometry and the moduli space of beta-deformed quiver gauge theories

TL;DR

The paper addresses how β-deformations of toric quiver gauge theories modify the moduli space and how these deformations are realized in their gravity duals via generalized complex geometry.It develops a unified framework using Lunin–Maldacena T-duality and pure spinor formalism to study D3/D5 probes and dual giants, relating probe moduli to the gauge-theory Coulomb and Higgs branches for both generic and rational β.A central result is that the abelian mesonic moduli space collapses to a set of d complex lines (d = number of external toric vertices) and that rational β opens extra Higgs branches isomorphic to CY/\mathbb{Z}_n\times\mathbb{Z}_n, which mirrors the gravity-side D5 dual giant moduli.The analysis confirms that generalized geometry provides a powerful, metric-independent approach to SUSY brane probes and to connecting gravity with the mesonic chiral ring, offering insights for extending the AdS/CFT correspondence to β-deformed toric theories.

Abstract

We consider a class of super-conformal beta-deformed N=1 gauge theories dual to string theory on $AdS_5 \times X$ with fluxes, where $X$ is a deformed Sasaki-Einstein manifold. The supergravity backgrounds are explicit examples of Generalised Calabi-Yau manifolds: the cone over $X$ admits an integrable generalised complex structure in terms of which the BPS sector of the gauge theory can be described. The moduli spaces of the deformed toric N=1 gauge theories are studied on a number of examples and are in agreement with the moduli spaces of D3 and D5 static and dual giant probes.

Figures (8)

• Figure 1: The toric diagram for $\mathbb{C}^3$ and the conifold consisting of the points $V_{\alpha}=(v_{\alpha},1)$ pictured in the plane $z=1$ in $\mathbb{R}^3$. The vectors $V_{\alpha}$ determine a rational polyedron in $\mathbb{R}^3$ with three and four sides, respectively, whose projection on the plane $z=1$ is shown in the Figure.
• Figure 2: The toric diagram $\mathcal{C}$ and the generators of the dual cone $\mathcal{C}^*$ with the associated mesonic fields for: (a) $\mathcal{N}=4$, (b) conifold. The $U(1)^3$ charges of the mesons are explicitly indicated; the first two entries of the charge vectors give the $U(1)^2$ global charge used to define the non commutative product.
• Figure 3: The toric diagram and the quiver of the $SPP$ singularity
• Figure 4: The toric diagram and the quiver of the $PdP_4$ singularity
• Figure 5: $\mathbb{C}^3 \rightarrow \mathbb{C}^3/\mathbb{Z}_n \times \mathbb{Z}_n$ in the toric picture, $b^5=1$.