The dual superconformal theory for Lpqr manifolds

TL;DR

The paper constructs and analyzes the superconformal gauge theories dual to AdS5 × L^{p,q,r} toric Sasaki–Einstein horizons using the brane tiling (dimer) formalism. It derives the toric data and computes volumes to perform a-maximization, obtaining R-charges and central charges that precisely match holographic predictions, thereby validating the dimer-toric correspondence for this infinite family. The work also extends to non-conformal regimes through fractional branes, discussing duality cascades and the obstructions to IR fixed points due to lack of complex deformations in smooth cases. Collectively, it provides a concrete, computable framework to connect toric geometry, dimers, and AdS/CFT for L^{p,q,r} spaces and highlights directions for future exploration of deformable horizons and non-conformal dynamics.

Abstract

We present the superconformal gauge theory living on the world-volume of D3 branes probing the toric singularities with horizon the recently discovered Sasaki-Einstein manifolds L^{p,q,r}. Various checks of the identification are made by comparing the central charge and the R-charges of the chiral fields with the information that can be extracted from toric geometry. Fractional branes are also introduced and the physics of the associated duality cascade discussed.

Figures (13)

• Figure 1: The toric diagram for $L^{p,q,r}$. $F$ is typically negative and it will be renamed $-k$ in the following. The toric diagram is given by the points $(0,0),(1,0),(P,s),(-k,q))$. The explicit example in the Figure is $L^{1,5;2,4}$.
• Figure 2: An alternative description for the toric diagram of $L^{p,q,r}$. The explicit example is still $L^{1,5;2,4}$. $A$ and $X$ satisfy the linear Diophantine equation $s\,A-p\,X-q=0$
• Figure 3: The brane tiling corresponding to $L^{p,q;r,s}$. The fundamental cell is composed by $n+m$ hexagons, $m$ of which are divided in two quadrilaterals. Each face (hexagon or quadrilateral) is a gauge group, each link a bi-fundamental field and each vertex is a term in the superpotential given by the product of all the fields associated with links meeting at the vertex.
• Figure 4: The dimer for $L^{1,5;2,4}$. In this case $n=4$, $m=1$ and $k=3$.
• Figure 5: The toric diagram for $L^{1,5;2,4}$.