Counting BPS Baryonic Operators in CFTs with Sasaki-Einstein duals

TL;DR

The paper develops a geometric framework to count 1/2-BPS baryonic operators in CFTs with Sasaki–Einstein AdS5 duals by analyzing D3-branes wrapped on nontrivial three-cycles. Using Beasley’s holomorphic-surface quantization and toric geometry, it constructs sector-wise partition functions $Z_D$ that count BPS states with fixed baryonic, flavor, and R-charges, and shows how these counts encode volumes of divisors and the horizon $H$. The authors provide explicit formulae and examples (conifold, $Y^{p,q}$, del Pezzo, $L^{p,q,r}$) to illustrate the mapping between sections of line bundles on the Calabi–Yau cone and gauge-invariant baryonic operators in the dual quiver theories, and they derive the relations between $Z_D$, the symmetric-product partition function $Z_{D,N}$, and volumes via fixed-point localization. This framework links the operator spectrum in the CFT to geometric data of the Sasaki–Einstein horizon, enabling holographic volume extraction and offering a path toward a full chiral-ring partition function in toric (and potentially non-toric) settings.

Abstract

We study supersymmetric D3 brane configurations wrapping internal cycles of type II backgrounds AdS(5) x H for a generic Sasaki-Einstein manifold H. These configurations correspond to BPS baryonic operators in the dual quiver gauge theory. In each sector with given baryonic charge, we write explicit partition functions counting all the BPS operators according to their flavor and R-charge. We also show how to extract geometrical information about H from the partition functions; in particular, we give general formulae for computing volumes of three cycles in H.

Figures (10)

• Figure 1: On the left: the fan for $\mathbb{P}^2$ with three maximal cones of dimension two which fill completely $\mathbb{R}^2$; there are three one dimensional cones in $\Sigma(1)$ with generators $\{(1,0),(0,1),(-1,-1)\}$. On the right: the fan for the conifold with a single maximal cone of dimension three; there are four one dimensional cones in $\Sigma(1)$ with generators $\{ (0,0,1),(1,0,1),(1,1,1),(0,1,1)\}$.
• Figure 2: (a) The polytope associated to the line bundle $\mathcal{O}(1) \rightarrow \mathbb{P}^2$. (b) The polytope associated to the line bundle $\mathcal{O}(3) \rightarrow \mathbb{P}^2$.
• Figure 3: A generic toric diagram with the associated trial charges $a_i$, homogeneous coordinates $x_i$ and divisors $D_i$.
• Figure 4: (1) Dimer configuration for the field theory dual to $C(Y^{2,1})$ with a given assignment of charges $a_i$ and the orientation given by the arrows connecting the gauge groups. We have drawn in green the bounds of the basic cell. For notational simplicity we have not indicated with different colors the vertices; the dimer is a bipartite graph and this determines an orientation. (2) Toric diagram for the singularity $C(Y^{2,1})$.
• Figure 5: (1) Dimer configuration for the field theory dual to $C(T^{1,1})$ with a given assignment of charges $a_i$ and the orientation given by the arrows linking the gauge groups. We have drawn in green the bounds of the basic cell. (2) Toric diagram for the singularity $C(T^{1,1})$.