Counting Chiral Operators in Quiver Gauge Theories

TL;DR

This work develops a comprehensive framework for counting BPS chiral operators in quiver gauge theories arising from D-branes on toric Calabi–Yau singularities. It introduces generating functions that separate mesonic and baryonic sectors, establishes a dual description via GKZ (secondary fan) decompositions of the Kähler moduli space, and uses localization to compute sector-by-sector contributions. The Plethystic Exponential then builds the full N-brane generating functions from the N=1 data, with explicit results for several geometries (C^3/Z_3, F_0, dP_1) and checks via the Molien formula. The analysis reveals multiplicities arising from the CY geometry and shows how refining the GKZ lattice with anomalous charges can remove these multiplicities, linking geometric data to field-theoretic determinant structures and advancing the understanding of the gauge/gravity correspondence in toric setups.

Abstract

We discuss in detail the problem of counting BPS gauge invariant operators in the chiral ring of quiver gauge theories living on D-branes probing generic toric CY singularities. The computation of generating functions that include counting of baryonic operators is based on a relation between the baryonic charges in field theory and the Kaehler moduli of the CY singularities. A study of the interplay between gauge theory and geometry shows that given geometrical sectors appear more than once in the field theory, leading to a notion of "multiplicities". We explain in detail how to decompose the generating function for one D-brane into different sectors and how to compute their relevant multiplicities by introducing geometric and anomalous baryonic charges. The Plethystic Exponential remains a major tool for passing from one D-brane to arbitrary number of D-branes. Explicit formulae are given for few examples, including C^3/Z_3, F_0, and dP_1.

Figures (22)

• Figure 1: Quiver and toric diagram for the conifold.
• Figure 2: The GKZ decomposition for the Kähler moduli space of the conifold, consisting of two one-dimensional cones connected by a flop. The coordinate $t$ on the moduli space is associated with the volume of the two-cycle in the resolution of the conifold. When $t$ goes to zero, the cycle vanishes and we can perform a flop on the variety by inflating a different two-cycle. A natural discretization of the GKZ fan is in correspondence with the decomposition of the $g_1$ generating function.
• Figure 3: Localization data for the $N=1$ baryonic generating functions. The vertices $V_i$ are in correspondence with homogeneous coordinates $x_i$ and with a basis of divisors $D_i$. Two different resolutions, related by a flop, should be used for positive and negative $B$, respectively. Each resolution has two fixed points, corresponding to the vertices of the pq-webs; the weights $m^{(I)}_i,\, i=1,2,3$ and $m^{(I)}_B$ at the fixed points are indicated in black and red, respectively.
• Figure 4: Quiver and toric diagram for $\mathbb{C}^3/\mathbb{Z}_3$.
• Figure 5: The hollow triangle $C(4)$ above $R=4$, i. e. the terms containing $t^4$. It gives the multiplicity $4\times 3 = 12$.