Baryonic Generating Functions

TL;DR

This work develops and applies a unified plethystic framework to count BPS operators in the chiral rings of quiver gauge theories on D-branes probing non-compact Calabi–Yau manifolds, with a focus on baryonic sectors. It shows how to compute baryonic generating functions from the single-brane function $g_{1,B}$ using the plethystic exponential, and interprets $g_{1,B}$ in geometric terms as holomorphic sections of line bundles $ ext{O}(B)$ on toric CYs, employing localization and toric data. The paper provides exact results for the conifold (including $N=1$ and $N=2$ cases, Chebyshev structures, and geometric quantization interpretations) and for the orbifold $ ext{C}^2/ ext{Z}_2 imes ext{C}$, including complete analyses of several truncations such as

Abstract

We show how it is possible to use the plethystic program in order to compute baryonic generating functions that count BPS operators in the chiral ring of quiver gauge theories living on the world volume of D branes probing a non compact CY manifold. Special attention is given to the conifold theory and the orbifold C^2/Z_2 times C, where exact expressions for generating functions are given in detail. This paper solves a long standing problem for the combinatorics of quiver gauge theories with baryonic moduli spaces. It opens the way to a statistical analysis of quiver theories on baryonic branches. Surprisingly, the baryonic charge turns out to be the quantized Kahler modulus of the geometry.

Figures (7)

• Figure 1: The dimer representation for the conifold. The fundamental domain is depicted in green. We have drawn some paths representing BPS operators in the chiral ring in the case $N=1$: in cyan $B_2A_2B_1$; in yellow $A_1B_1A_1B_1$, in magenta $A_1B_1A_1B_1A_2B_1A_2$.
• Figure 2: The $(p,q)$ webs for the two possible resolution of the conifold. The left figure corresponds to positive baryon number, $B>0$, while the right figure corresponds to negative baryon number, $B<0$. The vertices correspond to the fixed points of the torus action on the resolution.
• Figure 3: Localization data for the $N=1$ baryonic partition 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 $(p,q)$ 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: Examples of the "integral conical pyramid"s for the conifold in the cases $B<0$, $B=0$, $B>0$.
• Figure 5: Quiver for half the conifold.