E3-brane instantons and baryonic operators for D3-branes on toric singularities

TL;DR

The paper develops a dynamical framework in which E3-brane instantons wrapping holomorphic 4-cycles in toric Calabi–Yau geometries induce precise BPS operator insertions in the D3-brane quiver theories, thereby underpinning the AdS/CFT map between BPS operators and wrapped D3-branes on the horizon. For N=1, it shows a direct one-to-one-like correspondence between holomorphic 4-cycles and BPS operators via charged fermion zero modes, with the instanton amplitude producing insertions of the corresponding O_P; this is extended to general N through determinant operators det(O_P) and a Veronese-embedding structure that generates all single-particle BPS operators. The construction leverages orbifolds, partial resolution, and mirror symmetry (D6/E2 picture) to establish a general, constructive map between cycles and operators across all toric singularities, providing a principled explanation of the AdS/CFT dictionary for baryonic sectors. The results have implications for operator counting, holographic duals of baryons, and D-brane model-building, including potential roles for fractional instantons and master-space techniques in organizing the operator spectrum.

Abstract

We consider the couplings induced on the world-volume field theory of D3-branes at local toric Calabi-Yau singularities by euclidean D3-brane (E3-brane) instantons wrapped on (non-compact) holomorphic 4-cycles. These instantons produce insertions of BPS baryonic or mesonic operators of the four-dimensional ${\cal{N}}=1$ quiver gauge theory. We argue that these systems underlie, via the near-horizon limit, the familiar AdS/CFT map between BPS operators and D3-branes wrapped on supersymmetric 3-cycles on the 5d horizon. The relation implies that there must exist E3-brane instantons with appropriate fermion mode spectrum and couplings, such that their non-perturbative effects on the D3-branes induce operators forming a generating set for all BPS operators of the quiver CFT. We provide a constructive argument for this correspondence, thus supporting the picture.

Figures (13)

• Figure 1: The dimer diagram for the $\mathbb{C}^3/(\mathbb{Z}_2\times \mathbb{Z}_2)$ theory (a) and its partial resolution to the SPP singularity (b) and the conifold (c). In order to keep the relation to the unresolved orbifold theory, we have not integrated out the bifundamental fields in di-valent nodes (mass terms in the superpotential).
• Figure 2: The mirror Riemann surface for the conifold, with punctures shown as crosses. The two 1-cycles in green and blue correspond to the two D3-brane gauge groups and the pink 1-cycle connecting the two punctures corresponds to the E3-brane instanton. The disk leading to the coupling $\alpha A_1 \beta$ is painted in red stripes. The instanton amplitude thus produces an insertion of the field $A_1$ (for the $N=1$ theory) in the D3-brane field theory. The right hand part of the figure shows the toric diagram for the conifold and its $(p,q)$-web.
• Figure 3: The two E2-brane instantons (a) can be recombined to a single E2-brane instanton (b) with coupling $\alpha A_1 \gamma + \gamma \delta + \delta B_1 \beta$. After integrating over the two modes $\gamma$, $\delta$ the coupling in (b) is equivalent to $\alpha A_1 B_1 \beta$, as shown pictorially in (c). Integration over the remaining modes leads to the appearance of the mesonic BPS operator $A_1B_1$ in the instanton non-perturbative amplitude.
• Figure 4: The three E2-brane instantons (a) can be recombined to a single E2-brane instanton (b) with coupling $\alpha A_1 \gamma + \gamma \delta + \delta B_1 \mu + \mu \nu + \nu A_2 \beta$. After integrating over the four modes $\gamma$, $\delta$, $\mu$, $\nu$ the coupling in (b) is equivalent to $\alpha A_1 B_1 A_2 \beta$, as shown pictorially in (c). Integration over the remaining modes leads to the appearance of the baryonic BPS operator $A_1 B_1 A_2$ in the instanton non-perturbative amplitude.
• Figure 5: The 1-cycles describing instantons with couplings $\alpha_1 A_1 \beta_1 + \alpha_2 A_1 \beta_2$. The two-instanton process leads to the insertion of the operator $A_1^2$ for $N=1$, or $(\det ( A_1))^2$ for general $N$.