Seiberg Duality, 5d SCFTs and Nekrasov Partition Functions

TL;DR

This work demonstrates that Seiberg-like dualities in 5d SCFTs arise from Picard–Lefschetz moves of affine $E_N$ 7-brane backgrounds, leading to multiple Type IIB 5-brane webs and toric/pseudo del Pezzo duals describing the same fixed points. The authors conjecture and test a unifying relation between Nekrasov partition functions across toric and pseudo del Pezzo phases, $Z_{dP_k} = Z_{PdP_k^{p}} / Z^{PdP_k^{p}}_{ ext{extra}}$, where $Z^{PdP_k^{p}}_{ ext{extra}}$ captures non-full spin content from stacks of parallel external legs. Using refined topological vertex techniques, they perform explicit one-instanton (and some higher) checks for $E_1$ through $E_5$ theories, showing that the extra factors precisely account for the discrepancies and render the partition functions equivalent. The results classify 5d SCFTs with one-dimensional Coulomb branches into eight distinct theories, reveal deep links to the 4d Seiberg duality via 7-brane moves, and suggest promising directions toward non-toric vertex constructions and 5d AGT relations.

Abstract

It is known that a 4d N = 1 SCFT lives on D3-branes probing a local del Pezzo Calabi-Yau singularity. The Seiberg (or toric) duality of this SCFT arises from the Picard-Lefshetz transformation of the affine E_N 7-brane background that is associated with the Calabi-Yau threefold. In this paper we study the duality of the affine E_N background itself and a 5-brane probing it. We then find that many different Type IIB 5-brane webs describe the same SCFT in 5d. We check this duality by comparing the Nekrasov partition functions of these 5-brane web configurations.

Figures (37)

• Figure 1: The left hand side is the 5-brane web dual to the local $\mathbb{CP}^2$ geometry. We regularize an external leg by terminating it on a 7-brane. A colored circle is a 7-brane, and a dashed lines is branch cut arising from it. Moving these three 7-branes inside the 5-brane loop yields the right hand side through the Hanany-Witten effect.
• Figure 2: All the $GL(2,\mathbb{Z})$-inequivalent convex lattice polygons with single internal point and their dual web diagrams.
• Figure 3: Blowup in a toric web diagram. The local Calabi-Yau for the first del Pezzo ${\mathcal{B}}_1$ is the one point blowup of the local $\,\mathbb{P}^2$. The resulting three toric diagrams are equivalent up to the $SL(2,\mathbb{Z})$ symmetry transformation.
• Figure 4: The two toric phases of the local geometry of the another first del Pezzo $\tilde{\mathcal{B}}_1$.
• Figure 5: The left hand side is the 5-brane loop probe of $\,\,\bf \hat{E}_1$ 7-brane configuration. Since all the 7-branes are not collapsible, only a sub-algebra $E_1$ is realized on the 5-brane. The right hand side is the corresponding web diagram which is obtained by moving a 7-brane the outside of the loop along the associated geodesic.