Complete prepotential for 5d $\mathcal{N}=1$ superconformal field theories

TL;DR

The paper develops a complete prepotential for 5d ${ m N}=1$ SCFTs that extends the Intriligator–Morrison–Seiberg framework to nonperturbative regions and UV dual frames. By introducing invariant Coulomb moduli and organizing corrections with flop, GV, and Mori-cone data, it makes enhanced global symmetries manifest (e.g., $E_{N_f+1}$, $SO(20)$, $E_8 imes SU(2)$) and remains valid over the extended parameter space. Explicit constructions are given for rank-1 theories (e.g., ${ m SU}(2)$ with up to 7 flavors) and rank-2 theories (e.g., ${ m Sp}(2)$ with 9 flavors and with 1 antisymmetric plus 7 flavors), including checks via dualities, Higgsing, and RG flows. The results tie field-theoretic prepotentials to geometric data from Calabi–Yau threefolds, 5-brane webs, and topological-string/GV invariants, providing a unified framework for both perturbative and nonperturbative regimes. This complete prepotential thus acts as a robust tool for understanding UV dualities and symmetry enhancements in 5d SCFTs and offers a pathway to extend to higher ranks and other theories.

Abstract

For any 5d ${\cal N}=1$ superconformal field theory, we propose a "complete" prepotential which reduces to the perturbative prepotential for any of its possible gauge theory realizations, manifests its global symmetry when written in terms of the invariant Coulomb branch parameters, and is valid for the whole parameter region. As concrete examples, we consider $SU(2)$ gauge theories with up to 7 flavors, $Sp(2)$ gauge theories with up to 9 flavors, and $Sp(2)$ gauge theories with 1 antisymmetric tensor and up to 7 flavors, as well as their dual gauge theories.

Figures (13)

• Figure 1: (a) A 5-brane web for pure $SU(2)_0$ gauge theory (or $E_1$ theory). (b) A 5-brane web for pure $SU(2)_\pi$ gauge theory (or $\widetilde{E}_1$ theory).
• Figure 2: (a) $\widetilde{E}_1$ theory with $a+\frac{1}{2}m_0<0$. (b) $E_0$ theory
• Figure 3: All possible distinct phases of $E_2$ web diagram.
• Figure 4: The parameter space of the $E_2$ theory, which is identified as the Kähler cone of the corresponding geometry. The allowed parameter region is the space which is surrounded by the three planes represented by $OAB$, $OBC$, and $OCD$, respectively. If we fix the mass parameters $m_0$ and $m_1$, this allowed region is identified as the physical Coulomb moduli. This allowed parameter region is divided into five phases due to the three internal "walls", which are represented by $OAE$, $OCD$, and $ODE$, respectively. These walls are the place where the flop transition occurs. Especially, Phase ${\rm (I)}$ is identified as the region surrounded by these three internal walls, which we represent as $ODEF$.
• Figure 5: String generating BPS particle with spin $(j_L,j_R) = (0,\frac{1}{2})$