Discrete theta angle from an O5-plane

TL;DR

This work shows that the discrete theta angle in the 5d pure $Sp(1)$ gauge theory, realized via an O5-plane, is encoded in the Seiberg-Witten curves through boundary conditions on OM5-planes, yielding distinct curves for the $E_1$ ($\theta=0$) and $\tilde{E}_1$ ($\theta=\pi$) theories. By uplifting to M-theory and performing a circle decompactification, the authors reconstruct the corresponding $(p,q)$ 5-brane webs and reveal qualitatively different strong-coupling configurations, while the weak-coupling regimes appear visually similar. Extending the analysis to the $E_2$ theory (one flavor) one finds phase structures and generalized flop transitions that interpolate between $E_1$, $\tilde{E}_1$, and $E_0$ via flavor decoupling and strong-coupling flows. A unified description of the effective coupling $\tau_{\text{eff}}$ across phases is given, including an all-encompassing expression for the $E_2$ theory, which reduces to the known results in the appropriate decoupling limits. The results establish a concrete bridge between brane-web realizations with orientifold planes and the UV symmetry enhancements of 5d SCFTs, with implications for generalized flop transitions and higher-rank generalizations. Key technical innovations include introducing boundary conditions on OM5-planes to extract $E_1$ and $\tilde{E}_1$ Seiberg-Witten curves from webs with an O5-plane, and a systematic decompactification procedure that recovers physically distinct strong-coupling brane configurations, validated by matching known SW curves and phase structures across $E_1$, $\tilde{E}_1$, and $E_2$ theories.

Abstract

We consider 5d $\mathcal{N}=1$ $Sp(1)$ gauge theory based on a brane configuration with an O5-plane. At the UV fixed point, the theory with no matter enjoys enhanced global symmetry $SU(2)$ or $U(1)$ depending on the discrete theta angle $θ=0, π$ (mod $2π$). A naive brane configuration with an O5-plane, however, does not distinguish two different theories, as it describes the weak coupling region. We devise a technique for computing 5d Seiberg-Witten curve of the two theories from the brane web with an O5-plane. Their Seiberg-Witten curves show that their M5 configurations under the presence of OM5-planes are different. The decompactification limit of each Seiberg-Witten curve also shows distinct phase structures in their Coulomb branch leading to significantly different $(p,q)$ 5-brane configurations with an O5-plane in the strong coupling region.

Figures (36)

• Figure 1: Web diagram for 5d $SU(2)$ theory of enhanced $E_1=SU(2)$ and $\widetilde{E}_1=U(1)$ global symmetries
• Figure 2: Covering space web configuration for 5d $Sp(1)$ theory and corresponding toric-like diagram.
• Figure 3: The phase diagram of the $E_1$ theory. The line between the Region 1 and the Region 3 is $u=m_0$. The boundary of the physical parameter space is given by $u=\frac{1}{2}m_0$ for $m_0 \leq 0$ and $u=0$ for $m_0 \geq 0$.
• Figure 4: (a): The blue letter $A_{k, l}^{(m)}$ indicates the largest $A_{k, l}^{(m)}$ in the region. The black solid lines are the boundaries of those regions and give the 5-brane web in the decompactification limit of the $E_1$ curve in Region 1 \ref{['eq:E1region1']}. The thick line denotes the coincident 5-branes. (b): The toric-like diagram of the $E_1$ theory with a triangulation given by the web in Figure (a).
• Figure 5: The blue letter $A_{k, l}^{(m)}$ indicates the largest $A_{k, l}^{(m)}$ in the region. The black solid lines are the boundaries of those regions and give the 5-brane web in the decompactification limit of the $E_1$ curve in Region 2 \ref{['eq:E1region2']}.