Discrete theta and the 5d superconformal index

TL;DR

This work analyzes a $Z_2$-valued discrete theta parameter in 5d $Sp(N)$ gauge theories and clarifies its Type I' string-theory origin. It then uses the 5d superconformal index to distinguish between the $E_1$ and $ ilde{E}_1$ fixed points for the $Sp(1)=SU(2)$ theory, showing that $ ilde{E}_1$ lacks global symmetry enhancement. The authors present two complementary approaches: (i) an instanton-based index with explicit $ heta$-dependent projections for pure and flavored cases, and (ii) a CS-term viewpoint that maps the discrete theta to a level-$ abla$ structure, including a mechanism to restore modular invariance via a delta term. The results establish a concrete method to incorporate discrete 5d theta parameters into index computations and point toward broader implications for large-$N$ holography and Type I' interpretations of 5d instanton physics.

Abstract

5d Yang-Mills theory with an Sp(N) gauge group admits a discrete analog of the theta parameter. We describe the origin of this parameter in N=1 theories from Type I' string theory, and study its effect on the 5d superconformal fixed point theories with an Sp(1)=SU(2) gauge group by computing the superconformal index. Our result confirms the lack of global symmetry enhancement in the so-called \tilde{E}_1 theory.

Figures (4)

• Figure 1: Pure $SU(2)$: (a) $E_1$ theory, (b) $\tilde{E}_1$ theory, (c) another representation of $E_1$. The webs are drawn with the assumption that $g_s=1$ and $C_0 = 0$.
• Figure 2: $SU(2)$ with one flavor: (a) $E_2$ theory, (b) positive mass deformation, (c) negative mass deformation, (d) $E_0$ theory.
• Figure 3: Type I' description of the $\theta$ parameter: (a) A transverse D8-$\overline{\hbox{D8}}$ forms a $\theta$ domain wall, (b) adding a flavor D8-brane leads to an instability, (c) the branes break and reconnect, eliminating the domain wall.
• Figure 4: The instanton particle: (a) in the $E_1$ theory, (b) in the $\tilde{E}_1$ theory.