Spheres, Charges, Instantons, and Bootstrap: A Five-Dimensional Odyssey

TL;DR

This work develops and uses a unified framework combining supersymmetric localization on ${\rm S}^5$ with the conformal bootstrap to study five-dimensional ${\rm N}=1$ SCFTs. It classifies admissible counter-terms, uncovers a new 5D superconformal anomaly, and proposes a precise triple-factorization formula for the five-sphere partition function that incorporates instanton contributions and flavor-symmetry enhancement. By computing the central charges ${C_T}$ and ${C_J}$ for rank-one Seiberg and Morrison–Seiberg theories and applying bootstrap bounds, the authors find strong evidence that these theories saturate bootstrap constraints, enabling predictions of long-multiplet spectra. The results provide insight into RG flows, suggest a five-dimensional $F$-theorem, and establish connections between localization, instanton physics, flavor symmetry enhancement, and holographic duals, with promising directions for higher-rank generalizations and rigorous proofs of monotonicity theorems in 5D.

Abstract

We combine supersymmetric localization and the conformal bootstrap to study five-dimensional superconformal field theories. To begin, we classify the admissible counter-terms and derive a general relation between the five-sphere partition function and the conformal and flavor central charges. Along the way, we discover a new superconformal anomaly in five dimensions. We then propose a precise triple factorization formula for the five-sphere partition function, that incorporates instantons and is consistent with flavor symmetry enhancement. We numerically evaluate the central charges for the rank-one Seiberg and Morrison-Seiberg theories, and find strong evidence for their saturation of bootstrap bounds, thereby determining the spectra of long multiplets in these theories. Lastly, our results provide new evidence for the $F$-theorem and possibly a $C$-theorem in five-dimensional superconformal theories.

Figures (12)

• Figure 1: Renormalization group flows across various theories, starting from the Seiberg $E_8$ theories. On the right, fund and anti denote hypermultiplets transforming in the fundamental and antisymmetric representations of ${\rm USp}(2N)$. Note that for rank-one, there is no antisymmetric representation of ${\rm USp}(2)$.
• Figure 2: Lower bounds on $C_T$ and $C_J$ for general (free or interacting) theories with SU(2) flavor group and twist gap $\Delta - s \geq 6$, at various derivative orders $\Lambda$ and extrapolated to infinite order using the quadratic ansatz \ref{['QuadAnsatz']} (solid) and the linear ansatz \ref{['LinearAnsatz']} (dashed). Also shown are the values $C_T = 15$ and $C_J = {8 \over 3}$ for a free hypermultiplet (dotdashed).
• Figure 3: Left: The extremal functional optimizing the lower bound on $C_J$ for general (free or interacting) theories with SU(2) flavor group and twist gap $\Delta - s \geq 6$, acted on the contribution of the spin-zero long multiplet to the crossing equation, ${\alpha}_{{\cal D}[2],{\bf 3}}[{\cal K}^{{\cal L}[0]_{\Delta,0}}]$, in the 1 and 5 of ${\rm SU}(2)$, plotted in logarithmic scale. Increasing derivative orders $\Lambda = 24, 26, \dotsc, 48$ are shown from green to red. Right: The gaps at different $\Lambda$, and extrapolations to $\Lambda \to \infty$ using the quadratic ansatz \ref{['GapAnsatz']} for $\Lambda \in 4\mathbb{Z}$ and $\Lambda \in 4\mathbb{Z}+2$, separately (solid), and using the linear ansatz \ref{['GapAnsatzLinear']} (dashed).
• Figure 4: Allowed region in the $C_T-C_J$ plane for interacting theories with SU(2) flavor group, at derivative orders $\Lambda = 20, 24, \dotsc, 40$, shown from green to red. Also shown are the four-instanton values in the rank-one Seiberg $E_1$ theory, the perturbative values in the rank-two and three, and the values according to the large-rank formula (dashed line).
• Figure 5: Lower bounds on $C_T$ and $C_J$ for interacting theories with SU(2) flavor group, at various derivative orders $\Lambda$ and extrapolated to infinite order using the quadratic ansatz \ref{['QuadAnsatz']} (solid) and the linear ansatz \ref{['LinearAnsatz']} (dashed). Also shown are the values of $C_T$ and $C_J$ in the rank-one Seiberg $E_1$ theory (dotdashed).