The resurgence of errors in the localization of $\mathcal{N} = 2$ superconformal Yang-Mills

TL;DR

The paper addresses the analytic structure of the $S^4$ partition function for $SU(2)$ $\ ext{N}=2$ superconformal Yang–Mills and interprets the poles of the Coulomb-branch integrand as unstable two-dimensional configurations. It develops a Higgs-branch localization picture, uses resurgent analysis to relate Coulomb-branch poles to a residue sum, and proposes a 2d $(0,2)$ model inspired by chiral algebra techniques (Beem–Pan) that reproduces the same non-perturbative data via localization on $S^2$ with analytic continuation $g_{2d}=-i g_{YM}/2$. The main contributions are (i) a precise residue structure for the zeroth instanton sector with residues ${\ m Re}(\mathcal{R}(n))$ containing $\,\exp(16\pi^2 n^2/g_{YM}^2)$ and coefficients $f_{n,\ell}$, (ii) a demonstration that the full path integral equals the analytic continuation of the residue sum, encoded in a disc/discontinuity of a Borel-resummed function, and (iii) a concrete 2d model whose one-loop determinants and anomaly structure reproduce the 4d results and illuminate the physical origin of the non-perturbative terms through unstable 2d configurations. This provides a coherent 4d–2d picture of non-perturbative effects in a class of superconformal theories and suggests broad generalizations to other SCFTs and resurgent analyses.

Abstract

We give a physical interpretation for the analytic continuation of the partition function of superconformal SU$(2)$ $\mathcal{N}=2$ gauge theory on the four-sphere to all values of the Yang-Mills coupling. We show that a well-motivated 2d construction associates two-dimensional unstable instantons to the 4d complex saddles which appear as singularities in the integrand of the supersymmetric localization expression. The construction is based on the chiral algebra subsector, and aligns with the alternative Higgs branch localization.

Figures (2)

• Figure 1: Pictorial representation of rotating the contour of integration $a \in (-\infty, \infty)$. For contours within the shaded region, $\pi/4 < \text{arg}(a) < 3\pi/4$, $g_{\text{YM}}$ can be chosen such that these contours are equal to the sum over residues. An example of such a contour is highlighted in blue.
• Figure 2: Diagrams depicting the convergent behaviour of the positive contour defining $\mathcal{Z}(g_{\text{YM}},\phi)$. This corresponding rotation in the $a$--plane is shown in figure \ref{['fig:a_continuation']}. In \ref{['subfig:figures_of_8']} we rotate the contour by $\phi$, and see the divergent region rotate anti-clockwise and expand. The limiting case is shown in \ref{['subfig:infinite_8']}, when $\phi = \pi/4$: the divergent region is the entire 2nd and 4th quarter planes. If we continue to rotate arg$(a) > \pi/4$ the domains flip, and inside the figures of 8 is where the integral converges. The shaded red region is where the formal sum of residues converges. The analogous behaviour for the negative contour defining $\mathcal{Z}_{\text{ScYM}}(g_{\text{YM}},\phi)$ is given by reflecting these diagrams about the $y$--axis.