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The Schur limit of the superconformal index of four-dimensional $mathcal N=2$ superconformal field theories has been shown to equal the supercharacter of the vacuum module of their associated chiral algebra. Applying localization techniques to the theory suitably put on $S^3times S^1$, we obtain a direct derivation of this fact. We also show that the localization computation can be extended to calculate correlation functions of a subset of local operators, namely of the so-called Schur operators. Such correlators correspond to insertions of chiral algebra fields in the trace-formula computing the supercharacter. As a by-product of our analysis, we show that the standard lore in the localization literature stating that only off-shell supersymmetrically closed observables are amenable to localization, is incomplete, and we demonstrate how insertions of fermionic operators can be incorporated in the computation.
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We study the infrared renormalon in the gluon condensate in the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions (QCD(adj.)) on~$mathbb{R}^3times S^1$ with the $mathbb{Z}_N$ twisted boundary conditions. We rely on the so-called large-$beta_0$ approximation as a conventional tool to analyze the renormalon, in which only Feynman diagrams that dominate in the large-$n_W$ limit are considered while the coefficient of the vacuum polarization is set by hand to the one-loop beta function~$beta_0=11/3-2n_W/3$. In the large~$N$ limit within the large-$beta_0$ approximation, the W-boson, which acquires the twisted Kaluza--Klein momentum, produces the renormalon ambiguity corresponding to the Borel singularity at~$u=2$. This provides an example that the system in the compactified space~$mathbb{R}^3times S^1$ possesses the renormalon ambiguity identical to that in the uncompactified space~$mathbb{R}^4$. We also discuss the subtle issue that the location of the Borel singularity can change depending on the order of two necessary operations.
We present additional observations to previous studies on the infrared (IR) renormalon in $SU(N)$ QCD(adj.), the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions on~$mathbb{R}^3times S^1$ with the $mathbb{Z}_N$ twisted boundary condition. First, we show that, for arbitrary finite~$N$, a logarithmic factor in the vacuum polarization of the photon (the gauge boson associated with the Cartan generators of~$SU(N)$) disappears under the $S^1$~compactification. Since the IR renormalon is attributed to the presence of this logarithmic factor, it is concluded that there is no IR renormalon in this system with finite~$N$. This result generalizes the observation made by Anber and~Sulejmanpasic [J. High Energy Phys. textbf{1501}, 139 (2015)] for $N=2$ and~$3$ to arbitrary finite~$N$. Next, we point out that, although renormalon ambiguities do not appear through the Borel procedure in this system, an ambiguity appears in an alternative resummation procedure in which a resummed quantity is given by a momentum integration where the inverse of the vacuum polarization is included as the integrand. Such an ambiguity is caused by a simple zero at non-zero momentum of the vacuum polarization. Under the decompactification~$Rtoinfty$, where $R$ is the radius of the $S^1$, this ambiguity in the momentum integration smoothly reduces to the IR renormalon ambiguity in~$mathbb{R}^4$. We term this ambiguity in the momentum integration renormalon precursor. The emergence of the IR renormalon ambiguity in~$mathbb{R}^4$ under the decompactification can be naturally understood with this notion.
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