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Singularities in the complex Borel plane for Chern--Simons gauge theory on $L(2,1)$ with $t=4\pi$. The singularities in red are associated with the Gaussian or $c=1$ string contribution; see \citep{Pasquetti:2009jg} for details.
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Numerical analysis of the leading (left plot) and subleading (right plot) behavior of the ratio $Z_{g+1}^{(0)} / Z_{g}^{(0)}$, at large order $g$. In each case we plot the values of the ratio (the red line) and two of its corresponding Richardson transforms (of orders $2$ and $6$, shown in blue). The convergence towards the predicted values (the horizontal lines) is clearly very precise.
Singularities in the complex Borel plane for $\mathcal{N}=2$ superconformal Yang--Mills theory on $\mathbb{S}^4$, when considering small coupling $g_{\text{YM}}$ and rank of the gauge group $N=2$. All singularities lie along the negative real direction, giving rise to the corresponding single Stokes line of this example.
Numerical analysis of the coefficients multiplying the ``monomial'' $g^{-10}$, for two different exponentially--suppressed orders: the coefficient $c_{10}^{(2)}$ associated with $\left(-8A\right)^{-g}$ (left plot; corresponding to the pole at $s_2 = -8A$) and the coefficient $c_{10}^{(3)}$ associated with $\left(-18A\right)^{-g}$ (right plot; corresponding to the pole at $s_3 = -18A$). For each plot we present the values of the corresponding ratio as explained in the main text (in red) and of two of its corresponding Richardson transforms (of orders 2 and 6, in blue).
Singularities in the complex Borel plane for the $\mathcal{N}=2^{*}$ supersymmetric Yang--Mills theory on $\mathbb{S}^4$. There are infinite countable Stokes lines, each with only one pole. In the plot, the mass parameter was set to $M=3.2$, thus only six of the Stokes lines lie in the first and fourth quadrants (for $n=1,2,3$).
Numerical analysis of the leading large--order behavior of the coefficients $Z_g^{(0)}$, for values of the mass parameter $M=3.2$, and $A_1 = \frac{1}{2} A \left( 1+M^{2} \right)$. On the left, both the numerical value of these coefficients (the dots) and their predicted analytical behavior (the solid line) are shown. On the right, we plot the convergence towards $2 \cos \theta_{1,+}$ of a particular combination of the coefficients, the one given in \eqref{eq:2star-pertseries-conv-first-angle}, where $\theta_{1,+}$ is the angle associated with the first instanton action $s_1$. The agreement is excellent.
Numerical analysis of the exponentially subleading large--order behavior of the coefficients $Z_g^{(0)}$, for a value of the mass parameter $M=3.2$ and $A_2 = \frac{1}{2} A \left(4+M^{2}\right)$, corresponding to the \textit{leading} behavior of the redefined coefficients $Z_g^{\{1\}}$. Left and right plots show both the numerical value of these coefficients (the dots) as well as their analytically predicted large--order behavior (the solid line), for small and large $g$, respectively. The agreement is clearly excellent at large order.
Singularities in the complex Borel plane for Chern--Simons gauge theory on $L(2,1)$ with $t=4\pi$. The singularities in red are associated with the Gaussian or $c=1$ string contribution; see \citep{Pasquetti:2009jg} for details.
Singularities in the complex Borel plane for Chern--Simons gauge theory on $L(2,1)$ with $t=4\pi$. The singularities in red are associated with the Gaussian or $c=1$ string contribution; see \citep{Pasquetti:2009jg} for details.
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Singularities in the complex Borel plane for ABJM gauge theory on $\mathbb{S}^3$ (at large level $k$, and with rank of the gauge group $N=2$). Every vertical line in this figure should be seen as the very same imaginary axis, \textit{i.e.}, all singularities are along the same line (they are shown separately just to make manifest the existence of different ``instanton'' actions $A_\ell$, for $\ell=1,2,3,4,5$).
Numerical analysis of the leading (left plot) and subleading (right plot) behavior of the ratio $Z_{g+1}^{(0)} / Z_{g}^{(0)}$, at large order $g$. In each case we plot the values of the ratio (the red line) and two of its corresponding Richardson transforms (of orders $2$ and $6$, shown in blue). The convergence towards the predicted values (the horizontal lines) is clearly very precise.
Numerical analysis of the leading (left plot) and subleading (right plot) behavior of the ratio $Z_{g+1}^{(0)} / Z_{g}^{(0)}$, at large order $g$. In each case we plot the values of the ratio (the red line) and two of its corresponding Richardson transforms (of orders $2$ and $6$, shown in blue). The convergence towards the predicted values (the horizontal lines) is clearly very precise.
Singularities in the complex Borel plane for $\mathcal{N}=2$ superconformal Yang--Mills theory on $\mathbb{S}^4$, when considering small coupling $g_{\text{YM}}$ and rank of the gauge group $N=2$. All singularities lie along the negative real direction, giving rise to the corresponding single Stokes line of this example.
Singularities in the complex Borel plane for $\mathcal{N}=2$ superconformal Yang--Mills theory on $\mathbb{S}^4$, when considering small coupling $g_{\text{YM}}$ and rank of the gauge group $N=2$. All singularities lie along the negative real direction, giving rise to the corresponding single Stokes line of this example.
Numerical analysis of the coefficients multiplying the ``monomial'' $g^{-10}$, for two different exponentially--suppressed orders: the coefficient $c_{10}^{(2)}$ associated with $\left(-8A\right)^{-g}$ (left plot; corresponding to the pole at $s_2 = -8A$) and the coefficient $c_{10}^{(3)}$ associated with $\left(-18A\right)^{-g}$ (right plot; corresponding to the pole at $s_3 = -18A$). For each plot we present the values of the corresponding ratio as explained in the main text (in red) and of two of its corresponding Richardson transforms (of orders 2 and 6, in blue).
Numerical analysis of the coefficients multiplying the ``monomial'' $g^{-10}$, for two different exponentially--suppressed orders: the coefficient $c_{10}^{(2)}$ associated with $\left(-8A\right)^{-g}$ (left plot; corresponding to the pole at $s_2 = -8A$) and the coefficient $c_{10}^{(3)}$ associated with $\left(-18A\right)^{-g}$ (right plot; corresponding to the pole at $s_3 = -18A$). For each plot we present the values of the corresponding ratio as explained in the main text (in red) and of two of its corresponding Richardson transforms (of orders 2 and 6, in blue).
Singularities in the complex Borel plane for the $\mathcal{N}=2^{*}$ supersymmetric Yang--Mills theory on $\mathbb{S}^4$. There are infinite countable Stokes lines, each with only one pole. In the plot, the mass parameter was set to $M=3.2$, thus only six of the Stokes lines lie in the first and fourth quadrants (for $n=1,2,3$).
Singularities in the complex Borel plane for the $\mathcal{N}=2^{*}$ supersymmetric Yang--Mills theory on $\mathbb{S}^4$. There are infinite countable Stokes lines, each with only one pole. In the plot, the mass parameter was set to $M=3.2$, thus only six of the Stokes lines lie in the first and fourth quadrants (for $n=1,2,3$).
Numerical analysis of the leading large--order behavior of the coefficients $Z_g^{(0)}$, for values of the mass parameter $M=3.2$, and $A_1 = \frac{1}{2} A \left( 1+M^{2} \right)$. On the left, both the numerical value of these coefficients (the dots) and their predicted analytical behavior (the solid line) are shown. On the right, we plot the convergence towards $2 \cos \theta_{1,+}$ of a particular combination of the coefficients, the one given in \eqref{eq:2star-pertseries-conv-first-angle}, where $\theta_{1,+}$ is the angle associated with the first instanton action $s_1$. The agreement is excellent.
Numerical analysis of the leading large--order behavior of the coefficients $Z_g^{(0)}$, for values of the mass parameter $M=3.2$, and $A_1 = \frac{1}{2} A \left( 1+M^{2} \right)$. On the left, both the numerical value of these coefficients (the dots) and their predicted analytical behavior (the solid line) are shown. On the right, we plot the convergence towards $2 \cos \theta_{1,+}$ of a particular combination of the coefficients, the one given in \eqref{eq:2star-pertseries-conv-first-angle}, where $\theta_{1,+}$ is the angle associated with the first instanton action $s_1$. The agreement is excellent.
Numerical analysis of the exponentially subleading large--order behavior of the coefficients $Z_g^{(0)}$, for a value of the mass parameter $M=3.2$ and $A_2 = \frac{1}{2} A \left(4+M^{2}\right)$, corresponding to the \textit{leading} behavior of the redefined coefficients $Z_g^{\{1\}}$. Left and right plots show both the numerical value of these coefficients (the dots) as well as their analytically predicted large--order behavior (the solid line), for small and large $g$, respectively. The agreement is clearly excellent at large order.
Numerical analysis of the exponentially subleading large--order behavior of the coefficients $Z_g^{(0)}$, for a value of the mass parameter $M=3.2$ and $A_2 = \frac{1}{2} A \left(4+M^{2}\right)$, corresponding to the \textit{leading} behavior of the redefined coefficients $Z_g^{\{1\}}$. Left and right plots show both the numerical value of these coefficients (the dots) as well as their analytically predicted large--order behavior (the solid line), for small and large $g$, respectively. The agreement is clearly excellent at large order.


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