The round S4 supercharge square and its finite geometric Cartan data

Take the adopted massless $\mathcal{N}=4$ model on the round $S_r^4$ with its specified conformal Killing spinors, the trivial rank-seven Euclidean auxiliary bundle $E_{\mathrm{a}\mathrm{u}\mathrm{x}}=S_r^4\times {\mathbb{R}}^7$ with fixed global orthonormal basis $e_i$, and combined gauge-fixed differential $Q=s+\delta$. Use the spinor/R associated bundle $S_{ϵ}$ from the adopted round-$S^4$ postulate: its fibre is $W_{ϵ}=({\mathrm{\Delta }}_{+}\oplus {\mathrm{\Delta }}_{-})\otimes (V_R\oplus V_0)$, where ${\mathrm{\Delta }}_{\pm }={\mathbb{C}}^2$ are the defining $\mathrm{Spin}(4)$ factor modules, $V_R={\mathbb{C}}^2$ carries the defining $SU(2{)}_R$ representation, and $V_0={\mathbb{C}}^2$ carries the trivial representation; the product action is ${\rho }_{ϵ}(s,k)=({\pi }_{+}(s)\oplus {\pi }_{-}(s))\otimes (\tau (k)\oplus I_{V_0})$, with quotient convention $[(ph,z)]=[p,{\rho }_{ϵ}(h)z]$. The auxiliary field is $K\in \mathrm{\Gamma }(\mathrm{ad}P\otimes E_{\mathrm{a}\mathrm{u}\mathrm{x}})$ with components $K={\sum }_i{K}_i{e}_i$; the source's smooth spinor sections ${\nu }_i$ satisfy ${\sum }_i{\nu }_i\otimes e^i\in \mathrm{\Gamma }(S_{ϵ}\otimes E_{\mathrm{a}\mathrm{u}\mathrm{x}}^{\ast })$, $ϵ{\mathrm{\Gamma }}_M{\nu }_i=0$, $\frac{1}{2}(ϵ{\mathrm{\Gamma }}^Nϵ){\tilde{\mathrm{\Gamma }}}_{N,\alpha \beta }={\sum }_i{\nu }_{i,\alpha }{\nu }_{i,\beta }+{ϵ}_{\alpha }{ϵ}_{\beta }$, and ${\nu }_i{\mathrm{\Gamma }}_M{\nu }_j={\delta }_{ij}ϵ{\mathrm{\Gamma }}_Mϵ$. For every homogeneous adjoint-valued doublet coordinate $X$ and its opposite-parity partner $X^{\mathrm{\prime }}$, the source algebra gives $sX=X^{\mathrm{\prime }}$, $s{X}^{\mathrm{\prime }}={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(\varphi )X+{\delta }_{U(1)}(\omega )X$, $\delta X=-[c,X]$, and $\delta X^{\mathrm{\prime }}=-[c,X^{\mathrm{\prime }}]$, with ${\delta }^{2}X={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(a_0)X$ and $\{s,\delta \}X=-{\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(\varphi )X$. Here ${\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(u)X=[u,X]$ on adjoint fields and $D_{\mu }X={\mathrm{\nabla }}_{\mu }^{\mathrm{L}\mathrm{C}}X+[A_{\mu },X]$ on adjoint tensor components. For the connection, $sA=\psi$, its superpartner transforms by $s\psi ={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(\varphi )A+{\delta }_{U(1)}(\omega )A$, and the Faddeev--Popov rule is $\delta A_{\mu }=D_{\mu }c=-{\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(c)A_{\mu }$, with ${\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(u)A_{\mu }=-D_{\mu }u$. The generator $\omega$ is fixed by $v^{\mu }=ϵ{\mathrm{\Gamma }}^{\mu }ϵ$ without further rescaling. If ${\widehat{\alpha }}_{\theta }$ is the chosen lift of this flow to a field bundle, its geometric generator is ${\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}\xi ={\frac{d}{d\theta }\mid }_0{\widehat{\alpha }}_{\theta }^{\ast }\xi$. On physical and auxiliary fields, ${\delta }_{U(1)}(\omega )=-{\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}-{\delta }_R$: ${\delta }_R$ uses the source-ordered $R_{AB}^Q=2ϵ{\tilde{\mathrm{\Gamma }}}_{AB}\widetilde{ϵ}$, acting on scalars by $R_{AB}^Q{\mathrm{\Phi }}^B$ and on fermions by $\frac{1}{4}{R}_{AB}^Q{\mathrm{\Gamma }}^{AB}$, while the auxiliary-index part of ${\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}$ on $K_i$ is $v^{\mu }{\mathrm{\partial }}_{\mu }{K}_i+[{\beta }_{v,P},K_i]+{\mathrm{\Xi }}_{ij}{K}_j$, where ${\beta }_{v,P}$ is the fixed lift's local vertical coefficient and ${\mathrm{\Xi }}_{ij}={\nu }_{[i}{\mathrm{\Gamma }}^{\mu }{\mathrm{\nabla }}_{\mu }{\nu }_{j]}=\frac{1}{2}({\nu }_i{\mathrm{\Gamma }}^{\mu }{\mathrm{\nabla }}_{\mu }{\nu }_j-{\nu }_j{\mathrm{\Gamma }}^{\mu }{\mathrm{\nabla }}_{\mu }{\nu }_i)$. On the entire adjoint ghost and multiplier complex, including zero modes, ${\rho }_{\mathrm{g}\mathrm{h}}(\omega )=-{\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}$; the chosen lift fixes the zero-mode algebra. The global cohomological parameter is $\varphi =-{\kappa }_v$, with ${\kappa }_v=v^{\mu }{A}_{\mu }-{\beta }_{v,P}+v^A{\mathrm{\Phi }}_A$, so the source gauge-covariant Lie derivative is ${\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}+{\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}({\kappa }_v)$. The affine ghost rule is $\delta \varphi =-[c,\varphi ]+{\rho }_{\mathrm{g}\mathrm{h}}(\omega )c$, with $sc=\varphi$ and $s\varphi =0$. Thus the combined square is $Q^2={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(a_0)+{\delta }_{U(1)}(\omega )$: a gauge transformation by the zero-mode section $a_0$ and the selected combined spacetime/R action. In nontrivial gauge-bundle sectors $a_0\in \ker D_{A_{\ast }}$ is a parallel background zero mode, and the background-gauge functional $Q{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}$, ghost domains, and zero-mode complex are the sectorwise formal inputs specified for this model; no global gauge slice is asserted. The $U(1)$ base factor acts smoothly on the finite-dimensional manifold $S_r^4$; its ordinary Cartan complex is the invariant polynomial-valued differential-form complex for that base action. This finite Cartan complex concerns forms on $S_r^4$, not the infinite-dimensional field configuration space. The measure and noncommutative regulator extension remain formal assumptions of the adopted model.

[Adopted rigid S4 background postulate] Adopted rigid $N=4$ Yang–Mills localization background on the round four-sphere

[Flat-space $N=4$ model interface] Four-dimensional $N=4$ SU(2) super-Yang-Mills model

Pestun, §§2 and 4.1–4.3, Appendix C

Peelaers, §1