Finite-regulator Ward identity for the $N=2^{\ast }$ U(2) sphere model

Fix one bundle sector of the adopted $\mathcal{N}=2^{\ast }$ $U(2)$ model on round $S_r^4$, with $r>0$, real Euclidean adjoint-hypermultiplet mass $m_E$, and the source's gauge-fixed odd transformation including its mass-dependent flavor action. Use $c=\mathrm{\hslash }=1$. Suppose a regulator realization of THIS model is supplied as an oriented finite-dimensional smooth manifold $M_{\rho }$ of dimension $n$, positive volume form ${\mu }_{\rho }$, graded algebra ${\mathrm{\Omega }}^{\bullet }(M_{\rho };\mathbb{C})$, and top-degree integral $I_{\rho }(F)={\int }_{M_{\rho }}{F}_{[n]}$. Its smooth odd derivation $Q_{\rho }$ must realize the selected model transformation and the total square $Q^2={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(a_0)+\omega$ on regulated fields and ghosts, with the source-ordered $R_{AB}^Q=2ϵ{\tilde{\mathrm{\Gamma }}}_{AB}\widetilde{ϵ}$, the lift/gauge decomposition, the induced seven-auxiliary action, and the sectorwise $Q$-completed gauge fixing stated for that model; no massless $SU(2)$ regulator or regulator existence is inferred.

Let $J\subset \mathbb{R}$ be a nonempty open interval of dimensionless parameters, $S_{\rho },O_{\rho }$ smooth even forms and $V_{\rho }$ a smooth odd form, with dimensionless action $S_{\rho }+t{Q}_{\rho }{V}_{\rho }$. Put $E_t=e^{-S_{\rho }-t{Q}_{\rho }{V}_{\rho }}$. Require $Q_{\rho }{S}_{\rho }=Q_{\rho }{O}_{\rho }=0$, $(Q_{\rho }^2{V}_{\rho })E_t=0$, and the exact anomaly-free, measure/cycle/boundary identity $I_{\rho }(Q_{\rho }(O_{\rho }{V}_{\rho }{E}_t))=0$ for all $t\in J$. For $(O_{\rho }{E}_t{)}_{[n]}=f_t{\mu }_{\rho }$, assume each $f_t$ is integrable and there is one measurable null set $N\subset M_{\rho }$ such that for every $x\mathrm{\notin }N$, the map $t\mapsto f_t(x)$ is differentiable throughout $J$. For each $t\in J$, require the derivative $x\mapsto {\mathrm{\partial }}_t{f}_t(x)$, set to zero on $N$, to be measurable. For every compact $K\subset J$, assume one $g_K\in L^1(M_{\rho },{\mu }_{\rho })$ and a measurable null set $N_K$ bound both $\mathrm{\mid }{f}_t(x)\mathrm{\mid }$ and $\mathrm{\mid }{\mathrm{\partial }}_t{f}_t(x)\mathrm{\mid }$ by $g_K(x)$ for every $t\in K$ and every $x\mathrm{\notin }{N}_K$. Then

$Z_{\rho }(t)=I_{\rho }(O_{\rho }{E}_t),Z_{\rho }^{\mathrm{\prime }}(t)=-I_{\rho }(O_{\rho }(Q_{\rho }{V}_{\rho })E_t)=-I_{\rho }(Q_{\rho }(O_{\rho }{V}_{\rho }{E}_t))=0,$

and $Z_{\rho }$ is constant on $J$. This is conditional on the stated finite regulator, at finite parameters and fixed sector. It proves no $t\to \mathrm{\infty }$ saddle limit, noncompact ADHM pushforward, ADHM-to-Nekrasov pole equality, north/south regulator match, regulator removal or continuum functional measure. Those separate physical and analytic obligations remain necessary for a sphere partition-function formula.