Finite-Regulator Q-Exact Localization Ward Identity

Use natural units $c=\mathrm{\hslash }=1$ and fix one bundle sector of the adopted massless $\mathcal{N}=4$ $SU(2)$ Yang–Mills model on the round Euclidean four-sphere $S_r^4$, with $r>0$ a length. The adopted background fixes the action, contour, field content, and total gauge-fixed odd transformation $Q=s+\delta$; on its fields and ghosts the square is $Q^2={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(a_0)+{\delta }_{U(1)}(\omega )$, where $a_0\in {\mathfrak{h}}_P=\ker D_{A_{\ast }}$ is the declared parallel background zero mode and ${\delta }_{U(1)}(\omega )=-{\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}-{\delta }_R$ is the combined lifted spacetime/R action, not a second copy of a gauge-covariant Lie derivative. The nongauge action on ghosts is ${\rho }_{\mathrm{g}\mathrm{h}}(\omega )=-{\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}$, and its auxiliary-index action is the one fixed by the source off-shell square. The source-ordered bilinear is $R_{AB}^Q=2ϵ{\tilde{\mathrm{\Gamma }}}_{AB}\widetilde{ϵ}$ and the cohomological parameter is $\varphi =-(v^{\mu }{A}_{\mu }-{\beta }_{v,P}+v^A{\mathrm{\Phi }}_A)$, where ${\beta }_{v,P}$ is the fixed lift's vertical coefficient. The adopted affine ghost rule is $\delta \varphi =-[c,\varphi ]+{\rho }_{\mathrm{g}\mathrm{h}}(\omega )c$. Sectorwise gauge fixing is the $Q$-completed background-gauge insertion $Q{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}$ with its specified ghost and zero-mode domains. This is a physical model assumption, not a deduction that the formal continuum functional integral exists.

For a fixed regulator label $\rho$, assume that a finite-dimensional regulator of this model is supplied by an oriented smooth field-parameter manifold ${\mathfrak{F}}_{\rho }$ of finite dimension $n$, a smooth positive volume form ${\mu }_{\rho }$, and integration over the whole cycle ${\mathfrak{F}}_{\rho }$. Let ${\mathcal{A}}_{\rho }={\mathrm{\Omega }}^{\bullet }({\mathfrak{F}}_{\rho };\mathbb{C})$, let $I_{\rho }(F)={\int }_{{\mathfrak{F}}_{\rho }}{F}_{[n]}$ be the integral of the degree-$n$ part, and let $Q_{\rho }$ be a smooth odd derivation. Supply a smooth parity-preserving graded map ${\iota }_{\rho }:\mathrm{\Pi }T{\mathfrak{F}}_{\rho }\to \mathcal{U}$ into the model's local field chart, where functions on $\mathrm{\Pi }T{\mathfrak{F}}_{\rho }$ are differential forms. Its body map is smooth, and its pullback assigns to every field coordinate a form of the declared parity, with smooth graded observables pulled back by substitution and finite Taylor expansion in positive-degree forms. Require $Q_{\rho }{\iota }_{\rho }^{\ast }F={\iota }_{\rho }^{\ast }QF$ for every chart observable $F$. Applying this twice requires $Q_{\rho }^2{\iota }_{\rho }^{\ast }F={\iota }_{\rho }^{\ast }{Q}^{2}F$, so the parametrized fields preserve the stated total $Q$ action and its square. Existence of this compatible regulator is a hypothesis. Let $J\subseteq \mathbb{R}$ be a nonempty open interval of dimensionless deformation parameters. Let $S_{\rho },O_{\rho }\in {\mathcal{A}}_{\rho }$ be even, let $V_{\rho }\in {\mathcal{A}}_{\rho }$ be odd, and define $E_{\rho ,t}=e^{-S_{\rho }-t{Q}_{\rho }{V}_{\rho }}$. In units with $\mathrm{\hslash }=1$, assume the regulated action $S_{\rho }$ and deformation $t{Q}_{\rho }{V}_{\rho }$ are dimensionless. Assume $Q_{\rho }{S}_{\rho }=Q_{\rho }{O}_{\rho }=0$ and $(Q_{\rho }^2{V}_{\rho })E_{\rho ,t}=0$ for every $t\in J$. Assume the regulated measure and cycle are Q-invariant on the admissible integrands, in the exact sense that $I_{\rho }(Q_{\rho }(O_{\rho }{V}_{\rho }{E}_{\rho ,t}))=0$ for every $t\in J$, with no anomaly or boundary contribution.

Write $(O_{\rho }{E}_{\rho ,t}{)}_{[n]}=f_{\rho ,t}{\mu }_{\rho }$ and assume $f_{\rho ,t}\in L^1({\mathfrak{F}}_{\rho },{\mu }_{\rho })$ for every $t\in J$. Assume there is a measurable ${\mu }_{\rho }$-null set $N_{\rho }\subseteq {\mathfrak{F}}_{\rho }$ such that, for every $x\in {\mathfrak{F}}_{\rho }\setminus N_{\rho }$, the map $t\mapsto f_{\rho ,t}(x)$ is differentiable at every $t\in J$. For $x\mathrm{\notin }{N}_{\rho }$ define

${\mathrm{\partial }}_t{f}_{\rho ,t}(x):={\lim }_{\begin{array}{c}h\to 0\\ h\in \mathbb{R},h\mathrm{\ne }0,t+h\in J\end{array}}\frac{f_{\rho ,t+h}(x)-f_{\rho ,t}(x)}{h}\in \mathbb{C},$

and define ${\mathrm{\partial }}_t{f}_{\rho ,t}(x)=0$ for $x\in N_{\rho }$. Require $x\mapsto {\mathrm{\partial }}_t{f}_{\rho ,t}(x)$ to be measurable for every $t\in J$. For every compact $K\subset J$, assume one nonnegative $g_K\in L^1({\mathfrak{F}}_{\rho },{\mu }_{\rho })$ and one ${\mu }_{\rho }$-null set $N_{\rho ,K}$ such that, for every $x\mathrm{\notin }{N}_{\rho ,K}$ and every $t\in K$,

$\mathrm{\mid }{f}_{\rho ,t}(x)\mathrm{\mid }\le g_K(x),\mathrm{\mid }{\mathrm{\partial }}_t{f}_{\rho ,t}(x)\mathrm{\mid }\le g_K(x)\mathrm{.}$

Then

$Z_{\rho }(t):=I_{\rho }(O_{\rho }{E}_{\rho ,t})$

is constant on $J$, and for every $t\in J$,

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

The conclusion applies only to this fixed finite-dimensional regulator and fixed bundle sector at finite $t$. The adopted S4 postulate does not construct the finite-dimensional data above, and this result supplies no infinite-dimensional continuum identity, $t\to \mathrm{\infty }$ localization limit, or regulator-removal theorem.

Adopted S4 model and gauge-fixed Q-square

Gauge-fixed fluctuation and regulator data

Finite-dimensional Q-exact deformation theorem

Pestun, Section 3

Pestun and Zabzine, Section 1

Folland, Theorem 2.27(b)

Schwarz and Zaboronsky, Section 3

Peelaers, Sections 1.1–1.2