Finite-Dimensional Q-Exact Localization Data

Let $M$ be an oriented smooth manifold of finite dimension $n$, possibly with boundary, and set the integration cycle to be $\mathrm{\Gamma }=M$ with its given orientation. Let $\mu$ be a smooth positive $n$-form on $M$. The parity-reversed tangent bundle $\mathrm{\Pi }TM$ has even base coordinates $x^1,\dots ,x^n$ and odd fiber coordinates ${\theta }^1,\dots ,{\theta }^n$ that transform as the coordinate one-forms $d{x}^i$. Its smooth complex-valued functions are identified with the inhomogeneous complex differential forms $\mathcal{A}={\mathrm{\Omega }}^{\bullet }(M;\mathbb{C})$; a form has parity equal to its degree modulo two. Use the Berezin convention $\int d{\theta }^n\cdots d{\theta }^1{\theta }^1\cdots {\theta }^n=1$. For each $F\in \mathcal{A}$, its degree-$n$ component has a unique expression $F_{[n]}=f_F\mu$ with $f_F\in C^{\mathrm{\infty }}(M;\mathbb{C})$. If $f_F\in L^1(M,\mu )$, define

$I_{\mathrm{\Gamma }}(F):={\int }_M{f}_{F}d\mu ={\int }_M{F}_{[n]},$

where the complex integral is taken componentwise. This is equivalently Berezin integration over the odd fibers followed by integration over the oriented body $M$.

A supercharge $Q$ is a smooth odd derivation of $\mathcal{A}$: for homogeneous forms $\alpha ,\beta$, it satisfies $Q(\alpha \wedge \beta )=Q\alpha \wedge \beta +(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}\alpha \wedge Q\beta$. Its square $Q^2:=Q\circ Q$ is an even derivation. The canonical Berezinian density on $\mathrm{\Pi }TM$ has local expression ${\mathit{\mu }}_{\mathrm{\Pi }}=d{x}^1\cdots d{x}^{n}d{\theta }^n\cdots d{\theta }^1$ in oriented coordinates. Define divergence relative to this density by

${\mathcal{L}}_Q{\mathit{\mu }}_{\mathrm{\Pi }}=({\mathrm{div}}_{{\mathit{\mu }}_{\mathrm{\Pi }}}Q){\mathit{\mu }}_{\mathrm{\Pi }}\mathrm{.}$

The condition ${\mathrm{div}}_{{\mathit{\mu }}_{\mathrm{\Pi }}}Q=0$ is local: it says that $Q$ preserves this density. It does not impose any boundary condition. Choose a fixed class $\mathcal{C}\subseteq \mathcal{A}$ as part of the integration data. For every $F\in \mathcal{C}$, require the coefficients of $F_{[n]}$, $(QF{)}_{[n]}$, and $(({\mathrm{div}}_{{\mathit{\mu }}_{\mathrm{\Pi }}}Q)F{)}_{[n]}$, relative to $\mu$, to be integrable. Also require a specified finite complex-linear functional ${\mathrm{\Phi }}_{\mathrm{\partial }\mathrm{\Gamma }}(Q,\cdot )$ on $\mathcal{C}$, the signed boundary or decay-at-infinity contribution, such that the chosen integration-by-parts identity is

$I_{\mathrm{\Gamma }}(QF)={\mathrm{\Phi }}_{\mathrm{\partial }\mathrm{\Gamma }}(Q,F)-I_{\mathrm{\Gamma }}(({\mathrm{div}}_{{\mathit{\mu }}_{\mathrm{\Pi }}}Q)F)\mathrm{.}$

This identity is an additional hypothesis on the integration data; its validity must be established from a fully stated integration-by-parts result with its hypotheses checked, or by a direct calculation. No boundary formula for arbitrary manifolds with boundary or noncompact manifolds is asserted here. For $F\in \mathcal{C}$, call ${\mathrm{\Phi }}_{\mathrm{\partial }\mathrm{\Gamma }}(Q,F)$ the $Q$-boundary flux of $F$. The integration is $Q$-invariant on $\mathcal{C}$ when the integrated identity

$I_{\mathrm{\Gamma }}(QF)=0\text{for every }F\in \mathcal{C}$

holds. The displayed integration-by-parts identity implies that zero divergence and, separately, zero $Q$-boundary flux give the integrated $Q$-invariance identity. Zero divergence alone does not imply it. For a compact manifold without boundary the boundary contribution is absent in the usual compact-supermanifold formula; on a noncompact manifold or one with boundary, vanishing flux must be established from the chosen decay and boundary conditions.

An admissible finite-dimensional $Q$-exact deformation consists of even smooth forms $S,O\in \mathcal{A}$, an odd smooth form $V\in \mathcal{A}$, and a nonempty open interval $J\subseteq \mathbb{R}$ such that the integration is $Q$-invariant on $\mathcal{C}$ and $QS=QO=0$. The even form $QV$ is the $Q$-exact deformation, $S_t:=S+tQV$, and $E_t:=e^{-S_t}$, where the exponential is the pointwise exponential of the degree-zero part times the finite exterior-degree expansion of its positive-degree part. Require $(Q^{2}V)E_t=0$ for every $t\in J$; the stronger condition $Q^{2}V=0$ is sufficient. Require $O{E}_t$ and $OV{E}_t$ to belong to $\mathcal{C}$ for every $t\in J$, so $I_{\mathrm{\Gamma }}(Q(OV{E}_t))=0$. If this identity is established by graded integration by parts, its zero-divergence and zero-boundary-flux hypotheses are separate.

Write $(O{E}_t{)}_{[n]}=f_t\mu$. The deformation data are analytically admissible when $f_t\in L^1(M,\mu )$ for every $t\in J$, and for $\mu$-almost every $x$ the map $t\mapsto f_t(x)$ is differentiable at every $t\in J$ as a map from a real interval to $\mathbb{C}$, with

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

For each $t$, require $x\mapsto {\mathrm{\partial }}_t{f}_t(x)$ to be measurable, using zero on the common exceptional null set. For every compact $K\subset J$, require a single nonnegative $g_K\in L^1(M,\mu )$ that bounds both $\mathrm{\mid }{f}_t(x)\mathrm{\mid }$ and $\mathrm{\mid }{\mathrm{\partial }}_t{f}_t(x)\mathrm{\mid }$ for every $t\in K$ outside one $\mu$-null set. The underlying manifold is finite-dimensional; its smooth-function algebra need not be finite-dimensional as a vector space.

Schwarz and Zaboronsky, §2, p. 3: canonical Berezin volume and divergence identity for a compact boundaryless supermanifold

Pestun and Zabzine, §1

Folland, Theorem 2.27