A Q-exact deformation with a nonzero boundary term

Let $M=[0,1]\times (\mathbb{R}\mathrm{/}2\pi \mathbb{Z})$ be the cylinder as a smooth manifold with boundary, with dimensionless product coordinates $x,y$; $dx,dy$ denote the global product and angular one-forms. Orient $M$ by the volume form $dx\wedge dy$ and define the graded integral $I(F)={\int }_M{F}_{[2]}$.

The complex differential-form algebra is ${\mathrm{\Omega }}^{\bullet }(M;\mathbb{C})={\mathrm{\Omega }}^0\oplus {\mathrm{\Omega }}^1\oplus {\mathrm{\Omega }}^2$, where $C^{\mathrm{\infty }}(M;\mathbb{C})$ is the space of functions smooth up to $x=0,1$ and periodic in $y$, ${\mathrm{\Omega }}^0=C^{\mathrm{\infty }}(M;\mathbb{C})$, ${\mathrm{\Omega }}^1=\{adx+bdy:a,b\in C^{\mathrm{\infty }}(M;\mathbb{C})\}$, ${\mathrm{\Omega }}^2=\{cdx\wedge dy:c\in C^{\mathrm{\infty }}(M;\mathbb{C})\}$, and ${\mathrm{\Omega }}^k=0$ for $k>2$. The product is the exterior product, determined by $dx\wedge dx=dy\wedge dy=0$ and $dy\wedge dx=-dx\wedge dy$. The exterior derivative is the degree-one linear map $d$ defined by $df=({\mathrm{\partial }}_{x}f)dx+({\mathrm{\partial }}_{y}f)dy$, $d(adx+bdy)=({\mathrm{\partial }}_{x}b-{\mathrm{\partial }}_{y}a)dx\wedge dy$, and $d\alpha =0$ for $\alpha \in {\mathrm{\Omega }}^2$; these formulas give the graded Leibniz rule and $d^2=0$.

Take the odd derivation $Q=d$, dimensionless action $S=0$, even insertion $O=1$, and odd deformation $V=xdy$. For every dimensionless real $t$, $Z(t)=I(O{e}^{-S-tQV})=-2\pi t$. The change occurs because $I(Q(V{e}^{-S-tQV}))=2\pi \mathrm{\ne }0$: this cycle violates the zero-boundary Q-invariance condition and therefore is not an instance of the finite-dimensional Ward identity.