Beta independence of the physical Witten index

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ and natural units $c=\mathrm{\hslash }=1$. A physical application supplies its spacetime and spatial domain, global gauge group and global form, field bundles and representations, adopted action or other postulates, boundary conditions, and either a Hamiltonian/Hilbert construction or the regulator and measure used to produce its operator data. The conclusion is conditional on those inputs yielding the following exact data. Let $\mathcal{H}={\mathcal{H}}^{+}\oplus {\mathcal{H}}^{-}$ be a complex Hilbert space with grading $\mathrm{\Gamma }=+I$ on ${\mathcal{H}}^{+}$ and $-I$ on ${\mathcal{H}}^{-}$. Let $\mathsf{Q}$ be a densely defined self-adjoint odd operator with $\mathrm{\Gamma }$-invariant domain, and let $K={\mathsf{Q}}^2$ be nonnegative and self-adjoint on its composition domain. Assume an orthogonal pure-point decomposition

$\mathcal{H}={\mathcal{H}}_0^{+}\oplus {\mathcal{H}}_0^{-}\oplus {\widehat{⨁}}_{\lambda \in \mathrm{\Sigma }}({\mathcal{H}}_{\lambda }^{+}\oplus {\mathcal{H}}_{\lambda }^{-}),$

where $\mathrm{\Sigma }\subset (0,\mathrm{\infty })$ is countable, each parity eigenspace ${\mathcal{H}}_{\lambda }^{\pm }=\ker (K-\lambda I)\cap {\mathcal{H}}^{\pm }$ is finite-dimensional and contained in $D(K)$, ${\mathcal{H}}_0^{\pm }=\ker K\cap {\mathcal{H}}^{\pm }$ is finite-dimensional, and the union of compatible orthonormal bases is a Hilbert basis. Assume $e^{-\beta K}$ is trace class for each $\beta >0$. Then the physical Witten index defined by the supplied basis is

$W_E(\beta )={\sum }_{e\in E^{+}}⟨e^{-\beta K}e,e⟩-{\sum }_{e\in E^{-}}⟨e^{-\beta K}e,e⟩=\dim {\mathcal{H}}_0^{+}-\dim {\mathcal{H}}_0^{-}$

for every $\beta >0$. The conclusion concerns only this discrete trace-class operator datum; it does not assert a continuous-spectrum cancellation, a validity theorem for a regulator, a continuum limit, or an identification of the kernel with every physical BPS state.

Tong, §1.2.2

Kinney et al., §3.1