Pure-point graded trace-class index data

A pure-point graded trace-class index datum consists of a complex Hilbert space $\mathcal{H}={\mathcal{H}}^{+}\oplus {\mathcal{H}}^{-}$, its grading operator $\mathrm{\Gamma }$ equal to $+I$ on ${\mathcal{H}}^{+}$ and $-I$ on ${\mathcal{H}}^{-}$, and a densely defined self-adjoint operator $Q$ such that $D(Q)$ is invariant under $\mathrm{\Gamma }$ and $Q\mathrm{\Gamma }\psi =-\mathrm{\Gamma }Q\psi$ for every $\psi \in D(Q)$. Set $H=Q^2$ on $D(H)=\{\psi \in D(Q):Q\psi \in D(Q)\}$ and require this composition operator to be nonnegative and self-adjoint. Assume $H$ has the 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 a supplied countable set of distinct eigenvalues, ${\mathcal{H}}_0^{\pm }=\ker H\cap {\mathcal{H}}^{\pm }$ are finite-dimensional, and ${\mathcal{H}}_{\lambda }^{\pm }=\ker (H-\lambda I)\cap {\mathcal{H}}^{\pm }$ are finite-dimensional eigenspaces contained in $D(H)$. Supply orthonormal bases of these subspaces whose union $E=E^{+}\bigsqcup E^{-}$ is a Hilbert basis of $\mathcal{H}$ compatible with energy and parity. Assume Countable Choice as used by the imported Hilbert-space, adjoint and trace-class definitions. The local positive-energy-pairing lemma derives, rather than assumes, the unitary map $Q\mathrm{/}\sqrt{\lambda }:{\mathcal{H}}_{\lambda }^{+}\to {\mathcal{H}}_{\lambda }^{-}$ and equality of their finite dimensions.

For each real $\beta >0$, a bounded diagonal operator $T_{\beta }$ is supplied as part of the data, with $T_{\beta }e=e^{-\beta \lambda }e$ on $E_{\lambda }^{+}\cup E_{\lambda }^{-}$ and $T_{\beta }e=e$ on $E_0^{+}\cup E_0^{-}$. Require $T_{\beta }$ to be trace class for every $\beta >0$; it is denoted $e^{-\beta H}$. Its graded heat supertrace, called the Witten index at parameter $\beta$ in this data class, is

$W(\beta ):={\sum }_{e\in E^{+}}⟨T_{\beta }e,e⟩-{\sum }_{e\in E^{-}}⟨T_{\beta }e,e⟩,$

where each family sum is in the finite-subset sense for absolutely summable scalar families. The supplied trace-class Hilbert-basis definition makes these sums absolutely convergent. Here $\beta$ is only a positive real mathematical parameter and has no assigned physical units or temperature interpretation. The index datum is discrete: it contains no continuous spectral subspace or continuum regularization prescription, and makes no physical model identification.

Martin, §7.1, supersymmetry-algebra positivity and the zero-energy condition.

Gates et al., §8.4, mass-squared supertrace formulae.

Tong, §1.2.2, Witten-index definition and positive-energy cancellation.

Positive-energy eigenspaces pair under an odd self-adjoint supercharge: Positive-energy eigenspaces pair under an odd self-adjoint supercharge.

Hilbert space: Hilbert space.

Densely defined, closed and closable operators: Densely defined, closed and closable operators, and cores.

Unbounded linear operators: domain, graph and extension: Unbounded linear operators: domain, graph and extension.

Symmetric, self-adjoint and essentially self-adjoint operators: Symmetric, self-adjoint and essentially self-adjoint operators.

Countable Choice: The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

Real and complex inner-product spaces and their induced length: Real and complex inner-product spaces and their induced length.

Orthonormal families, complete orthonormal systems and Hilbert bases: Orthonormal families, complete orthonormal systems and Hilbert bases.

Trace class operator: Trace class operator.

Trace of a trace class operator: Trace of a trace class operator.

The real exponential function and the number $e$ by a power series: The real exponential function and the number $e$ by a power series.

The exponential is a continuous bijection from $\mathbb{R}$ onto $(0,\mathrm{\infty })$: The exponential is a continuous bijection from $\mathbb{R}$ onto $(0,\mathrm{\infty })$.

The exponential function is strictly increasing: The exponential function is strictly increasing.