Positive-Supercharge Pair Cancellation in the Superconformal Index

Let $\mathcal{H}={\mathcal{H}}^{\bar{0}}\oplus {\mathcal{H}}^{\bar{1}}$ be a ${\mathbb{Z}}_2$-graded Hilbert space in a unitary representation of the selected four-dimensional superconformal algebra. Let $Q$ be a densely defined closed odd operator, and let $S=Q^{†}$ be its Hilbert adjoint. The grading preserves $D(Q)$ and $D(S)$, and $Q$ and $S$ interchange the two parity subspaces on their domains. Assume a common invariant dense domain $D$ on which $Q^2=S^2=0$, and let $\delta$ be a nonnegative self-adjoint operator. For compositions, set $D(QS)=\{\psi \in D(S):S\psi \in D(Q)\}$ and $D(SQ)=\{\psi \in D(Q):Q\psi \in D(S)\}$. Require

$D(QS)\cap D(SQ)\subseteq D(\delta ),\delta \psi =2(SQ+QS)\psi (\psi \in D(QS)\cap D(SQ)),$

not merely as an extension of an identity on test vectors, and also require

$\ker \delta \subseteq D(QS)\cap D(SQ)\mathrm{.}$

Assume that $D_Q=\sqrt{2}(Q+S)$, with domain $D(Q)\cap D(S)$, is self-adjoint and that $D_Q^2=\delta$ as a closed-operator equality. Let $B$ be a countable set and suppose

$\mathcal{H}={\widehat{⨁}}_{b\in B}{\mathcal{H}}_b,$

where each ${\mathcal{H}}_b={\mathcal{H}}_b^{\bar{0}}\oplus {\mathcal{H}}_b^{\bar{1}}$ is a finite-dimensional graded subspace. Write $P_b$ for its orthogonal projection. Require $P_{b}D(Q)\subseteq D(Q)$ and $Q{P}_b=P_{b}Q$ on $D(Q)$, and likewise $P_{b}D(S)\subseteq D(S)$ and $S{P}_b=P_{b}S$ on $D(S)$. Define $Q_b=Q{\mathrm{\mid }}_{{\mathcal{H}}_b}$ and $S_b=S{\mathrm{\mid }}_{{\mathcal{H}}_b}$; require ${\mathcal{H}}_b\subseteq D(QS)\cap D(SQ)$, $Q_b^2=S_b^2=0$, and $S_b=Q_b^{\ast }$ as finite-dimensional operators. Require the exact maximal direct-sum domains

$D(Q)=\{\psi ={⨁}_b{\psi }_b\in \mathcal{H}:{\sum }_b\mathrm{\parallel }{Q}_b{\psi }_b{\mathrm{\parallel }}^2<\mathrm{\infty }\},D(S)=\{\psi ={⨁}_b{\psi }_b\in \mathcal{H}:{\sum }_b\mathrm{\parallel }{S}_b{\psi }_b{\mathrm{\parallel }}^2<\mathrm{\infty }\}\mathrm{.}$

Let $\delta$ be the diagonal self-adjoint operator $\delta {\mathrm{\mid }}_{{\mathcal{H}}_b}={\lambda }_{b}I$ with maximal domain $D(\delta )=\{\psi ={⨁}_b{\psi }_b\in \mathcal{H}:{\sum }_b{\lambda }_b^2\mathrm{\parallel }{\psi }_b{\mathrm{\parallel }}^2<\mathrm{\infty }\}$, where ${\lambda }_b\ge 0$. Repeated values of ${\lambda }_b$ are allowed, so full gap eigenspaces and $\ker \delta$ may be infinite-dimensional. Let $U={⨁}_b{U}_b$ be an even operator with maximal direct-sum domain $D(U)=\{\psi ={⨁}_b{\psi }_b\in \mathcal{H}:{\sum }_b\mathrm{\parallel }{U}_b{\psi }_b{\mathrm{\parallel }}^2<\mathrm{\infty }\}$, and require each finite-dimensional restriction to commute with $Q_b$ and $S_b$. Require $e^{-\beta \delta }U$ to extend to a trace-class operator for every $\beta >0$. This includes the bounded-insertion case when each full gap eigenspace is finite-dimensional and its spectral projection reduces $Q$ and $S$ with the stated domain conditions. It also includes the full chemical-potential character in the earlier definition when each common $(\delta ,{\mathbf{J}}^{\mathrm{\prime }})$ eigenspace is grouped together with both parity subspaces as one finite-dimensional graded block reducing $Q$ and $S$; its finite dimension follows from the nonzero exponential weight and trace-class assumption. Define

$\mathcal{I}(\beta )={\mathrm{Tr}}_{\mathcal{H}}\text{\hspace{0.17em}⁣}((-1{)}^F{e}^{-\beta \delta }U),$

where $(-1{)}^F$ acts as $+1$ on ${\mathcal{H}}^{\bar{0}}$ and $-1$ on ${\mathcal{H}}^{\bar{1}}$. Then

$\mathcal{I}(\beta )={\mathrm{Tr}}_{\ker \delta }\text{\hspace{0.17em}⁣}((-1{)}^{F}U)$

for every $\beta >0$, where the right-hand side is the trace of the trace-class extension of $U{\mathrm{\mid }}_{\ker \delta }$ supplied by the trace-class hypothesis. In particular, this trace is independent of $\beta$, and $\ker \delta =\ker Q\cap \ker S$, with each kernel taken on the domain of its operator. The conclusion applies to the superconformal index only when its Hilbert-space, operator-domain, reducing-block, commuting-charge, and trace hypotheses above hold; it makes no conclusion for continuous spectrum or a non-trace-class formal path integral.

Kinney, Maldacena, Minwalla, and Raju, §3, equations (3.1)–(3.3)

David Tong, Supersymmetric Field Theory, §§2.2–2.3 and 6.4.2