Four-dimensional $N=1$ superconformal index

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ and natural units $c=\mathrm{\hslash }=1$. A complex Hilbert space is a complete complex inner-product space, with inner product linear in its first variable. A self-adjoint operator $T$ is densely defined and satisfies $D(T)=D(T^{\ast })$ and $T\psi =T^{\ast }\psi$ there; nonnegative means $⟨T\psi ,\psi ⟩\ge 0$ for all $\psi \in D(T)$. A grading operator $\mathrm{\Gamma }$ is a bounded self-adjoint involution; its $+1$ and $-1$ eigenspaces are orthogonal, and a densely defined operator $T$ is odd if $D(T)$ is $\mathrm{\Gamma }$-invariant and $T\mathrm{\Gamma }=-\mathrm{\Gamma }T$ there. For operators $A,B$, $D(AB)=\{\psi \in D(B):B\psi \in D(A)\}$ is the composition domain. A Hilbert basis is a complete orthonormal family. Let $S^3=\{x\in {\mathbb{R}}^4:\mathrm{\mid }x\mathrm{\mid }=1\}$ have its round metric and standard orientation. The radial Hilbert space ${\mathcal{H}}_{S^3}$ is the complex Hilbert space supplied by the chosen physical model after quantization on this unit spatial three-sphere; its construction is model input, not proved here. The model record names its gauge group and global form, matter representations, action or other adopted postulates, and the Hilbert-space construction or regulator and measure; the spatial sphere has no boundary, and any other boundary or domain conditions are stated as input. The rotation group is $\mathrm{Spin}(4)=SU(2{)}_1\times SU(2{)}_2$. Let $E$ be the dimensionless self-adjoint conformal Hamiltonian, $J_1,J_2$ the self-adjoint Cartan generators of the two rotation factors, and $R$ the self-adjoint $U(1{)}_R$ charge. Normalize $R$ so that the chosen right-handed Poincare supercharge $Q={\tilde{Q}}_{\dot{-}}$ has $R$-charge $+1$ and $J_2$-weight $-\frac{1}{2}$; it has $J_1$-weight zero. Its actual Hilbert adjoint in radial quantization is the conformal supercharge $Q^{†}={\tilde{S}}^{\dot{-}}$. The model supplies a closed densely defined operator $Q$ and its actual Hilbert adjoint $Q^{†}$, the physical fermion-parity grading $\mathrm{\Gamma }=(-1{)}^F$, and a dense linear domain $\mathcal{D}$ invariant under $\mathrm{\Gamma },Q,Q^{†}$ and contained in $D(Q^2)\cap D((Q^{†}{)}^2)\cap D(Q{Q}^{†})\cap D(Q^{†}Q)$. Here $F$ is the model's fermion number modulo two: $\mathrm{\Gamma }$ acts as $+1$ on physical bosonic states and $-1$ on physical fermionic states, and the vacuum is even. This physical parity identification is part of the model input and fixes the sign of the index. Require $Q,Q^{†}$ to be odd, $Q^2=(Q^{†}{)}^2=0$ on $\mathcal{D}$, and the exact positive self-adjoint operator relation

$\mathrm{\Delta }=\frac{1}{2}\{Q,Q^{†}\}=E-2J_2-\frac{3}{2}R\ge 0.$

The protected sector is $\ker \mathrm{\Delta }=\ker Q\cap \ker Q^{†}$, since on the common domain $⟨\mathrm{\Delta }\psi ,\psi ⟩=\frac{1}{2}(\mathrm{\parallel }Q\psi {\mathrm{\parallel }}^2+\mathrm{\parallel }{Q}^{†}\psi {\mathrm{\parallel }}^2)$. Define the two dimensionless commuting charges

$J_{+}=J_1+J_2+\frac{1}{2}R,J_{-}=J_2-J_1+\frac{1}{2}R\mathrm{.}$

On a separately supplied dense domain $D(\mathsf{Q})\subseteq D(Q)\cap D(Q^{†})$, define the odd operator $\mathsf{Q}=(Q+Q^{†})\mathrm{/}\sqrt{2}$ and require it to be self-adjoint and to satisfy ${\mathsf{Q}}^2=\mathrm{\Delta }$ as a closed-operator equality. Its domain is $\mathrm{\Gamma }$-invariant and $\mathrm{\Gamma }\mathsf{Q}=-\mathsf{Q}\mathrm{\Gamma }$ there. Supply a simultaneous pure-point decomposition for $\mathrm{\Delta },J_{+},J_{-}$ and $\mathrm{\Gamma }$, with a compatible complete orthonormal basis $\mathcal{B}={\mathcal{B}}^{+}\bigsqcup {\mathcal{B}}^{-}$. More explicitly, the basis vectors have real charge eigenvalues $(\lambda ,a,b)$, with $\lambda \ge 0$, and span the joint subspaces ${\mathcal{H}}_{\lambda ,a,b}^{\pm }$; the distinct joint labels form a supplied countable set. Each closed self-adjoint charge has its maximal diagonal domain: for eigenvalues $d_e$ its domain consists of the coefficient sequences $\psi ={\sum }_e{c}_{e}e$ with ${\sum }_e\mathrm{\mid }{d}_e{c}_e{\mathrm{\mid }}^2<\mathrm{\infty }$. This specifies the joint spectral calculus, not merely formal commutators on test vectors. The total eigenspaces of $\mathrm{\Delta }$, including positive ones, may be infinite-dimensional.

For each joint label $(\lambda ,a,b)$, let $P_{\lambda ,a,b}$ be the orthogonal projection onto ${\mathcal{H}}_{\lambda ,a,b}^{+}\oplus {\mathcal{H}}_{\lambda ,a,b}^{-}$. Require reducing-domain commutation with the closed supercharges: $P_{\lambda ,a,b}D(T)\subset D(T)$ and $T{P}_{\lambda ,a,b}\psi =P_{\lambda ,a,b}T\psi$ for $T=Q,Q^{†},\mathsf{Q}$ and $\psi \in D(T)$. The grading preserves these subspaces and anticommutes with the odd operators on their domains. Thus the given commuting fugacity charges preserve the actual closed supercharge domains and the finite-span joint eigenvector domain; no integrated spectral conclusion is inferred from a common-domain commutator alone.

For positive real fugacities $p,q\in (0,1)$, define ${\mathrm{\Xi }}_{p,q}=p^{J_{+}}{q}^{J_{-}}$ by the supplied joint diagonal calculus, with eigenvalue $p^a{q}^b$ and its maximal diagonal domain. Define $T_{\beta ,p,q}$ first on the finite span of $\mathcal{B}$ by $T_{\beta ,p,q}e=p^a{q}^b{e}^{-\beta \lambda }e$. An admissible pair is one for which this positive diagonal operator extends to a bounded trace-class operator for every dimensionless $\beta >0$; denote that extension by ${\mathrm{\Xi }}_{p,q}{e}^{-\beta \mathrm{\Delta }}$. This notation specifies the closed diagonal product, without assuming that an arbitrary product of unbounded operators is everywhere defined. Its supplied-basis diagonal sum is then absolutely convergent. Every joint charge block is finite-dimensional for an admissible pair: its common eigenvalue $p^a{q}^b{e}^{-\beta \lambda }$ is strictly positive, so infinitely many orthonormal vectors of that weight would contradict trace-class summability. Infinitely many joint blocks, an infinite protected kernel, and infinite total multiplicity at fixed $\lambda$ are allowed. The right-handed four-dimensional $N=1$ superconformal index is

${\mathcal{I}}_R(p,q;\beta )={\sum }_{e\in {\mathcal{B}}^{+}}⟨{\mathrm{\Xi }}_{p,q}{e}^{-\beta \mathrm{\Delta }}e,e⟩-{\sum }_{e\in {\mathcal{B}}^{-}}⟨{\mathrm{\Xi }}_{p,q}{e}^{-\beta \mathrm{\Delta }}e,e⟩,$

where $\mathcal{B}={\mathcal{B}}^{+}\bigsqcup {\mathcal{B}}^{-}$ is the supplied joint energy/charge/parity eigenbasis. The positive supercharge gap singles out $\ker \mathrm{\Delta }$ as the protected sector, with signed fugacity weight ${\mathrm{\Xi }}_{p,q}$; equality of the full heat supertrace with a trace over that sector is a separate conditional theorem. Other supercharges or central-charge sectors are not silently included. If the weighted trace-class or absolute-convergence condition fails, the expression is only a formal series until a separate regularization and its validity are supplied. No analytic convergence is asserted for all $p,q\in (0,1)$.

Kinney et al., §3

Rastelli and Razamat, §2.1

The Axiom of Countable Choice

Trace class means a compact operator with summable singular values

A trace is the absolutely convergent diagonal sum in a supplied Hilbert basis

A rigid background fixes the spin/R geometry and model-specific spinor equation; it asserts no arbitrary-background existence