Spectral Invariant of Supercharge

• Spectral invariant of supercharge is defined as a function of Q's eigenvalues, capturing complete spectral data beyond mere zero-mode counting.
• It is computed via a modular integration over the torus domain using regularized traces of the partition function to ensure convergence and modular invariance.
• This invariant classifies SQFTs by detecting refined topological, geometric, and anomaly cancellation features, extending traditional elliptic genus methods.

A spectral invariant of a supercharge is a numerical or cohomological invariant derived from the spectrum of the (super)symmetry generator Q in a supersymmetric quantum system, quantum field theory, or family of Dirac-type operators. These invariants provide distinguished probes into the structure of supersymmetric states, constraints on anomaly cancellation, the classification of SQFTs, and quantitative measures of geometric or topological data encoded in the supercharge’s spectrum.

1. Definition and Conceptual Foundations

A spectral invariant of a supercharge is a function of the eigenvalues (and possibly eigensections) of the supercharge operator Q. In minimal supersymmetric quantum field theories—such as 2d $\mathcal{N}{=}(0,1)$ SQFTs—Q is typically an odd (fermionic) generator whose anticommutator squares to the Hamiltonian. The spectral invariants generalize classical ideas such as the Atiyah–Patodi–Singer (APS) η-invariant for Dirac operators to the context of supercharges, encoding refined spectral information not limited to index-theoretic (zero-mode counting) aspects.

In the context of two-dimensional SQFTs, the spectral invariant of the supercharge is defined by integrating a regulated expectation value of Q (often a trace over the Hilbert space weighted by suitable modular factors and insertions) over the fundamental region F of $SL(2,\mathbb{Z})$ in the upper half-plane. Explicitly, if $Z(\tau)$ is the partition function and $\langle Q \rangle$ is the (appropriately regularized) expectation value of Q, the invariant takes the form

$$\eta_{(S)} = \kappa \int_{F} \frac{d\tau\,d\bar{\tau}}{2 i \tau_2^{3/2}}\,\eta(\tau)^3 \langle Q \rangle,$$

where $\eta(\tau)$ is the Dedekind eta function, $\kappa$ a normalization depending on the CPT structure, and $F$ the modular fundamental domain (Tachikawa et al., 6 Aug 2025).

This formalism ensures that the spectral invariant captures the entire eigenvalue distribution of Q, not just the net number of zero modes, and can exhibit sensitivity to torsion or refined global structures beyond what is accessible to index or genus invariants.

2. Construction: Integration over Moduli and Partition Function Structure

The core construction of the spectral invariant in (0,1) SQFTs involves the following steps:

• The partition function $Z(\tau)$ of the SQFT is assembled to be modular-invariant up to the gravitational anomaly and is typically weighted by an inverse Dedekind eta factor, $Z(\tau) = \eta(\tau)^{-\nu} \sum_n q^n (\dots)$.
• The expectation value of Q, $\langle Q \rangle$, is defined via a regularized trace, typically as $$\eta(\tau)^{-\nu} \sum_n q^n \mathrm{Tr}_n[|q|^{2Q^2} Q]$$, ensuring convergence and appropriate modular transformation properties.
• The spectral invariant is computed as the (regularized) integral of $\langle Q \rangle$ (multiplied by modular-covariant measures) over the modular fundamental domain $F\subset \mathbb{H}$, i.e., all inequivalent tori or conformal structures.
• Normalization $\kappa$ is adjusted according to whether the theory is ordinary or possesses, for instance, Kramers degeneracy.
• If Q has zero modes, an additional correction term proportional to their count is included (Tachikawa et al., 6 Aug 2025).

This integral essentially "averages" the spectrum of Q over the space of torus complex structures, a process tightly connected to both physical modular invariance (consistency of the toroidal quantum theory) and topological invariants in the Stolz–Teichner program relating SQFTs to topological modular forms (TMF).

3. Spectral Pairing and Detection of Invariants

A central innovation enabled by the spectral invariant is the notion of "spectral pairing": a bilinear pairing on bordism classes of $(0,1)$ SQFTs whose gravitational anomalies sum to $-22$. For two such classes, $[\varphi]$ and $[\psi]$, the pairing $( [\varphi], [\psi] ) \in \mathbb{R}/\mathbb{Z}$ is defined by constructing a product theory, extending it to a higher-dimensional "bulk" theory $\mathcal{L}$, and extracting the $q^0$ term from the spectral invariant of this bulk: $$( [\varphi], [\psi] ) = \frac{1}{k}\left[ \eta(\mathcal{L}) \right]_{q^0},$$ where $k$ is the multiplicity of the extension required to render the boundary null (Tachikawa et al., 6 Aug 2025).

This pairing is designed to recover all previously known invariants:

• Primary: elliptic genus, mod-2 elliptic genus;
• Secondary: APS $\eta$-invariant analog when continuous variation vanishes; by "sampling" from the full Q-spectrum. The spectral data thereby exhaust the known invariant content of $(0,1)$ SQFTs and, in particular, reveal "new" (tertiary) invariants otherwise inaccessible to index-theoretic techniques alone.

The construction is closely parallel to Anderson duality in TMF and is conjectured in the Stolz–Teichner program to classify field theories up to deformation.

4. Comparison with Primary/Secondary Invariants

The spectral invariant is sharply distinguished from classical invariants:

• Elliptic genus: a (holomorphic) index counting BPS (zero-energy) states, insensitive to the detailed supercharge spectrum.
• Mod-2 elliptic genus: a similar mod-2 index (detecting Kervaire or Arf invariants).
• Secondary/APS $\eta$-invariant analogues: detect spectral asymmetry; only when the "primary" invariants vanish.

The spectral invariant, via integration over $\langle Q \rangle$ and its modular properties, is sensitive to the full supercharge spectrum—including continuous deformations and torsion information that escapes (co)homological invariants. Its construction also allows a systematic reduction: when primary and secondary invariants vanish, the spectral invariant (and the spectral pairing) detect the residual "tertiary" data parameterizing finer deformation classes of SQFTs (Tachikawa et al., 6 Aug 2025).

Its advantages include:

• Uniform detection of all known and conjectured invariants,
• A modular-invariant and "spectral" methodology,
• Natural incorporation into the holomorphic anomaly equation, which relates non-holomorphic dependence on $\tau$ to topological data.

5. Geometric and Physical Applications

Spectral invariants of the supercharge have several implications:

• Moduli space constraints: They classify distinct (0,1) SQFTs modulo deformation, directly probing topological modular forms and their torsion components.
• Anomaly cancellation in heterotic string theory: In theories with gravitational anomaly $-22$, spectral pairing reproduces the discrete phase of the Green–Schwarz mechanism, giving a precise tool for global anomaly computation (Tachikawa et al., 6 Aug 2025).
• Index theorems and holomorphic anomaly: The derivation via integrating the non-holomorphic derivative of the partition function (holomorphic anomaly equation) parallels generalized index theorems, blending analytic and topological perspectives.
• Extensions to quantum mechanics, BPS state counting, and moduli spaces: Similar spectral invariants are expected to play a role in the spectral geometry of supersymmetric quantum mechanics and microscopic counting in supersymmetric gauge and string theories (suggested by analogy to Dirac operators and spectral flow constructions).

6. Broader Mathematical Significance

Mathematically, the spectral invariant of supercharge provides a bridge between spectral geometry (encoding eigenvalue data of Dirac-type operators or supercharges) and refined algebraic topology (e.g., TMF, Anderson duality, and secondary or tertiary invariants in quantum field theory). The integration over modular space and the connection to global anomalies reinforce its role as a geometrically and physically robust invariant.

The construction and systematic use of such spectral invariants broaden the toolkit for distinguishing and classifying supersymmetric quantum field theories, detecting subtle anomalies, and understanding the interplay of geometry, topology, and spectral theory in contemporary mathematical physics.

Summary Table

Invariant Type	Definition / Construction	Sensitivity
Elliptic genus	Tr$[(–1)^F q^{H_L}]$ (zero-mode trace)	BPS/zero-energy sector
Mod-2 elliptic genus	Mod-2 Witten index	$\mathbb{Z}_2$ torsion, Arf/Kervaire
APS $\eta$-invariant	Spectral asymmetry (Dirac operator, traditional setup)	Spectral flow across zero
Spectral invariant	$\int_F (...) \langle Q \rangle$ over $SL(2,\mathbb{Z})$; modular	Full Q-spectrum, deformation, torsion
Spectral pairing	Bilinear form from bulk extension and spectral invariant	Systematic detection of all above

The spectral invariant of the supercharge, especially as constructed via $SL(2,\mathbb{Z})$-modular integration, is thus a refined analytic-topological tool, unifying and extending the known arsenal of deformation and anomaly invariants in supersymmetric quantum field theory (Tachikawa et al., 6 Aug 2025).