Non-Graviton Operators in Quantum Theories

• Non-Graviton Operators are defined as operators in QFT and string theory that do not create or annihilate graviton states, characterized by unique algebraic and topological features.
• They encompass higher-dimensional contact terms, protected BPS cohomologies, and topological surface charges, which are crucial in effective field theory and holographic analyses.
• Their study advances precision tests of EFT, reveals black hole microstructure, and addresses challenges in nonperturbative gauge-gravity duality.

Non-graviton operators are quantum field theory and string theory constructions that do not create or annihilate graviton states. They comprise a wide spectrum of objects: higher-dimensional contact terms in effective field theories, protected cohomologies dual to black-hole microstates, topological surface charges in gravity, and symmetry generators not associated with metric fluctuations. These operators play crucial roles in precision tests of effective field theory, holography, nonperturbative gauge-gravity duality, and the algebraic structure of quantum gravity.

1. Classification and Definitions

Non-graviton operators can be organized by physical context and structural properties:

• Local Effective Operators in Gauge Theory and EFT: These include Standard Model Effective Field Theory (SMEFT) couplings, higher-dimensional contact terms, and photon-matter interactions that are not associated with graviton exchange or creation.
• Protected Cohomologies and Black Hole Microstates: In supersymmetric gauge theories, non-graviton operators may represent BPS states not accounted for by multi-graviton combinatorics and instead probe the onset of black hole microstructure and finite-$N$ effects (Choi et al., 2022, Gadde et al., 16 Jun 2025).
• Topological Surface Charges in Linearized Gravity: These operators, such as Wilson-'t Hooft analogues supported on surfaces or cycles, generate higher-form symmetries, commute with all local graviton Fock space generators, and act as probes of topological sectors (Hull et al., 13 Dec 2024).
• Dressed Gauge-Invariant Operators in Quantum Gravity: Diffeomorphism-invariant observables constructed via matter dressings (Wilson-line, Coulomb or worldline cloud), which create proper Dirac observables but do not exhaust the full set of graviton creation operators (Donnelly et al., 2015).

The unifying feature is the absence of overlap with the Fock-space of graviton states—a consequence reflected in their algebraic, cohomological, or geometric properties.

2. Local Non-Graviton Operators in Effective Field Theory

In four-dimensional local EFT, only specific operators modify soft theorems or enter photon-matter interactions without graviton involvement:

• Pauli Dipole Operator (Charged Fermion):

$$\mathcal{L} \supset c_\chi\,\overline\chi\,\sigma^{\mu\nu}F_{\mu\nu}\chi$$

Dimension: $d=5$, chirality: $h_\chi = \pm \tfrac{1}{2}$. Generates $\mathcal{O}(1/\epsilon)$ anomalies in soft photon theorems at subleading order.

• Scalar–Photon and Pseudoscalar–Photon Operators:

$$\mathcal{L} \supset c_s\,\phi\,F_{\mu\nu}F^{\mu\nu} \qquad \mathcal{L} \supset c_p\,\tilde{\phi}\,F_{\mu\nu}\tilde F^{\mu\nu}$$

Both dimension five, CP-even and CP-odd. Only these three operators (and their CP conjugates) can produce universal kinematic factors in the subleading soft photon expansion: $[sk]/\langle sk\rangle$ times an $n$-point amplitude with a matter replacement (Elvang et al., 2016).

Higher-derivative ($d \geq 6$) local operators, such as $(\partial F)^2F$ or $F^3$, do not contribute at leading or subleading order in the $\epsilon$ expansion, being forbidden by helicity and dimension selection rules.

3. Non-Graviton Operators in Holographic Gauge Theories

Non-graviton operators arise in protected cohomologies, especially in theories with a gravity dual:

• Threshold Non-Graviton BPS Operators (SU(2) SYM): At finite $N=2$, the first gauge-invariant operator not reducible to multi-graviton structure appears in the charge sector $E=19$, $(R_1,R_2,R_3)=(3,3,3)$, $(J_1,J_2)=(\tfrac{1}{2},\tfrac{1}{2})$ (Choi et al., 2022). Its explicit form leverages the SU(2) trace relation and is not exact in $Q$-cohomology:

$$\mathcal{O}(X, Y, Z, \psi) \propto (\psi_1\cdot X-\psi_2\cdot Y) (\psi_3\cdot X)(\psi_4\cdot Z)(\psi_1\times\psi_4) + \text{cyclic perms}$$

This operator is interpreted as a finite-$N$ BPS microstate beyond the supergraviton tower, marking the onset of black-hole degeneracy in the index.

• BMN Non-Graviton Generators and S-Duality: In $\mathcal{N}=4$ SYM, poles $(1-t^q)^{-1}$ in the BMN sector index (with $q > 2N+4$) reveal bosonic towers of protected operators incompatible with graviton cohomology, suggesting a novel Fock space for black hole microstates (Gadde et al., 16 Jun 2025). S-duality breaking in BMN sectors for SO$(2N+1)$ vs. Sp$(N)$ identifies explicit non-graviton cohomologies and necessitates new subrings for restored duality at finite coupling.
• Giant-Graviton and Heavy Operators (ABJM): In SU(2) subsectors, restricted Schur polynomials built from $A_i, B_j$ fields and double-coset ansatz yield non-graviton heavy operators dual to giant gravitons and their open-string excitations (Koch et al., 2014). Their anomalous dimensions depend explicitly on the giant's size and angular momentum, and integrability fails at finite $N$.

4. Topological and Surface Operators in Linearized Gravity

Purely topological non-graviton operators are constructed as surface charges in the linearized graviton or dual-graviton formulation:

• Gauge-Invariant Surface Operators: Improved ADM charges $Q[K]$ (electric) and $\widetilde{Q}[K]$ (magnetic, in dual graviton) are integrals over closed $(d-2)$-surfaces, determined by conformal Killing–Yano tensors $K_{\rho\sigma}$:

$$O_E(S)=\exp(i Q[K]), \qquad O_B(S)=\exp(i \widetilde{Q}[K])$$

These operators generate higher-form global symmetries and affect only topological sectors. They commute with all graviton Fock operators but may have nontrivial mutual algebra (linking phases), analogous to Maxwell theory Wilson–'t Hooft operators (Hull et al., 13 Dec 2024).

• Physical Action: Topological operators act only on global features—curvature flux through a surface or cycle—and do not create propagating graviton states.

5. Construction and Algebra of Diffeomorphism-Invariant Non-Graviton Operators

In quantum gravity, non-graviton operators can also be identified as gauge-invariant “dressed” matter fields:

• Wilson-Line and Coulomb Dressed Matter Fields: Operators of the form

$$\Psi_W(x) = \phi(x+V_W(x)), \quad \Psi_C(x) = \phi(x+V_C(x))$$

where $V_W(x)$, $V_C(x)$ encode gravitational dressing, create a particle together with its inseparable gravitational (Schwarzschild or Coulomb) field. These operators reduce to standard matter fields in the weak-gravity limit and do not overlap with pure metric fluctuation (graviton) creation operators (Donnelly et al., 2015).

• Nonlocal Algebra: The commutators of such observables are nonlocal, decaying as a power of separation and becoming significant near strong-field or high-energy thresholds. This is intrinsic to any theory with dynamical gravity and marks a departure from QFT locality.

6. Enumeration of Non-Graviton Contact Operators in SMEFT

A systematic classification via on-shell amplitudes yields all independent non-graviton operators built from Standard Model fields only (no gravitons) through mass dimension $d=8$ (Durieux et al., 2019). Examples include:

Dimension	Operator Structure	Spinor/Helicity Amplitude
5	Weinberg mass	$\langle 12\rangle$
6	Triple gauge ($X^3$)	$[12][13][23]$
6	Scalar-photon	$[12]^2$
6	Dipole ($\psi\psi HX$)	$[12][13]$
8	Quartic gauge ($X^4$)	$[12]^2[34]^2 + \cdots$

Minimal spinor-helicity structures are algorithmically fixed by little-group covariance, locality, and selection rules (gauge invariance, statistics). Higher-dimensional operators parameterize new-physics corrections beyond graviton exchange.

7. Distinguishing Graviton-like and Non-Graviton Spin-Two Operators

Spin-two resonances in collider experiments may arise from extra dimensions (KK graviton) or from new strongly-coupled sectors (impostor spin-2). Both couple to the SM only through dimension-five stress-tensor operators, but no gauge-invariant, Lorentz-invariant dimension-four interactions exist due to Fierz–Pauli constraints (Fok et al., 2012). Experimental discrimination requires measurement of ratios of branching fractions, such as

$$R_{g/\gamma} = \frac{\text{Br}(X \to gg)}{\text{Br}(X \to \gamma\gamma)}$$

where $R_{g/\gamma}=8$ for a true KK graviton, but generically not for a composite impostor. A deviation from $1/8$ in $\text{Br}(X\to\gamma\gamma)/\text{Br}(X\to jj)$ is a direct signature of non-graviton origin.

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Non-graviton operators thus encompass a broad range of quantum field, holographic, and topological constructs, unified only by their exclusion from pure graviton creation, propagation, or metric perturbations. Their paper yields nontrivial tests of dualities, reveals new black-hole microstructure, and exposes intrinsic nonlocality in quantum gravity and effective field theory.