Mixed Gauge-Gravitational Global Anomalies

• Mixed gauge-gravitational global anomalies are topological obstructions where the fermion partition function becomes a nontrivial section over the gauge and spacetime bundles.
• They are diagnosed using mathematical tools like bordism groups and the Atiyah-Patodi-Singer η-invariant, with descent formalism yielding effective Chern-Simons terms.
• These anomalies impact theoretical consistency in QFT and can manifest in observable phenomena such as non-linear current responses in Weyl semimetals.

A mixed gauge-gravitational global anomaly is a subtle obstruction to consistently defining the quantum theory of fermions and gauge or conformal fields when both gauge and gravitational (diffeomorphism) background fields are present. Unlike perturbative (local) anomalies, these global anomalies are topological in origin and sensitive to the full global structure of the gauge and spacetime bundles, often requiring analysis via bordism groups, η-invariants, or extended symmetry frameworks such as 2-group symmetries.

1. Definition and Diagnostic Criteria

Mixed gauge-gravitational global anomalies arise when the quantum path integral for a system with gauge and gravitational backgrounds is not invariant under large gauge transformations and large diffeomorphisms, despite the apparent absence of local (perturbative) anomalies. The formal diagnostic is the failure of the fermion determinant or partition function to be globally well-defined as a function of the background fields; it is instead a section of a line bundle whose nontriviality encodes the anomaly.

Mathematically, in even spacetime dimension $d$, fermion global anomalies are characterized by the spin bordism group $\Omega_{d+1}^{\mathrm{Spin}}(BG)$, where $BG$ is the classifying space of the gauge group, and the anomaly phase is given by the exponentiated Atiyah-Patodi-Singer (APS) $\eta$-invariant on a $(d+1)$-manifold with boundary, coupled to the background fields (Lee et al., 2022, Davighi et al., 2019). A nontrivial character $\Omega_{d+1}^{\mathrm{Spin}}(BG)\to U(1)$ signals a global anomaly.

2. Anomaly Polynomials and Descent Formalism

Local anomalies are encoded in the characteristic class anomaly polynomial (e.g., for chiral fermions in 4d: $I_6 = k_m F \wedge \mathrm{tr} R^2$). In the descent procedure, mixed terms in the anomaly polynomial descend to inflow actions in one higher dimension (e.g., $A\wedge \mathrm{tr} R^2$ Chern-Simons terms in 5d for 4d mixed anomalies), which measure the global anomaly phase under large gauge/diffeomorphism transformations (Golkar et al., 2015, Cranganore, 2021).

The signature of a mixed anomaly is a nontrivial phase: $\exp\left(i\int_{M_{d+1}} I_{d+1}\right)$ where $I_{d+1}$ is the descendant Chern-Simons term pulled back over a mapping torus construction implementing the large transformation. For example, in 4d,

$$I_6^{\mathrm{mixed}} = \frac{q}{(2\pi)^3 3!} F \wedge \mathrm{tr}(R\wedge R)$$

descends to a 5d action whose nontriviality under large transformations detects the global anomaly (Golkar et al., 2015, Cranganore, 2021).

3. Classification in Specific Dimensions and Models

3.1 Two Dimensions: WZW Models and Discrete Symmetries

In 1+1d Wess-Zumino-Witten (WZW) models for simply-connected simple Lie groups $G$, mixed global anomalies obstruct gauging certain discrete symmetries (the centers of $G$). The anomaly is diagnosed by the phase picked up under modular transformations of the twisted partition function or, equivalently, by the nonexistence of conformal boundary states invariant under the action of the center. The absence of the anomaly restricts the allowed levels $k$ for which the orbifold is consistent and the center can be gauged (Numasawa et al., 2017):

Group $G$	Center $Z_N$	Consistent Level $k$
$SU(n)$	$\mathbb{Z}_n$	$n$ odd: any $k$; $n$ even: $k \in 2\mathbb{Z}$
$Spin(2n+1)$	$\mathbb{Z}_2$	Any $k$
$USp(2n)$	$\mathbb{Z}_2$	$n$ even: any $k$; $n$ odd: $k \in 2\mathbb{Z}$
$Spin(4\ell+2)$	$\mathbb{Z}_4$	$k \in 2\mathbb{Z}$
$E_6$	$\mathbb{Z}_3$	Any $k$
$E_7$	$\mathbb{Z}_2$	$k \in 2\mathbb{Z}$

This divisibility is derived from modular transformation phases or Cardy-state boundary analysis (Numasawa et al., 2017).

3.2 Four Dimensions: Bordism and Effective Field Theory

For 4d chiral gauge theories, global mixed anomalies are classified by $\Omega_5^{\mathrm{Spin}}(BG)$. For Standard Model groups and realistic extensions, explicit spectral sequence computations show that $\Omega_5^{\mathrm{Spin}}(BG)$ is generically trivial or $\mathbb{Z}_2$ (Witten anomaly), with no exotic mixed gauge-gravitational global anomalies beyond this (Davighi et al., 2019). Mixed anomalies are reflected in the structure of effective Chern-Simons couplings and in the non-renormalization of certain hydrodynamic coefficients (e.g., the chiral vortical effect coefficient) (Golkar et al., 2015).

Explicitly, for a single Weyl fermion,

$$\nabla_\mu J^\mu = \frac{1}{16\pi^2} \left(F_{\alpha\beta} \tilde F^{\alpha\beta} - \frac{1}{24} R_{\alpha\beta\gamma\delta} \tilde R^{\alpha\beta\gamma\delta}\right)$$

with the mixed term signified by the $R\tilde R$ piece (Larue et al., 2023). Dimensional regularization and anomaly consistency force the absence of separate $R\tilde R$ or $F\tilde F$ terms in the trace anomaly.

3.3 Six Dimensions: 2-Group Structures and Little String Theories

In 6d, nontrivial mixed gauge-gravitational anomalies for 1-form (instanton) symmetries are present only in little string theories (LSTs), not in superconformal field theories (SCFTs) (Cordova et al., 2020). The anomaly polynomial includes a term $I_8 \supset c_2(f) p_1(T)$ linking the 2-form instanton symmetry current to the Pontryagin class of the tangent bundle, corresponding to a nontrivial 2-group global symmetry encoded by a Postnikov class. The quantization condition for the coefficient $k_{g^2 p_1}$ follows from anomaly inflow and large Lorentz/gauge transformation consistency. In genuine 6d SCFTs, such anomalies must vanish; the would-be 1-form symmetry is always gauged via Green-Schwarz couplings on the tensor branch (Cordova et al., 2020).

3.4 Eight Dimensions: Bordism and Anomaly Cancellation Mechanisms

For 8d spin manifolds and supergravity, the relevant global anomalies are classified by $\Omega_9^{\mathrm{spin}}(BG)$ and may arise for gauge, gravitational, or mixed cases. Explicit computations show only $\mathbb{Z}_2$ or $\mathbb{Z}_2^2$ classes, with cancellation mechanisms involving dynamical 2-form (or in certain cases, 3-form $\mathbb{Z}_2$) gauge fields via generalized Green-Schwarz inflow. The modified Bianchi identity for the 2-form field strength absorbs the anomaly for specific quantized coefficients (Lee et al., 2022). In some “rank-2” models, cancellation necessitates additional topological (Wu structure) degrees of freedom and constraints on spacetime topology.

4. Operator Algebra, Physical Signatures, and Transport

Mixed gauge-gravitational anomalies manifest in operator commutators and physical observables. For Weyl fermions in curved backgrounds, quantum corrections from the anomaly modify the Lorentz algebra at the operator level, producing nontrivial (order-sensitive) Schwinger terms in the Lorentz commutator algebra; these central extensions directly track the non-closure due to the anomaly (Cranganore, 2021).

In condensed matter realizations such as Weyl semimetals, mixed anomalies yield distinctive nonlinear current responses (e.g., chiral charge nonconservation driven by emergent curvature due to applied electric fields), which are observable as anomaly-induced nonlinear conductivity, insensitive to scattering time and with unique tensor structure (Holder et al., 2021). These effects are direct physical consequences of the underlying mixed anomaly and provide experimental probes in solid state systems.

5. Anomaly Matching, Inflow, and Cancellation Mechanisms

Any consistent quantum field theory must either ensure that all local and global mixed anomalies trivialize (via matter content) or provide dynamical mechanisms for their inflow cancellation.

• Green-Schwarz-type mechanisms use axion-like couplings or dynamical $p$-forms whose classical shifts compensate the anomalous variation, enforced by quantization of the Chern-Simons-like inflow terms (Golkar et al., 2015, Lee et al., 2022).
• In higher dimensions and string frameworks, additional topological degrees of freedom (e.g., 3-form TQFTs, Wu structures) may be required when ordinary differential-form inflow fails (Lee et al., 2022).
• Effective Chern-Simons couplings (e.g., $A \wedge \mathrm{tr} R^2$) in reduced dimensions encode the underlying mixed anomaly and ensure that anomaly matching across energy scales is maintained (Chang et al., 2019, Golkar et al., 2015).
• In the presence of higher-form symmetries and 2-group global symmetry structures, the anomaly-cancellation conditions generalize to include Postnikov data and interrelations between symmetry generators (Cordova et al., 2020).

6. Summary Table: Example Occurrences and Conditions

Theory/Dimension	Diagnostic/Structure	Cancellation or Consistency Condition
1+1d WZW models	Center symmetry, modular phase	Level divisibility condition from modular/BCFT analysis
4d chiral gauge theories (SM, GUTs)	$\Omega_5^{\mathrm{Spin}}(BG)$	No new global anomalies beyond Witten anomaly for $SU(2),Sp(M)$
6d Yang-Mills, LSTs	2-group symmetry, $c_2(f)\,p_1(T)$	$k_{g^2 p_1}$ quantized, vanishes for SCFTs
8d supergravity	$\Omega_9^{\mathrm{Spin}}(BG)$	Green-Schwarz inflow/3-form TQFT; odd Chern-Simons coefficients
QFT in $d$-dimensions	$\Omega_{d+1}^{\mathrm{Spin}}(BG)$	Trivialization via matter, inflow, or topology

The study of mixed gauge-gravitational global anomalies has clarified constraints on gauge group choices, informed effective field theory construction (fixing non-renormalizable couplings and transport coefficients), and established connections between algebraic topology, operator algebras, and observable physical phenomena (Numasawa et al., 2017, Golkar et al., 2015, Davighi et al., 2019, Cranganore, 2021, Holder et al., 2021, Lee et al., 2022, Larue et al., 2023, Cordova et al., 2020, Chang et al., 2019).