Anomaly Polynomial TFTs in QFT and M-Theory

• Anomaly Polynomial TFTs are invertible topological quantum field theories that encode ’t Hooft anomalies through gauge-invariant polynomial integration and descent methods.
• They implement a geometric inflow mechanism in M-theory and holographic settings by translating higher-dimensional actions into QFT anomaly cancellations.
• Bordism classifications and symmetry TFT frameworks extend these theories to address higher-form and non-invertible symmetries, providing clear obstructions to gauging.

Anomaly Polynomial Topological Field Theories (TFTs) are a class of invertible topological quantum field theories that provide a precise and universal encoding of ’t Hooft anomalies in quantum field theories (QFTs). The key insight is that the anomaly polynomial of a $d$-dimensional QFT, a gauge- and diffeomorphism-invariant $(d+2)$-form constructed from background characteristic classes, can be integrated to define the action of a $(d+1)$-dimensional invertible TFT whose variation under gauge transformations exactly cancels the anomalous variation of the QFT partition function. This perspective not only unifies local and global aspects of anomalies but also provides a geometric inflow mechanism and a rigorous mathematical classification via bordism.

1. Formalism for Anomaly Polynomial TFTs

Anomaly polynomials $I_{d+2}$ are constructed as gauge-invariant polynomials in the background gauge field strengths $F$, R-symmetry curvatures, and spacetime curvature $R$. From the modern perspective, each such polynomial determines an invertible $(d+1)$-dimensional TFT (i.e., anomaly field theory) via the descent procedure. For a closed $(d+1)$-manifold $W^{d+1}$ with given backgrounds, the action and partition function are

$$S_{\mathrm{anom}}[W]=2\pi i \int_W I_{d+1}(A,\omega), \qquad Z_{\mathrm{anom}}(W;A,\omega)=\exp\left(2\pi i \int_W I_{d+1}\right)$$

where $(d+2)$0 is a Chern–Simons–like form satisfying $(d+2)$1 and is related to $(d+2)$2 by the descent equations ($(d+2)$3 for the boundary contribution). The nontriviality of $(d+2)$4 signifies the presence of an anomaly in the $(d+2)$5-dimensional boundary QFT, which is then understood as a relative field theory valued in the anomaly line $(d+2)$6 associated to the boundary $(d+2)$7 (Monnier, 2019).

This structure captures both local and global anomalies. For global consistency, the constructed TFT must extend to a group homomorphism from the relevant $(d+2)$8-bordism group to $(d+2)$9. This bordism classification underpins the modern treatment of invertible phases and anomaly detection.

2. Anomaly Inflow and Geometric Engineering in M-theory

Anomaly polynomial TFTs admit an elegant geometric realization via anomaly inflow in M-theory and string theory constructions. For QFTs engineered from stacks of $(d+1)$0 M5-branes, anomalies are derived from inflow formulas in 11 dimensions. The bulk M-theory action localizes to boundary data capturing the anomaly: $(d+1)$1 where $(d+1)$2 is a closed, gauge-invariant 4-form on the normal bundle (Bah et al., 2019). Integrating $(d+1)$3 over the internal space $(d+1)$4 yields the QFT’s anomaly polynomial: $(d+1)$5 The anomaly polynomial $(d+1)$6 is interpreted as the curvature of an invertible $(d+1)$7-dimensional TFT, with action $(d+1)$8. This direct geometric picture not only computes the anomaly polynomials efficiently but also links finite-$(d+1)$9 corrections to higher-derivative terms and normal bundle data in the M-theory background. In holographic settings, the same inflow formula reproduces the dual CFT anomalies, including subleading terms (Bah et al., 2019).

3. The Symmetry TFT and Anomalies for Higher-Form and Non-invertible Symmetries

Symmetry TFTs (SymTFTs) generalize this framework to discrete and higher-form global symmetries. The basic construction involves a BF-type action in $I_{d+2}$0 dimensions with additional Dijkgraaf–Witten (higher-form Chern–Simons) twists: $I_{d+2}$1 where $I_{d+2}$2 and $I_{d+2}$3 are $I_{d+2}$4 $I_{d+2}$5- and $I_{d+2}$6-form gauge fields (Zotto et al., 2024). The boundary variation of the action under large gauge transformations gives rise to the ’t Hooft anomaly for a $I_{d+2}$7-form symmetry, specifically an anomaly polynomial $I_{d+2}$8 for the boundary background gauge field. These higher-form symmetry anomalies can act as obstructions to gauging, precisely captured by the topology of defect correlators (e.g., higher link numbers) in the bulk SymTFT.

SymTFTs are also crucial in geometric engineering, where topological branes wrapping torsion cycles in the link of a singularity manifest as operators in the topological field theory, and their linking correlators encode gaugeability obstructions for higher symmetries (Zotto et al., 2024).

4. Explicit Examples and Key Computations

The formalism admits concrete realization in diverse settings:

1. 6d (2,0) Theories: The anomaly polynomial for $I_{d+2}$9 M5-branes is

$F$0

capturing leading $F$1 behavior and including gravity, R-symmetry, and mixed terms. Finite-$F$2 corrections in related backgrounds arise from higher-derivative inflow terms (Bah et al., 2019).

2. 4d $F$3 from M5 on Riemann surfaces: The polynomial

$F$4

encodes both leading and subleading terms, including center-of-mass multiplet effects (Bah et al., 2019).

3. Chiral U(1) Anomalies: For a generic 4d theory, the anomaly polynomial is $F$5, with the corresponding 5d TFT having action

$F$6

The BF+CS structure realizes both the symmetry and its anomalies (Antinucci et al., 2024).

4. Non-invertible Symmetry TFTs: Non-invertible defects and symmetry fractionalization phenomena (e.g., $F$7 chiral symmetry and fusion rules in 4d) are realized by dynamically gauging in BF-CS SymTFTs, where continuum defects require dressing by auxiliary TQFTs to be genuine. Triple linking correlators in the bulk reflect noninvertibility and anomaly data (Antinucci et al., 2024).

5. Algebraic and Cobordism Classification

Invertible anomaly polynomial TFTs are classified by bordism invariants: equivalence classes under local deformations correspond to group homomorphisms from the relevant $F$8-dimensional cobordism group to $F$9: $R$0 Allowed anomaly polynomials $R$1 must map to global invariants on all closed $R$2-manifolds. The infinitesimal variation of the partition function (“local anomaly data”) arises from integration of $R$3, but only those classes globally extending to bordism invariants give consistent anomaly field theories. In practice, this corresponds to global anomaly cancellation conditions, a central aspect in 6d supergravity and string constructions (Monnier, 2019).

6. Physical and Mathematical Implications

Anomaly polynomial TFTs provide a rigorous and universal language for analyzing anomalies:

• Anomaly Inflow: The $R$4-dimensional TFT yields a bulk action whose anomalous boundary variation cancels that of the QFT partition function. This geometric inflow mechanism is realized concretely in M-theory and string setups, where brane worldvolumes source the bulk couplings (Bah et al., 2019).
• Obstructions to Gauging: Mixed terms and higher powers in the polynomial (e.g., $R$5 or $R$6) specify precise obstructions to gauging the associated global (higher) symmetries. Nontrivial bulk correlators (such as higher linking in SymTFTs) mathematically encode these obstructions (Zotto et al., 2024).
• Dualities and Defects: The structure controls the construction and fusion rules for defects, including non-invertible duality defects, symmetry fractionalization, and 2-group symmetry extensions.

A plausible implication is that the anomaly polynomial TFT formalism not only unifies diverse manifestations of anomalies but also provides computationally effective tools, particularly when geometric engineering or holography is available.

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References:

(Monnier, 2019, Bah et al., 2019, Antinucci et al., 2024, Zotto et al., 2024)