Symmetry TFTs, Anomaly Inflow, and Boundary Realizations

A symmetry TFT is a $(d+1)$-dimensional topological theory whose boundary conditions encode choices of generalized symmetry, global form, gauging, and anomaly for a $d$-dimensional QFT. The boundary QFT is a theory relative to that bulk. The bulk determines topological defect and inflow data, but not the boundary Hamiltonian, local operator dimensions, or OPE coefficients. Distinct boundary dynamics can therefore share the same symmetry TFT.

Required background. Invertible Field Theories and Generalized Cohomology supplies topological anomaly theories. Anomalies as Relative and Invertible Field Theories supplies the relative-theory map. Categorical Symmetries, Higher Representations, and Charges supplies defect actions. Symmetry TFT: Encoding Symmetry, Anomaly, and Gauging supplies the physical construction. Helpful background. Higher-Group Operators, Gauging, and Anomalies gives mixed higher-form examples.

Relative boundary data

Let $\alpha$ be an extended $(d+1)$-dimensional field theory. In one orientation convention, a $d$-dimensional theory relative to $\alpha$ is a morphism

$$F:\mathbf1\longrightarrow\tau_{\le d}\alpha.$$

For a closed $d$-manifold $X$, the partition function of $F$ is then a vector in the state space $\alpha(X)$, rather than canonically a number. For a closed $(d-1)$-manifold, $F$ selects an object or functor associated with the category $\alpha(Y)$. Freed and Teleman give this definition and its boundary interpretation in Freed and Teleman 2014, Definition 2.1 and §2, printed pp. 3–7 (PDF).

When $\alpha$ is invertible, this is the usual anomaly theory: $\alpha(X)$ is a line, and the boundary partition function has an anomalous phase ambiguity controlled by that line. A general symmetry TFT need not be invertible. Its state spaces can have several sectors, and choosing a topological boundary condition amounts to choosing a polarization or gauging frame. Interfaces between such boundary conditions implement gauging operations or changes of global form.

Bulk topological defects can end on the boundary and act on boundary operators. Their fusion and junctions encode the generalized symmetry category. Yet $F$ is extra data: the same $\alpha$ can admit inequivalent boundary conditions and many inequivalent non-topological theories attached to one boundary type.

A five-dimensional Abelian model

For a four-dimensional theory with electric and magnetic $\mathbb Z_N$ one-form symmetries, a continuum presentation of the symmetry TFT uses two $U(1)$ two-form gauge fields $B_e,B_m$ with quantized periods and action

$$S_{\mathrm{sym}}= \frac{N}{2\pi}\int_{M_5} B_e\wedge\mathrm d B_m.$$

This formula is shorthand for a differential-cohomology or discrete-cochain theory; the global quantization data are essential. Wilson surfaces for $B_e$ and $B_m$ have a linking phase $e^{2\pi i/N}$. On a manifold with boundary, a gauge transformation produces a boundary term. That term is the inflow representative of the mixed electric–magnetic one-form anomaly.

Different topological boundary conditions fix one member of the conjugate pair or a maximal isotropic combination. A boundary that treats $B_e$ as background realizes the electric one-form symmetry; exchanging the polarization exchanges electric and magnetic descriptions. Summing over an admissible boundary background implements gauging. These alternatives encode global-form and gauging choices without changing the bulk topological pairing.

The exact application belongs to Symmetry TFT: Encoding Symmetry, Anomaly, and Gauging. The five-dimensional action records the mixed pairing, while the four-dimensional page specifies the actual gauge theory, line spectrum, global form, and anomaly matching. Gaiotto and collaborators explain higher-form backgrounds, gauging, and anomalies in Gaiotto et al. 2015, §§3 and 7, printed pp. 7–11 and 41–48 (PDF).

Three checks protect the normalization. Exchanging two linked basic surfaces gives an $N$th root of unity; fusing $N$ identical surfaces is transparent; and reversing one orientation complex-conjugates the phase. These are topological checks and do not test any boundary spectrum.

Inflow is not reconstruction

Suppose two four-dimensional boundary theories have the same one-form symmetry, the same mixed anomaly, and the same allowed gauging polarizations. They may still have different gauge couplings, massless spectra, correlation functions, and renormalization-group flows. The symmetry TFT forgets precisely these local dynamical data.

The adversarial inference tries to extract a boundary scaling dimension or Hamiltonian from the bulk BF pairing. A continuous family of boundary couplings—or two different boundary QFTs with the same anomaly—gives identical $S_{\mathrm{sym}}$ but different spectra. The strongest licensed conclusion is equality of the specified topological symmetry and anomaly data. It is not equivalence of boundary QFTs.

Even anomaly cancellation has limited reach. A boundary condition that trivializes inflow is necessary for an absolute boundary partition function, but it does not prove locality, reflection positivity, or continuum existence. Those require independent boundary-theory arguments.

A polarization must also be globally admissible. Choosing electric data on one boundary component and magnetic data on another creates a gauging interface between the corresponding boundary realizations. Gluing such interfaces is controlled by the finite pairing of flux sectors. If the chosen subgroup is not isotropic for the anomaly pairing, gauge variation remains on the interface and the supposed boundary condition is obstructed.

Exercises

What happens to the basic linking phase when the orientation of the magnetic surface is reversed?

Solution

The oriented linking number changes sign, so $e^{2\pi i/N}$ becomes $e^{-2\pi i/N}$. This equals the action of the inverse magnetic defect, as required by invertibility.

References

• Freed, Daniel S., and Constantin Teleman. “Relative Quantum Field Theory.” Communications in Mathematical Physics 326 (2014): 459–476. DOI; Open PDF.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, no. 2 (2015): 172. DOI; Open PDF.