Background Responses and Invertible Phases

A quantized functional of fixed background fields defines an invertible response only when it is a globally defined, local, gluing-compatible field theory and has an inverse under stacking. On a closed spacetime its normalized amplitude is a nonzero phase. On a spatial slice it assigns a line—or a superline in the fermionic setting—and its bordism maps must be invertible. A pointwise phase formula on closed manifolds is therefore necessary but not, by itself, the full definition.

Keeping a field fixed also does not prove that the whole phase is invertible. The same response can multiply a theory with intrinsic topological order. Conversely, integrating or summing a field does not automatically make the result noninvertible: an invertible dynamical topological theory is possible. The field role and the stacking test are two independent questions.

Required background. When Is a Topological Term Well Defined? supplies the global phase, filling-independence, and tangential-structure tests. Coupling Background Gauge Fields and Bundles supplies the distinction between a fixed covariant source and a field that is integrated or summed. Anomaly Polynomials and Inflow supplies the counterterm quotient and the boundary-orientation convention for inflow.

Helpful background. Global and Torsion Anomalies explains why a vanishing differential-form response does not exclude a finite global phase.

A background response is an invertible functional

Let $M^d$ be a closed spacetime with an admitted tangential structure—such as orientation, spin, pin$^\pm$, or framing. Let $B$ denote all fixed background data: bundles or higher cocycles, connections, metric data, and allowed transformations. The microscopic dynamical fields are denoted by $\phi$ and are integrated out; $B$ is not:

$$Z[M;B] = \int\mathcal D\phi\,e^{iS[\phi;B]}.$$

In a gapped regime, after declaring which nonuniversal local terms have been removed, a normalized topological response sector may be written on a closed manifold as

$$\mathcal Z_\alpha[M;B] = e^{iS_{\mathrm{eff},\alpha}[M;B]} \in U(1).$$

The subscript $\alpha$ labels the response. This equation records only the closed-manifold value. The field-theoretic object must also assign lines or superlines to closed $(d-1)$-manifolds and compatible maps to bordisms. Freed formulates an effective exponentiated action precisely after the fluctuating fields have been integrated out, and distinguishes an invertible theory by its one-dimensional state spaces and nonzero maps Freed 2014, § 2.1, arXiv v2, printed pp. 2–3, eqs. (2.1)–(2.3), PDF. The modern line-valued formulation and tensor-product stacking appear in Freed 2023, § 3, arXiv v1, printed pp. 11–13, Definition 3.15 and eqs. (3.16)–(3.18) and (3.24), PDF.

Stacking two decoupled response theories multiplies their amplitudes and tensors their state spaces:

$$\begin{aligned} \mathcal Z_{\alpha\otimes\beta}[M;B] &= \mathcal Z_\alpha[M;B]\, \mathcal Z_\beta[M;B], \\ \mathcal H_{\alpha\otimes\beta}[\Sigma;B] &= \mathcal H_\alpha[\Sigma;B] \otimes \mathcal H_\beta[\Sigma;B]. \end{aligned}$$

The response $\alpha$ is invertible when another theory $\alpha^{-1}$ exists with

$$\alpha\otimes\alpha^{-1}\simeq\mathbf 1.$$

For a unitary theory the inverse has complex-conjugate amplitudes on closed manifolds, but complex conjugation of numbers is only the visible part of the statement: the boundary lines, state spaces, and every bordism map must also pair with those of $\alpha$ to give the trivial theory. Tensor invertibility forces

$$\dim\mathcal H_\alpha(\Sigma;B)=1$$

for every admitted connected closed spatial slice $\Sigma$, and all assigned maps must be nonzero and invertible. One-dimensional state spaces are a powerful necessary check, not a general converse for an incompletely specified extended theory. This stacking criterion is stated in Freed and Hopkins 2021, § 5.2, arXiv v6, printed pp. 33–34, eq. (5.4), Example 5.3, and Definition 5.9, PDF.

Because $B$ remains fixed, the response functional itself introduces no additional bundle sum, gauge orbit, or dynamical superselection sector. But a microscopic phase may have an extra universal factor,

$$\mathcal Z_{\mathrm{IR}}[M;B] = \mathcal Z_\alpha[M;B]\, Z_{\mathrm{top}}[M;B],$$

where $Z_{\mathrm{top}}$ is a noninvertible topological field theory. The response $\mathcal Z_\alpha$ remains invertible while the full infrared phase does not. Calling the phase invertible therefore requires the absence of such a noninvertible universal sector, not merely the existence of a useful background response.

Local counterterms identify response representatives

An equivalence relation must be declared before one says that two responses are the same. Let $\mathcal C_{\mathrm{scheme}}$ be a specified subgroup of counterterms $C[M;B]$ that are

• local in the backgrounds and finitely many derivatives;
• globally defined on every admitted bundle and tangential structure;
• properly quantized so that $e^{iC}$ is single valued; and
• compatible with the manifold, bundle, tangential, and boundary domain.

Gauge invariance under the transformation whose anomaly is being compared is not assumed: the counterterm’s variation is precisely what can move an anomaly representative. It must, however, preserve any other structures and symmetries that the comparison declares fixed.

Multiplication by such a term gives another representative,

$$\mathcal Z_\alpha[M;B] \longmapsto \mathcal Z_\alpha[M;B]\,e^{iC[M;B]}.$$

For an anomalous $d$-dimensional theory, write the finite background transformation law as

$$Z_X[B^g] = e^{i\mathcal A(B,g)}Z_X[B].$$

The counterterm changes the anomaly representative by a coboundary,

$$\mathcal A(B,g) \longmapsto \mathcal A(B,g) +C[B^g]-C[B] \quad\pmod{2\pi}.$$

If no admissible $C$ makes the transformed phase trivial for every allowed background and transformation, the anomaly class remains nonzero. A local formula on one trivializing patch is not an admissible global counterterm, and a total derivative on a manifold with boundary creates new boundary data rather than disappearing. The locality, globality, and quantization conditions are made explicit in Witten and Yonekura 2021, § 3.3, arXiv v3, printed pp. 36–38, eqs. (3.10)–(3.11), PDF.

There are two related but different quotients:

1. Anomaly representatives. A permitted boundary counterterm changes a trivialization or moves a mixed anomaly between Ward identities. The invariant object is the remaining anomaly class.
2. Invertible phases. A quantized, symmetry-invariant bulk functional may itself be the physical response that distinguishes two phases. It should be quotiented out only if the chosen phase equivalence declares that stacked response trivial—for example through an allowed deformation or a specified atomic phase.

Thus “it is a local counterterm” is not enough to erase a response from a phase classification. One must name the subgroup regarded as trivial for that classification. If $\mathcal C_{\mathrm{triv}}$ denotes that subgroup, then two responses are equivalent only when

$$\frac{\mathcal Z_\alpha[M;B]} {\mathcal Z_\beta[M;B]} = e^{iC[M;B]} \qquad \text{for some }C\in\mathcal C_{\mathrm{triv}}$$

on every admitted background, with compatible boundary-line maps and a gluing-compatible natural equivalence. Taking $\mathcal C_{\mathrm{triv}}=\mathcal C_{\mathrm{scheme}}$ without further qualification can incorrectly discard the very quantized responses one is trying to classify.

A boundary turns invariance into anomaly inflow

Now let an invertible $(d+1)$-dimensional response live on $Y$ with $\partial Y=X$. Its value is no longer naturally an absolute number. For a fixed boundary background $B_X$, it lies in a one-dimensional anomaly line,

$$\mathcal Z_{\mathrm{bulk}}[Y;B] \in \mathcal L_\alpha(X;B_X).$$

An anomalous boundary partition function is valued in the dual line,

$$Z_X[B_X] \in \mathcal L_\alpha(X;B_X)^{-1}.$$

Their canonical pairing is a number:

$$\left\langle \mathcal Z_{\mathrm{bulk}}[Y;B], Z_X[B_X] \right\rangle \in\mathbb C.$$

Equivalently, a background transformation acts on the two factors by inverse phases. In a local representative,

$$\begin{aligned} Z_X[B^g] &=e^{i\mathcal A(B,g)}Z_X[B], \\ \mathcal Z_{\mathrm{bulk}}[B^g] &=e^{-i\mathcal A(B,g)} \mathcal Z_{\mathrm{bulk}}[B]. \end{aligned}$$

The product is invariant, while neither factor need be an invariant number by itself. This is the field-theoretic content of anomaly inflow, rather than a claim that the boundary variation is a defect of the combined system. Freed develops the anomaly as an invertible theory in one higher dimension and the boundary theory as a relative theory in Freed 2014, §§ 2.2–2.3, arXiv v2, printed pp. 4–6, eqs. (2.5)–(2.10), PDF.

In a common high-energy-theory usage, an SPT response is an invertible TQFT whose zero-background amplitude is normalized to one and whose fixed-background partition function is $U(1)$-valued. At the full-theory level, this is a trivialization after forgetting the protecting symmetry; the closed-manifold normalization is only its shadow. This is a useful operational convention, not a claim that group cohomology or one response formula classifies every microscopic phase. The convention and the inverse boundary–bulk variations are reviewed in Bhardwaj et al. 2024, §§ 4.2.2–4.2.3, arXiv v2, printed pp. 74 and 77–79, Definitions 4.5–4.6 and eqs. (4.62)–(4.64) and (4.84)–(4.95), PDF.

The inflow class constrains a boundary, but it does not choose a unique boundary dynamics. Depending on dimension and hypotheses, matching may use gapless modes, symmetry breaking, boundary topological order, or a relative theory with additional bulk data. A nonzero local boundary variation does not, by itself, prove the existence of one particular edge action.

First application: fixed probes versus dynamical topological fields

Work first on a closed connected oriented three-manifold $M^3$. Compact $U(1)$ connections have unit minimal electric charge, with $[\mathrm dA/(2\pi)]$ and $[\mathrm dB/(2\pi)]$ integral whenever the corresponding field is present. All displayed weights use the Lorentzian convention $e^{iS}$; their intrinsic differential-cohomology refinements and the relevant spin or framing data are understood. Boundaries are introduced only for the local variation tests.

Compact Abelian Chern–Simons response

For a fixed compact background connection $A$,

$$\mathcal R_k[M;A] = \exp\!\left( \frac{ik}{4\pi}\int_M A\wedge\mathrm dA \right).$$

The ordinary one-component oriented bosonic response requires even $k$; with spin structure, every integer $k$ is allowed and odd $k$ is spin-dependent. Stacking adds levels, and the inverse response has level $-k$:

$$\mathcal R_k\,\mathcal R_{-k}=1.$$

On a manifold with boundary, the local small-gauge transformation $A\mapsto A+\mathrm d\lambda$ gives, with the outward-normal-first boundary orientation,

$$\Delta_\lambda S_{\mathrm{CS}} = \frac{k}{4\pi} \int_{\partial M}\lambda\,\mathrm dA.$$

The intrinsic global transformation law belongs to the differential refinement; the displayed formula is its local boundary representative.

If $A$ is instead integrated and $k\ne0$, the result is the nondegenerate compact $U(1)_k$ Chern–Simons theory. Its genus-$g$ state-space dimension is

$$\dim\mathcal H(\Sigma_g)=\lvert k\rvert^g.$$

Therefore $\lvert k\rvert>1$ rules out invertibility immediately. The standard nondegenerate $k=\pm1$ theory is instead an invertible spin Chern–Simons theory; “dynamical” is not synonymous with “noninvertible.” The case $k=0$ is degenerate and is not assigned zero states by this formula. The level lattice and determinant count are given in Belov and Moore 2005, §§ 1–2 and § 5.3, arXiv v1, printed pp. 3–4, 7–8, and 26, especially eqs. (1.1)–(1.3), (2.2)–(2.7), and the prose after eq. (5.17), PDF.

Compact BF cross-response

For two fixed compact $U(1)$ backgrounds $A$ and $B$, take

$$\mathcal R_N[M;A,B] = \exp\!\left( \frac{iN}{2\pi} \int_M B\wedge\mathrm dA \right), \qquad N\in\mathbb Z_{>0}.$$

This is an invertible mixed response; its stacking inverse changes $N\mapsto-N$. If $M$ has boundary, the local transformation $B\mapsto B+\mathrm d\chi$ produces

$$\Delta_\chi S_{\mathrm{BF}} = \frac{N}{2\pi} \int_{\partial M}\chi\,\mathrm dA.$$

If both compact fields are integrated, the same local density—together with its global sector sum and measure—defines the untwisted $\mathbb Z_N$ gauge TQFT. Its torus state space has dimension

$$\dim\mathcal H(T^2)=N^2,$$

so it is noninvertible for $N>1$; $N=1$ is the trivial invertible limit. The compact action, integer level, global completion, and boundary qualification are developed in Kapustin and Seiberg 2014, §§ 3 and 5, arXiv v2, printed pp. 9–13 and 20–21, eqs. (3.1)–(3.16) and (5.1)–(5.3), PDF. The torus count follows from the Abelian $K$-matrix determinant formula in Belov and Moore 2005, § 5.3, arXiv v1, printed p. 26, prose after eq. (5.17), PDF.

A finite-bundle response and its gauging

Let $G$ be finite, $P\to M$ a fixed flat principal $G$-bundle with classifying map $f_P:M\to BG$, and

$$[\omega]\in H^3(BG;\mathbb R/\mathbb Z).$$

Evaluation on the fixed bundle gives the invertible phase

$$\mathcal R_\omega[M;P] = \exp\!\left( 2\pi i\, \bigl\langle f_P^*\omega,[M]\bigr\rangle \right),$$

with inverse represented by $-[\omega]$. On a boundary the value is line-valued and needs a compatible trivialization or boundary theory.

Gauging is a different operation: sum over the finite bundle groupoid with its automorphism weights,

$$Z_{G,\omega}(M) = \sum_{[P]\in\pi_0\operatorname{Bun}^{\mathrm{flat}}_G(M)} \frac{\mathcal R_\omega[M;P]} {\lvert\operatorname{Aut}(P)\rvert}.$$

Each summand is a phase, but their sum is not another pointwise invertible response. It defines Dijkgraaf–Witten gauge theory and is generally noninvertible for nontrivial $G$. The identity-flux contribution to the torus state space already contains the irreducible representations of $G$; the explicit $\mathbb Z_2$ example has four torus states. These state-space checks appear in Dijkgraaf and Witten 1990, §§ 6.3 and 6.6, printed pp. 416–426, especially eq. (6.41) on p. 424 and the $\mathbb Z_2$ example on p. 426, PDF. The finite-bundle sum and cocycle weight are constructed in Dijkgraaf and Witten 1990, § 6.2, printed pp. 415–416, eqs. (6.8)–(6.10); the boundary line, automorphism-weighted measure, and gluing law are developed in Freed and Quinn 1993, §§ 1–2, printed pp. 438–445, especially eqs. (1.1)–(1.2), (2.1), and (2.9), Theorem 2.13, and eq. (2.17), arXiv v3 PDF.

The table collects the field-role change. It compares the three examples in one dimension and convention package; it does not claim that their local actions or completed quantum theories are equivalent.

Fixed-background response versus field-summed quantum theory in the threaded three-model comparison; the rows are not equivalent theories

Fixed datum	Response and stacking inverse	Boundary signal	If the datum is summed	Invertibility witness or ceiling
Compact U(1) connection A	Chern–Simons level k; inverse level minus k	Local variation proportional to lambda dA	Compact U(1) level-k Chern–Simons theory	For nonzero k, torus dimension is absolute k; zero is degenerate, while levels plus or minus one give the invertible spin cases
Compact U(1) connections A and B	BF level N; inverse level minus N	Local variation proportional to chi dA	Untwisted Z_N gauge TQFT after the global sum and measure	Torus dimension N squared forbids invertibility for N greater than one
Flat finite-G bundle P	Cocycle class omega; inverse class minus omega	A line-valued cocycle phase on the boundary	Automorphism-weighted Dijkgraaf–Witten bundle sum	Multiple bundle and operator sectors generally require a noninvertible state-space theory

For $G=\mathbb Z_N$ and trivial $[\omega]$, the finite gauge theory matches compact $BF_N$ only after the global sectors, line operators, and measure are identified. A local flatness equation is not enough. More generally, the inverse of a fixed cocycle phase is obtained by $[\omega]\mapsto-[\omega]$; that statement does not make the gauged Dijkgraaf–Witten theory invertible.

What invertibility does—and does not—exclude

For a unitary gapped system with one local vacuum and no topological excitations, the universal infrared theory is invertible and its partition function records quantized background response. It can still be physically nontrivial: a torsion phase or gravitational response may survive even when all ordinary curvature response vanishes. Choi and Ohmori state this invertible-infrared criterion and the torsional possibility in Choi and Ohmori 2022, Abstract (unpaginated), § 1.1, printed p. 2 and n. 1, and § 1.2, printed p. 4, final paragraph, arXiv v2 PDF.

The converse inference from one measured response is invalid. A noninvertible topological sector may carry the same symmetry response as an invertible phase. Response data can distinguish phases under a declared equivalence relation, but need not classify all operator sectors, ground-state degeneracies, or boundary realizations.

Likewise, a protected boundary response does not universally force a particular gapless particle or edge CFT. Current general results remain hypothesis-dependent. Córdova, Freed, and Teleman prove special cases of the statement that an invertible theory admitting a gapped, projectively topological boundary has finite-order anomaly; they do not promote that implication to an unrestricted classification in every dimension Córdova, Freed, and Teleman 2024, Abstract and Introduction, arXiv v1, printed pp. 1–3, especially Theorems A and B and the Rationality Criterion, PDF.

Finally, neither group cohomology nor a differential-form action is a universal classification of invertible phases. The answer can depend on spin or pin structure, global symmetry form, reflection positivity, interactions, and torsion data. Kapustin’s cobordism proposal was motivated precisely by bosonic symmetry-protected phases beyond group cohomology Kapustin 2014, arXiv v3, printed p. 3 and pp. 14–15, PDF. The theorem-first bordism and generalized-cohomology classifications are beyond this page’s physical definition and finite-model checks.

Source versions and official publication records used here were checked through 10 August 2026. The stacking definition and the three finite-model tests are established within their stated structures; completeness claims outside those hypotheses remain deliberately excluded.

Common pitfalls

A phase-valued formula proves full invertibility. It proves only a closed-manifold response candidate. One must still supply gluing, boundary lines, state spaces, and invertible bordism maps—and check that no extra noninvertible infrared sector is present.

Every local counterterm is physically trivial. Counterterms that change an anomaly representative and bulk responses that distinguish phases can be the same kind of local functional used under different equivalence relations. State the subgroup being quotiented before discarding one.

Integrating a topological field always produces a noninvertible TQFT. The $U(1)_{\pm1}$ spin Chern–Simons theories are invertible. The state-space and stacking tests, not the word “dynamical,” decide.

A boundary variation uniquely determines the edge theory. It fixes an anomaly that the boundary must match. Gapless modes, symmetry breaking, topological order, or a relative completion may realize that requirement under different hypotheses.

A vanishing anomaly polynomial means a trivial response. Differential forms forget torsion. Mapping-torus phases, eta invariants, and other global data can remain nontrivial.

Check your understanding

1. Why is complex conjugation not the whole inverse?

Suppose $\mathcal Z_{\alpha^{-1}}(M)=\overline{\mathcal Z_\alpha(M)}$ for every closed $M$. Explain what else must be checked before concluding that $\alpha^{-1}$ is a stacking inverse.

Solution

The closed amplitudes are only the top-dimensional values of the field theory. The state spaces on every closed codimension-one manifold must tensor to a line, the boundary anomaly lines must pair to the trivial line, and every bordism map and admitted defect datum must compose to that of the trivial theory. Without these checks, equal closed amplitudes need not determine an equivalence of extended field theories.

2. Separate a response from the full phase

Let $Z_{\mathrm{IR}}[B]=\mathcal Z_\alpha[B]Z_{\mathrm{toric}}[B]$, where $\mathcal Z_\alpha$ is invertible and $Z_{\mathrm{toric}}$ has four states on $T^2$. Is the response invertible? Is the full infrared phase invertible?

Solution

$\mathcal Z_\alpha$ is an invertible response by assumption. The full phase is not invertible because its toric-code factor has a four-dimensional torus state space, which cannot tensor with another finite-dimensional state space to give a line.

3. Test fixed and dynamical Chern–Simons fields

Take spin Chern–Simons level $k=4$. Give the stacking inverse when $A$ is a fixed background, then decide whether the theory obtained by integrating $A$ is invertible.

Solution

The fixed-background inverse has level $-4$, so the two response phases multiply to one. After integrating $A$, the compact $U(1)_4$ theory has $\dim\mathcal H(T^2)=4$. That state-space dimension rules out invertibility.

4. Diagnose compact BF at N = 3

Compute the local boundary variation under $B\mapsto B+\mathrm d\chi$ and compare the fixed response with the theory obtained by integrating both fields.

Solution

The variation is

$$\Delta_\chi S_{\mathrm{BF}} = \frac{3}{2\pi} \int_{\partial M}\chi\,\mathrm dA.$$

With $A$ and $B$ fixed, the level-three phase is invertible and its inverse has level $-3$. Integrating both compact fields gives the untwisted $\mathbb Z_3$ gauge TQFT, whose torus state space has dimension $9$; it is noninvertible.

5. Why does changing omega to minus omega not invert the gauged theory?

For a finite group $G$, compare the fixed-bundle phase with the automorphism-weighted bundle sum.

Solution

At fixed $P$, the phases for $[\omega]$ and $-[\omega]$ multiply pointwise to one. Gauging replaces evaluation at one $P$ by a sum over inequivalent bundles. Multiplying two such sums does not collapse them to the trivial theory; the gauged theories retain state spaces and operator sectors that are generally noninvertible.

6. Which counterterm quotient is being used?

An invariant quantized functional $e^{iI[B]}$ distinguishes two candidate gapped phases but is also a local functional of $B$. May it be discarded as a counterterm?

Solution

Not without specifying the equivalence relation. In an anomaly calculation, a permitted local term may change a representative. In a phase classification, the same quantized invariant response remains physical unless the declared trivial subgroup includes it—for example because it is connected to an atomic phase by an allowed deformation. Locality alone does not decide.

Continue to phase applications and classifications

Time-Reversal-Invariant $\mathbb Z_2$ Topological Insulators will apply background responses to band matter, protected surfaces, and material diagnostics. Invertible Field Theories and Generalized Cohomology will give the theorem-first bordism and generalized-cohomology classification under its stated tangential, positivity, and deformation hypotheses. Symmetry TFT: Encoding Symmetry, Anomaly, and Gauging will package fixed backgrounds, anomaly inflow, boundary conditions, and gauging in one topological framework. These continuations require the response-versus-field-sum distinction established here.

References

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• Choi, Yichul, and Kantaro Ohmori. “Higher Berry Phase of Fermions and Index Theorem.” Journal of High Energy Physics 2022, no. 9 (2022): 022. DOI. Open PDF, arXiv v2.
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