Topological Terms and Invertible Responses

A topological density becomes a physical action term only when its exponentiated action is a single, globally defined phase on every admitted field configuration. That test comes before coefficient quantization, periodicity, boundary inflow, or any claim about invertibility. The chapter therefore follows one durable order: specify the global fields and manifold, make the phase independent of representatives and fillings, determine its period or level, supply boundary and tangential data, decide which fields are fixed or summed, and only then apply the stacking and state-space tests.

Choose the global-definition route when a local total derivative or metric-independent density is being promoted to an action. Choose an action-specific route for theta, Chern–Simons, Wess–Zumino, or compact BF terms. Choose the response route when fixed backgrounds, boundary anomaly, and invertibility are the issue. A dynamical topological gauge theory is the next chapter’s question: integrating the fields is not a synonym for taking an invertible response.

Helpful background. Characteristic Classes and Chern–Weil Theory supplies integral characteristic numbers and transgression. Anomaly Polynomials and Inflow supplies the boundary-orientation and counterterm conventions used when a bulk response is not invariant by itself. Neither is required merely to choose an entry route.

Parent volume. Symmetry and Gauge Structure

Jump to: separate the tests · choose a route · follow the action thread · open the exact guide · review the chapter

The exponentiated phase is the common object

Let $M^d$ carry the declared orientation and any required spin, pin, framing, or other tangential structure. Let $\Phi$ denote the complete global field data: bundles or higher cocycles, connections, defects, boundary conditions, and charge or flux lattices. The invariant object is

\mathcal W_M[\Phi]=\exp\!\bigl(iS_M[\Phi]\bigr)\in U(1),

not a preferred real-valued branch of $S$ and not merely its local density. If a local formula is defined using a filling $X^{d+1}$ , two choices $X$ and $X'$ glue to a closed manifold $Y=X\cup_M(-X')$ , and filling independence is the condition

\frac{\mathcal W_M[X]}{\mathcal W_M[X']} = \exp\!\left(i\kappa\int_Y\Omega_{d+1}\right) =1

for every admitted closed $Y$ and extended field. The period lattice of $\Omega_{d+1}$ therefore fixes the allowed coefficient $\kappa$ or, when the coefficient is a continuous angle, its identification. On a boundary the same calculation leaves uncancelled boundary data; it does not justify discarding the variation.

Local exactness, metric independence, and global descent answer different questions. Chern–Simons transgression is a standard example: the characteristic form is global while its primitive is generally only local Nakahara 2003, § 11.5.1, eqs. (11.100)–(11.102). The chapter’s governed five-row comparison gives the decisive question, required input, and nonimplication for every claim. It is linked rather than copied so that the local-density, global-action, response, and TQFT distinctions have one canonical table.

Two decisions follow only after the phase exists:

Field role: a fixed background produces a response functional; a field integrated or summed over is dynamical.
Invertibility: an invertible theory has a tensor inverse under stacking, so in the finite topological setting it assigns lines or superlines to spatial slices and invertible maps to bordisms.

A dynamical theory can be invertible, and a fixed response can multiply a noninvertible theory. Field role and stacking are independent axes. The line-valued criterion and its full-functor scope are developed in Freed and Hopkins 2021, § 5.2, arXiv v6, printed pp. 33–34, eq. (5.4), Example 5.3, and Definition 5.9, PDF.

Check your preparation

Use these questions to repair the first missing capability, not to assign a score.

Observable preparation for the three chapter routes
Can you do this?	Ready	If unsure	Repair route
Distinguish a local gauge potential from a global connection	Enter the global-definition or Chern–Simons route	Check patch transformations and integral flux before integrating a local form	Local Potentials and Global Gauge Configurations
Use orientation, pullback, and Stokes' theorem with the correct form degree	Enter the Wess–Zumino or BF route	Check whether the displayed integrand is a top form and what survives on a boundary	Differential Forms, Integration, Orientation, and Stokes' Theorem
Read a coefficient from the actual integral period lattice	Enter the theta or level-quantization route	Distinguish a real cohomology class from its integral refinement and torsion data	Characteristic Classes and Chern–Weil Theory
Distinguish a fixed background from a variable in the path integral	Enter the response and inflow route	Ask what is transformed, what is integrated, and which counterterms are admissible	Coupling Background Gauge Fields and Bundles

Choose a route from the global question

The arrows below are suggested reading orders. The exact guide later in the page names every hard prerequisite; chapter order alone does not create one.

Four routes from a proposed term to a controlled conclusion
Starting question	Route	Hard inputs along the route	Licensed conclusion
Does this local expression define a global phase?	Global definition and quantization tests	Global gauge configurations and differential forms	A pass/fail verdict for the exponentiated action, with boundary and tangential assumptions visible
What is the period of a continuous topological angle?	Global phase → theta terms and vacuum sectors	Large transformations and the actual admitted topological-charge lattice	The exact parameter identification, including global-form, boundary, and background-counterterm qualifications
Which coefficient is allowed for Chern–Simons, Wess–Zumino, or BF?	Global phase → choose Chern–Simons, Wess–Zumino/WZW, or compact BF	Connections for Chern–Simons; coset fields for Wess–Zumino; complementary higher forms for BF	A coefficient lattice or periodic identification in the declared global theory, not a dynamical phase diagram
Is the result an invertible response, an anomaly inflow term, or a dynamical TQFT?	Global phase + field role + inflow → background responses and invertible phases	Fixed-background Ward identities, counterterm quotient, and stacking data	A response/invertibility verdict and the exact point where a dynamical TQFT becomes a separate construction

One global-definition test runs through four actions

The four model families are useful because they change different pieces of the same decision problem. They should not be identified merely because each contains a metric-independent local expression.

Theta terms test the admitted charge lattice

For sectors with topological charge $Q(\gamma)$ ,

Z_M(\theta)= \sum_{\gamma\in\Gamma_M} e^{i\theta Q(\gamma)}Z_{M,\gamma}.

Let $\Lambda_Q\subset\mathbb R$ be the additive subgroup generated by every admitted charge. The period group is

\operatorname{Per}(Q)= \left\{ \Delta\theta\in\mathbb R \ \middle|\ e^{i\Delta\theta q}=1 \text{ for every }q\in\Lambda_Q \right\}.

Thus integer charge gives a $2\pi$ identification, while fractional sectors can enlarge it or make a shift return the theory only after a background counterterm is changed. Orientation reversal sends $Q$ to $-Q$ and theta to $-\theta$ ; it does not determine the vacuum energy, number of branches, or spontaneous CP breaking.

Chern–Simons and Wess–Zumino terms test transgression and fillings

A Chern–Simons local form transgresses an integral characteristic class. With the Yang–Mills coupling absorbed into the Hermitian connection, $\widehat A:=g_{\mathrm{YM}}A$ , a basic non-Abelian normalization is

S_{\mathrm{CS}}[\widehat A] = \frac{k}{4\pi} \int_{M^3} \operatorname{Tr}\!\left( \widehat A\wedge\mathrm d\widehat A -\frac{2i}{3}\widehat A\wedge\widehat A\wedge\widehat A \right),

but the allowed $k$ is determined by the global group, trace normalization, and tangential structure. A bosonic compact Abelian diagonal level is even in the convention used by this volume; an integral odd diagonal level defines a spin theory. Quantization does not remove the possible quantum framing anomaly. The translated lattice normalization and the framing-sensitive vacuum factor are given in Belov and Moore 2005, § 1, arXiv v1, printed pp. 3–5, eqs. (1.1)–(1.7), and § 5.6.1, printed p. 35, eq. (5.45), PDF.

A Wess–Zumino term instead starts on a closed oriented $\Sigma^d$ with a field $\phi:\Sigma^d\to X$ , an oriented $B^{d+1}$ with $\partial B=\Sigma$ , an extension $\widetilde\phi:B\to X$ satisfying $\widetilde\phi|_{\Sigma}=\phi$ , and a closed form $H_{d+1}$ ,

\mathcal W_H[\phi;B] = \exp\!\left( i\int_B\widetilde\phi^{*}H_{d+1} \right).

Comparing two fillings gives the period condition. Nonbounding fields, torsion, gerbe data, and boundaries require an intrinsic global completion; the extension formula alone is not universal. The group-valued WZW control and its quantized coefficient are developed in Witten 1984, printed pp. 458–459, eqs. (12)–(15), PDF.

Compact BF tests field role and global sectors

For compact $U(1)$ higher connections $A_p$ and $B_{D-p-1}$ with $1\leq p\leq D-2$ and $2\pi$ -integral curvatures on a closed oriented $D$ -manifold $M$ ,

S_E^{\mathrm{loc}}[A,B] = \frac{iN}{2\pi} \int_M B_{D-p-1}\wedge\mathrm dA_p, \qquad N\in\mathbb Z_{>0}.

The local equations impose flatness; when the fields are summed, the compact global measure at level $N$ projects the surviving holonomies to $\mathbb Z_N$ . Linked electric and magnetic supports acquire an $N$ th-root phase. If $A$ and $B$ are fixed backgrounds, the expression is a cross-response candidate; if they are integrated with the global measure, it defines a dynamical compact BF theory. Kapustin and Seiberg give the compact global completion, integer level, and operator algebra in 2014, § 3, arXiv v2, printed pp. 9–13, eqs. (3.1)–(3.16), PDF.

Finite Dijkgraaf–Witten theory makes the last distinction sharp. For a finite group $G$ , a class $[\omega]\in H^D(BG;U(1))$ , and a closed oriented $D$ -manifold $M$ , the evaluation $\operatorname{Hol}_\omega(P)$ on a principal $G$ -bundle is one action weight. Summing the field gives the automorphism-weighted partition function

Z_{G,\omega}(M) = \sum_{[P]\in\pi_0\operatorname{Bun}_G(M)} \frac{\operatorname{Hol}_\omega(P)} {\lvert\operatorname{Aut}(P)\rvert}.

The weight and completed TQFT are not interchangeable. The finite sum and lattice cocycle construction are given in Dijkgraaf and Witten 1990, §§ 6.2–6.5, printed pp. 415–421, eqs. (6.8)–(6.27), Open PDF; the action line and groupoid measure are made functorial in Freed and Quinn 1993, §§ 1–2, printed pp. 438–445, especially eqs. (1.1)–(1.2), (2.1), and (2.9), Open PDF.

Across the thread, the invariant order is: global fields, exponentiated phase, periods, boundary completion, field role, and stacking or state-space test. What changes is whether the coefficient is a continuous angle, an integer level, a finite cohomology class, or part of a dynamical sum.

Exact chapter guide

The six pages appear in chapter order. Every row names its actual hard preparation rather than treating the preceding page as sufficient by itself.

Six pages from the global phase test to invertible response and inflow
Page	Question and capability	Required background	Stop condition
When Is a Topological Term Well Defined?	Separate local exactness, metric independence, global descent, coefficient quantization, field role, and invertibility	Global gauge configurations; differential forms and Stokes' theorem	A globally defined phase still need not be an invertible response or a dynamical TQFT
Theta Terms, Periodicity, and Vacuum Sectors	Read the theta identification from the admitted charge lattice and qualify it by global form, boundaries, and background counterterms	Global phase tests; large transformations and sectors	Periodicity alone does not determine vacuum branches, tunneling, or CP realization
Chern–Simons Actions and Level Quantization	Derive the level lattice from global connections and distinguish spin, boundary, and framing data	Global phase tests; connections and curvature	Classical level quantization does not determine the quantum TQFT, framing anomaly, or boundary dynamics
Wess–Zumino and WZW Terms	Compare two extensions, quantize the normalized periods, and relate boundary variation to anomaly data	Global phase tests; cosets and nonlinear realizations	The extension formula does not classify nonbounding, torsion, spin-refined, or boundary completions
BF Couplings and Discrete Topological Data	Use compactness and integer level to obtain finite holonomy, complementary linking, and the fixed-versus-dynamical distinction	Global phase tests; differential forms and Stokes' theorem	The local equation “flat” does not mean globally trivial, and de Rham forms miss torsion
Background Responses and Invertible Phases	Test stacking invertibility, local-counterterm equivalence, boundary inflow, and the response of a complete phase	Global phase tests; background coupling; anomaly inflow	A normalized phase on closed manifolds is not by itself a full invertible field theory or proof that every other sector is trivial

Keep the action-specific nonimplications visible

The governed five-row comparison already carries the chapter’s general logical distinctions. The specialized actions add their own stop conditions:

a theta identification does not determine the vacuum branches, tunneling rate, or realization of CP;
an integral Chern–Simons level does not remove spin, framing, global-form, or boundary dependence;
a Wess–Zumino period test for bounding fields does not construct the intrinsic phase on every nonbounding or torsion configuration;
a flat compact BF field is not globally trivial, and finite holonomy appears only after the level- $N$ global sum is included;
one Dijkgraaf–Witten cocycle evaluation is an action weight, not the automorphism-weighted dynamical theory; and
a local inflow cancellation does not establish a globally defined, extension-independent bulk response on every admitted background.

These are action-level qualifications. The general distinction among a local density, global action, invertible response, and dynamical TQFT remains in the canonical five-row table.

What the chapter establishes—and where it stops

Within its declared geometric and compact-gauge examples, the chapter establishes a repeatable workflow:

declare the global field and manifold data;
construct the exponentiated phase rather than trusting a local density;
determine the period, level, or finite cohomology class;
open a boundary and identify the missing relative data;
decide which fields are fixed backgrounds and which are dynamical; and
test invertibility and gluing without inferring them from one closed amplitude.

It does not calculate instanton dynamics, vacuum energies, axion phenomenology, WZW current algebra, material response coefficients, or a bordism classification of all invertible phases. It also does not construct the exact dynamical TQFT associated with every action. Those theories require state spaces, measures, operators, and gluing, which begin in What Is a Topological Field Theory?.

An invertible response is naturally relative at a boundary: a bulk state or phase and boundary covector live in dual one-dimensional spaces and only their pairing is a number. Invertible field theories and anomaly data are related through this line-valued structure Freed 2014, §§ 2.1–2.3, arXiv v2, printed pp. 2–6, eqs. (2.1)–(2.10), PDF. The overview stops before the theorem-level classification of those lines and their fully extended data.

Review the chapter

The prompts below are retrieval and transfer checks, not a separate formal assessment. A successful response states its domain and passes the listed invariant; the repair link identifies the characteristic missing step.

Eight ways to demonstrate the chapter's central capabilities
Mode and task	Owner pages	Successful response and invariant	Characteristic repair
Retrieval — state the ordered tests for a proposed topological term	When Is a Topological Term Well Defined?	Names global fields, exponentiated descent, period, boundary/tangential completion, field role, and stacking/state-space test without merging them	Return to the five-claim comparison if local exactness is promoted directly to TQFT status
Explanation — explain why theta is a character of the admitted charge lattice	Theta Terms, Periodicity, and Vacuum Sectors	Derives the annihilator condition for every admitted charge and distinguishes parameter identification from vacuum dynamics	Repair global-form assumptions at Global Form, Matter Representations, and the Faithful Gauge Group
Derivation check — compare two fillings of a Wess–Zumino term	Wess–Zumino and WZW Terms	Glues the fillings with opposite orientation, evaluates the closed period, and identifies the coefficient condition needed for a unit phase	Return to de Rham periods and torsion if a closed form is assumed automatically integral
Representation change — translate a local Chern–Simons or BF density into global data	Chern–Simons levels; compact BF	Replaces a fictitious global potential by bundle or differential-cocycle data while preserving the exponentiated phase and integral lattice	Repair patching at Local Potentials and Global Gauge Configurations
Comparison — distinguish a theta angle, Chern–Simons level, and finite Dijkgraaf–Witten class	Theta terms; Chern–Simons; BF and finite data	Identifies respectively a continuous periodic character, an integral level lattice, and a finite cocycle label; no shared word “topological” erases the difference	Return to periods determine quantization or periodicity
Transfer — analyze exp(i theta Q) for a compact U(1) connection on a closed oriented surface with all line-bundle sectors admitted	Theta terms; global phase tests	Uses Q=(2π)⁻¹∫F ∈ ℤ and the fact that every integer first Chern number is admitted to obtain theta modulo 2π, then states separately whether the connection is fixed or integrated	Repair the flux normalization at Characteristic Classes and Chern–Weil Theory
Failure diagnosis — locate the error in “metric independent implies globally defined invertible TQFT”	Global tests; invertible responses	Names the missing descent, period, boundary, field-role, gluing, and stacking checks	Repair at the governed comparison table
Synthesis — decide whether a boundary variation describes inconsistency, inflow, or an allowed relative theory	Background responses; anomaly inflow; Wess–Zumino boundary data	Specifies the bulk phase, boundary orientation, admissible counterterms, field role, and compatible boundary trivialization before claiming cancellation	Return to What Is an Anomaly? if a removable representative is confused with the anomaly class

Continue from the global phase

For state spaces, gluing, and the distinction between an action and a full dynamical theory, continue to What Is a Topological Field Theory?.
For the exact compact Abelian theory whose action was quantized here, continue to Abelian Chern–Simons Theory.
For compact higher fields, finite holonomy, and boundary polarizations, continue to BF Theory as a Topological Gauge Theory.
For the one-higher-dimensional organization of response, anomaly, and gauging choices, continue to Symmetry TFT: Encoding Symmetry, Anomaly, and Gauging.

References

Belov, Dmitriy M., and Gregory W. Moore. “Classification of Abelian Spin Chern–Simons Theories.” arXiv:hep-th/0505235v1 [hep-th] (2005). Stable record.
Dijkgraaf, Robbert, and Edward Witten. “Topological Gauge Theories and Group Cohomology.” Communications in Mathematical Physics 129, no. 2 (1990): 393–429. DOI. Open PDF.
Freed, Daniel S. “Anomalies and Invertible Field Theories.” Proceedings of Symposia in Pure Mathematics 88 (2014): 25–46. DOI. Open PDF, arXiv v2.
Freed, Daniel S., and Michael J. Hopkins. “Reflection Positivity and Invertible Topological Phases.” Geometry & Topology 25, no. 3 (2021): 1165–1330. DOI. Open PDF, arXiv v6.
Freed, Daniel S., and Frank Quinn. “Chern–Simons Theory with Finite Gauge Group.” Communications in Mathematical Physics 156, no. 3 (1993): 435–472. DOI. Open published PDF.
Kapustin, Anton, and Nathan Seiberg. “Coupling a QFT to a TQFT and Duality.” Journal of High Energy Physics 2014, no. 4 (2014): 001. DOI. Open PDF, arXiv v2.
Nakahara, Mikio. Geometry, Topology and Physics. 2nd ed. Graduate Student Series in Physics. Bristol: Institute of Physics Publishing, 2003. ISBN 978-0-7503-0606-5. Publisher record.
Witten, Edward. “Non-Abelian Bosonization in Two Dimensions.” Communications in Mathematical Physics 92, no. 4 (1984): 455–472. DOI. Open PDF.