Chern–Simons Actions and Level Quantization

A Chern–Simons density is only a local representative. Its coefficient is allowed precisely when the exponentiated action descends to one phase on the space of global connections. In the basic normalization for a connected, simply connected compact simple group, a large gauge transformation changes the action by $2\pi k$ times an integer, so $k\in\mathbb Z$. For a general compact group the primary datum is instead an integral class in $H^4(BG;\mathbb Z)$; the global form of $G$, torsion data, and the tangential structure can change the level lattice without changing the displayed local three-form.

This page begins on a closed smooth oriented three-manifold. It then opens a boundary, distinguishes ordinary oriented from spin theories, and finally quantizes around an acyclic flat connection. Classical level quantization, the quantum framing anomaly, and perturbative BV obstruction classes are different tests. Fixed background connections define response phases; integrating the same connections defines a dynamical theory and requires additional state-space and gluing data.

Required background. When Is a Topological Term Well Defined? supplies the global-phase and filling-independence tests used below. Bundle Connections, Curvature, and the Bianchi Identity supplies the patch law for a connection and explains why its local potential is not generally a global one-form.

Helpful background. Characteristic Classes and Chern–Weil Theory supplies transgression and the integral characteristic-class test. Anomaly Polynomials and Inflow supplies the local bulk–boundary descent used when the three-manifold has a boundary.

The Chern–Simons phase is the global action

Let $M^3$ be closed, smooth, and oriented, let $P\to M$ be a principal bundle for a compact group $G$, and let $A$ be a connection. To match the site’s Hermitian-generator convention, absorb the Yang–Mills coupling into the matrix-valued potential,

$$\widehat A:=g_{\mathrm{YM}}A, \qquad D=\mathrm d-i\widehat A, \qquad \widehat F=\mathrm d\widehat A-i\widehat A\wedge\widehat A.$$

For $G=SU(N)$ use the fundamental trace with $\operatorname{tr}_F(T^aT^b)=\tfrac12\delta^{ab}$. In a bundle trivialization the corresponding Chern–Simons form is

$$\operatorname{CS}_3(\widehat A) = \operatorname{tr}_F\!\left( \widehat A\wedge\mathrm d\widehat A -\frac{2i}{3}\widehat A\wedge\widehat A\wedge\widehat A \right), \qquad \mathrm d\operatorname{CS}_3(\widehat A) = \operatorname{tr}_F(\widehat F\wedge\widehat F).$$

Thus one often writes the Lorentzian local functional

$$S_k^{\mathrm{loc}}[\widehat A] = \frac{k}{4\pi}\int_M\operatorname{CS}_3(\widehat A).$$

The superscript is consequential: the local potential and the three-form can change between bundle charts. If $(P,\widehat A)$ extends to a connection on $X^4$ with $\partial X=M$, the filling diagnostic is

$$\mathcal W_k[M;\widehat A] = \exp\!\left[ \frac{ik}{4\pi}\int_X \operatorname{tr}_F(\widehat F\wedge\widehat F) \right].$$

This presentation must be independent of $X$ and of the extension. It is not an intrinsic definition when the bundle does not extend. In that case the global phase is constructed by a differential character or equivalent bundle-and-cocycle data. For a general compact $G$, the invariant pairing used in the local form must be the real image of an integral level

$$\lambda\in H^4(BG;\mathbb Z),$$

possibly with a spin refinement. Torsion information in $\lambda$ can be invisible to the differential form. Freed’s global construction makes the integral lattice, differential character, boundary line, and gluing law explicit Freed 2002, §§ 1.2–2.3, journal pp. 297–303, especially eqs. (1.14)–(1.18).

If $\widehat A$ is fixed, $\mathcal W_k$ is a background response. If it is integrated over, the same phase is part of a Chern–Simons path integral. Nothing in the local density alone decides which role is intended.

Large gauge transformations select the level lattice

For the simply connected $SU(N)$ normalization above, a closed four-manifold $Z$ obeys

$$Q_Z := \frac{1}{8\pi^2} \int_Z\operatorname{tr}_F(\widehat F\wedge\widehat F) \in\mathbb Z.$$

Two fillings of the same boundary data glue to such a $Z$, and their phases differ by

$$\frac{\mathcal W_{k,X_1}}{\mathcal W_{k,X_2}} = \exp(2\pi i kQ_Z).$$

Equivalently, on a closed $M$ a finite gauge transformation $u$ changes a local representative by

$$S_k[\widehat A^{\,u}]-S_k[\widehat A] = 2\pi k\,\nu(u), \qquad \nu(u)\in\mathbb Z,$$

where the sign used to define $\nu$ follows the convention for the gauge action. A transformation of unit winding then forces $k\in\mathbb Z$. Witten uses the anti-Hermitian connection $A_{\mathrm W}=-i\widehat A$. With the ordinary matrix trace, his local three-form is minus the site form above, so the same orientation uses $k_{\mathrm W}=-k_{\mathrm{site}}$; reversing the pairing or orientation is an equivalent translation. After that sign bridge, his winding calculation gives the displayed shift and the same integrality test Witten 1989, § 1, printed pp. 353–354, eqs. (1.1)–(1.4), Open PDF.

The slogan “the level is an integer” has three hidden hypotheses:

• a generator of the free part of $H^4(BG;\mathbb Z)$ has been chosen;
• the invariant trace has been normalized to that generator; and
• no additional torsion or spin-refined datum is being suppressed.

For a quotient group, a representation trace that was basic on the simply connected cover may describe only a sublattice of allowed levels. Conversely, two global phases can have the same real Chern–Weil form while differing by torsion. The globally safe statement is therefore $\lambda\in H^4(BG;\mathbb Z)$, not a universal assertion about a bare number $k$. Freed’s first construction derives the closed-manifold gauge shift and the boundary-valued action under its connected, simply connected hypotheses Freed 1995, § 2, printed pp. 14–21, especially Proposition 2.7 and Theorem 2.19, Open PDF.

Spin structure changes the Abelian lattice

The one-component compact $U(1)$ theory displays the tangential-structure dependence with no group-theory complications. Normalize $F/(2\pi)=c_1\in H^2(X;\mathbb Z)$. The filling phase is

$$\mathcal W_k[M;a] = \exp\!\left[ i\pi k\left\langle c_1^2,[X]\right\rangle \right].$$

On a general oriented four-manifold the integer $\langle c_1^2,[X]\rangle$ can be odd: $\mathbb{CP}^2$ supplies the basic test. Independence of the filling then requires

$$k\in2\mathbb Z \qquad \text{for an ordinary oriented bosonic theory.}$$

On a closed spin four-manifold, the intersection form is even, so every integer $k$ passes:

$$k\in\mathbb Z \qquad \text{for the spin theory.}$$

For $n$ compact Abelian connections this becomes

$$S_K[a] = \frac{1}{4\pi} \int_M K_{IJ}a^I\wedge\mathrm da^J.$$

The matrix $K$ is symmetric and integral. An ordinary oriented bosonic theory requires even diagonal entries, while a spin theory permits arbitrary integral diagonal entries. Belov and Moore use curvatures with integral periods and a level $k_{\mathrm{BM}}$ for which $k_{\mathrm{site}}=2k_{\mathrm{BM}}$; their integer and half-integer cases therefore become the site-even bosonic and site-odd spin cases, respectively Belov and Moore 2005, § 1, arXiv v1, printed pp. 3–4, eqs. (1.1)–(1.3), Open PDF.

Four independent axes should not be merged:

• orientation supplies the sign of the integral, and reversal sends $S_K\mapsto-S_K$ or $K\mapsto-K$;
• spin structure enlarges the classical Abelian level lattice;
• global gauge form determines which bundles and integral classes are allowed; and
• framing enters the regulated quantum partition function below.

A spin structure does not remove the quantum framing anomaly, and a framing does not change the classical level lattice.

A boundary turns the phase into relative data

Now let $M$ have boundary and keep the induced boundary orientation. In the site convention, variation of the non-Abelian local representative gives

$$\delta S_k = \frac{k}{2\pi}\int_M \operatorname{tr}_F(\delta\widehat A\wedge\widehat F) - \frac{k}{4\pi}\int_{\partial M} \operatorname{tr}_F(\widehat A\wedge\delta\widehat A).$$

Flatness, $\widehat F=0$, is the bulk equation only after the boundary term is made compatible with the variational problem. For $\delta_\epsilon\widehat A=\mathrm d\epsilon-i[\widehat A,\epsilon]$, the same convention gives the local boundary variation

$$\delta_\epsilon S_k = \frac{k}{4\pi}\int_{\partial M} \operatorname{tr}_F(\epsilon\,\mathrm d\widehat A),$$

with the overall sign reversed if the boundary-orientation or gauge-action convention is reversed.

This is not a contradiction with gauge invariance on a closed manifold. On a boundary the exponentiated action is naturally a vector in a line over the boundary fields, and gauge transformations act on that line. A standalone system must choose a completion, for example:

• a boundary condition that kills the offending variation;
• a boundary counterterm on a restricted field space;
• boundary degrees of freedom with the opposite variation; or
• a relative bulk–boundary definition supplied by inflow.

The local variation licenses none of these choices automatically, and it does not by itself derive a Wess–Zumino–Witten edge theory. Freed’s Chern–Simons line and gluing theorem give the global form of this statement Freed 1995, § 2, printed pp. 16–21, especially Proposition 2.17 and Theorem 2.19, Open PDF.

Quantization introduces a framing-sensitive phase

Classical metric independence does not imply that a regulated quantum partition function is an invariant of an unframed oriented three-manifold. Gauge fixing introduces a metric. The magnitude and phase of the Gaussian determinant behave differently: analytic torsion supplies a topological magnitude, while the spectral $\eta$ invariant has a local metric variation. A gravitational Chern–Simons counterterm cancels that variation only after a framing convention is chosen.

If $f$ is a tangent framing and $s\in\mathbb Z$ changes it by $s$ units, the partition function transforms as

$$Z(M,f+s) = \exp\!\left(\frac{2\pi i s\,c_-}{24}\right)Z(M,f).$$

At positive level, the one-loop compact-group saddle has coefficient $\dim G$ in this formula. Negative level complex-conjugates the spectral and framing phases, equivalently inserting $\operatorname{sgn}(k)$ in their exponents. In the standard positive-level non-Abelian theory, the exact current-algebra value is

$$c_- = \frac{k\,\dim G}{k+h^\vee}.$$

Witten derives the one-loop gravitational counterterm and the framing law in Witten 1989, § 2, printed pp. 360–361, eqs. (2.18)–(2.25), Open PDF. The familiar $k_{\mathrm{eff}}=k+h^\vee$ is likewise tied to Witten’s positive-level, basic-trace regulator and to a bare-versus-renormalized convention. It must not be applied again when $k$ already denotes a renormalized level.

This three-manifold framing anomaly is also distinct from the self-linking framing of an individual Wilson line. Both appear in Chern–Simons theory, but they concern different objects and have different transformation laws.

First application: Chern–Simons in the three-model thread

The shared $K$-matrix notation makes the compact Abelian Chern–Simons, BF, and finite-gauge cases comparable without declaring them identical. On a closed oriented $M^3$, take compact $U(1)$ connections $a,b$ and write

$$S_{N,p}[a,b] = \frac{1}{4\pi}\int_M \left( p\,a\wedge\mathrm da +2N\,b\wedge\mathrm da \right), \qquad K_{N,p} = \begin{pmatrix} p & N\\ N & 0 \end{pmatrix}.$$

Compactness and large transformations require $N\in\mathbb Z_{>0}$ and $p\in\mathbb Z$. Odd $p$ requires spin; an ordinary bosonic theory has $p=2r$. The field redefinition $b\mapsto b-a$ identifies $p$ with $p-2N$, so the bosonic twist is $r\in\mathbb Z_N$. The cases are:

• one compact field with $K=(k)$: compact $U(1)_k$ Chern–Simons theory;
• $p=0$: compact $BF_N$ and, with both fields integrated, untwisted $\mathbb Z_N$ gauge theory; and
• $p=2r$: a continuum presentation of the bosonic $\mathbb Z_N$ Dijkgraaf–Witten twist $r$.

For the last two cases $\lvert\det K_{N,p}\rvert=N^2$, so canonical quantization gives $\dim\mathcal H(T^2)=N^2$. The twist changes spins and braiding, not this count. Belov and Moore give the general genus-$g$ result $\dim\mathcal H(\Sigma_g)=\lvert\det K\rvert^g$ Belov and Moore 2005, § 5.3, arXiv v1, printed p. 26, after eq. (5.17), Open PDF.

Three compact topological models in one normalization; the rows are comparisons, not claims of equivalence

Model and field role	Global coefficient test	Extra structure	Established here	Next treatment
Compact U(1) Chern–Simons; fixed or integrated connection	Even integer level for an ordinary bosonic theory; any integer level for spin	Bundle data and differential refinement; spin for odd level; quantum framing if integrated	The level lattice and the separation of response from dynamical theory	Abelian Chern–Simons TQFT and K-matrix observables
Compact BF; both fields integrated	Integer N from compact large-gauge pairing	Compact differential-cocycle and torsion sectors	The untwisted K-matrix and N-squared torus-state count	BF couplings and discrete topological data
Finite Z_N gauge theory; bundles summed with twist r	r is a class modulo N; in the continuum matrix p equals 2r modulo 2N	Bundle groupoid measure and global normalization	The twisted continuum representative and its field-role ceiling	Finite-gauge and symmetry-TFT constructions

Kapustin and Seiberg derive the compact global completion, the integer BF coefficient, the $p\sim p+2N$ identification, and the spin qualification 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), Open PDF. Holding $a,b$ fixed produces a phase; integrating them produces a TQFT. Compact BF and a finite bundle sum agree only after their global sectors and normalization are matched; the local equations alone do not prove the equivalence.

Acyclic flat connections isolate the one-loop test

The connection to obstruction–deformation theory can now be made without confusing three separate questions. Let $M$ be a closed oriented three-manifold, let $G$ be compact, simple, and simply connected, and choose a positive integral basic level. Let $A_0$ be an irreducible flat connection satisfying

$$H^\bullet(M;\operatorname{ad}A_0)=0.$$

This acyclicity condition removes stabilizer, ghost, and moduli zero modes. It does not prove convergence of the full path integral or remove the quantum framing anomaly. The linearized fields and ghosts form the twisted de Rham complex

$$0\longrightarrow \Omega^0(M;\operatorname{ad}P) \xrightarrow{\mathrm d_{A_0}} \Omega^1(M;\operatorname{ad}P) \xrightarrow{\mathrm d_{A_0}} \Omega^2(M;\operatorname{ad}P) \xrightarrow{\mathrm d_{A_0}} \Omega^3(M;\operatorname{ad}P) \longrightarrow0.$$

After choosing a gauge-fixing metric, the relevant odd-signature operator is

$$L_-:=*\mathrm d_{A_0}+\mathrm d_{A_0}*,$$

on odd-degree forms. In the convention used here, the one-loop factor splits as

$$Z_1(A_0;g) = \tau_{\mathrm{RS}}(M;\operatorname{ad}A_0)^{1/2} \exp\!\left[ \frac{i\pi}{4}\eta(L_-;g) \right].$$

Ray–Singer torsion $\tau_{\mathrm{RS}}$ supplies the metric-independent magnitude. The $\eta$ invariant supplies the phase and has a local metric variation. Wernli derives this separation in Wernli 2022, §§ 3.3.2–3.3.3, printed pp. 86–89, eqs. (3.38)–(3.46), Open PDF. Witten’s $\eta$ is one half of the standard signed spectral sum, so his apparently different exponent is the same convention after translation Witten 1989, § 2, printed pp. 357–361, eqs. (2.8) and (2.12)–(2.25), Open PDF.

The gauge-BV obstruction class vanishes in the semisimple control

The first question is the ordinary local gauge-BV quantum master equation. Work on a contractible bulk chart $U\subset M$, trivialize the bundle, gauge the flat $A_0$ to zero on $U$, and hold the metric and framing backgrounds fixed. This is the universal ultraviolet counterterm problem; it is not an identification of the full global deformation complex on the closed three-manifold.

Let $c=c^aT_a$ be the ghost and $[T_a,T_b]=f_{ab}{}^cT_c$. In the configuration-space regularization, the only elementary one-loop gauge tadpole contracts two legs of the cubic vertex. Up to a scheme-dependent coincident-point density $\rho_{\mathrm{reg}}$ and constant $C_{\mathrm{reg}}$, its local representative is

$$\mathcal O_{1}^{\mathrm{gauge}}(c) = C_{\mathrm{reg}} \int_U c^a f_{ab}{}^b\,\rho_{\mathrm{reg}} =0.$$

The last equality is pointwise: compact semisimple $\mathfrak g$ is unimodular, so $f_{ab}{}^b=\operatorname{tr}(\operatorname{ad}_{T_a})=0$. The representative therefore vanishes independently of the unused scheme-dependent factor. Wernli proves the corresponding effective quantum master equation in Wernli 2022, §§ 3.1.4 and 3.4.2, printed pp. 74–75 and 90–93, Open PDF.

The universal translation-invariant local deformation complex on such a chart is the reduced Chevalley–Eilenberg complex shifted by three. Its one-loop obstruction class lies in

$$\mathcal O_1\in Z^1(\operatorname{Def}_{\mathrm{CS}}), \qquad [\mathcal O_1] \in H^1(\operatorname{Def}_{\mathrm{CS}}) \cong H^4_{\mathrm{Lie}}(\mathfrak g).$$

For semisimple $\mathfrak g$, $H^4_{\mathrm{Lie}}(\mathfrak g)=0$, in agreement with the zero representative just computed. At one loop, after the lower-order action and equivalence relation have been fixed, the choices of local lift form a torsor for

$$H^0(\operatorname{Def}_{\mathrm{CS}}) \cong H^3_{\mathrm{Lie}}(\mathfrak g) \cong \operatorname{Sym}^2(\mathfrak g)^{\mathfrak g}.$$

For simple $\mathfrak g$ this space is one-dimensional and represents the freedom to renormalize the invariant pairing. Order by order, the formal all-loop family is a torsor for $\hbar H^3_{\mathrm{Lie}}(\mathfrak g)[[\hbar]]$. This is a perturbative vector space, not the integral global lattice $H^4(BG;\mathbb Z)$ derived above. The local obstruction-complex identification is stated in Costello, Francis, and Gwilliam 2026, § 4.3, preliminary arXiv v1, printed pp. 33–34, Open PDF; the torsor of ordinary Chern–Simons quantizations is also summarized in Gwilliam and Williams 2020, § 5, arXiv v2, printed p. 20, Open PDF.

The framing anomaly is a different cocycle

The second question is whether changing the gauge-fixing metric changes the answer. Let $\operatorname{Met}(M)$ denote the space of gauge-fixing metrics. In an oriented orthonormal frame, let $\omega$ be the Levi-Civita $\mathfrak{so}(3)$ connection. Use Wernli’s normalized invariant pairing $\langle-,-\rangle_{\mathrm W}$ and define

$$\operatorname{cs}_{\mathrm{grav}}^{\mathrm W}(\omega) := \frac12\left\langle\omega,\mathrm d\omega\right\rangle_{\mathrm W} +\frac16\left\langle\omega,[\omega,\omega]\right\rangle_{\mathrm W}, \qquad S_{\mathrm{grav}}^{\mathrm W}(g,f) := \int_M\operatorname{cs}_{\mathrm{grav}}^{\mathrm W}(\omega).$$

The pairing is fixed operationally, including its sign, by the unit-framing law

$$S_{\mathrm{grav}}^{\mathrm W}(g,f+1) -S_{\mathrm{grav}}^{\mathrm W}(g,f) =4\pi^2.$$

This normalization is not the raw ordinary trace in the defining vector representation; translating to the site’s representative of $p_1$ requires an additional sign and factor. Keeping the normalized pairing explicit avoids inserting that conversion twice. The one-loop metric-variation cocycle is

$$\mathcal A_{\mathrm{fr}}^{(1)} := \mathrm d_{\operatorname{Met}(M)}\log Z_1 = -\frac{i\,\dim G}{48\pi} \mathrm d_{\operatorname{Met}(M)}S_{\mathrm{grav}}^{\mathrm W}.$$

Define the Wernli-normalized Pontryagin transgression on the space of metrics by

$$\tau_{\mathrm W} := \frac{1}{4\pi^2} \mathrm d_{\operatorname{Met}(M)}S_{\mathrm{grav}}^{\mathrm W}.$$

The invariant local-cohomology class of the metric cocycle is then

$$\left[ \frac{\mathcal A_{\mathrm{fr}}^{(1)}}{2\pi i} \right] = -\frac{\dim\mathfrak g}{24}\,[\tau_{\mathrm W}],$$

with the sign reversed if the orientation or path-integral convention is reversed. This is the Pontryagin class in Wernli’s normalized pairing, not an asserted coefficient multiplying the site’s raw $p_1$ representative. A framing supplies the local primitive, and consequently

$$Z_1^{\mathrm{fr}} := Z_1\, \exp\!\left( \frac{i\,\dim G}{48\pi}S_{\mathrm{grav}}^{\mathrm W} \right)$$

is metric-independent after the framing is fixed. A unit change of framing multiplies the one-loop result by $\exp(2\pi i\,\dim G/24)$. These coefficients are derived in Wernli 2022, § 3.5.1, printed pp. 94–96, eqs. (3.53)–(3.56), Open PDF.

At higher orders, the same dependence appears as a local Pontryagin cocycle in the homotopy variation. With a chosen orthonormal frame, subtracting its gravitational Chern–Simons transgression gives a corrected construction that is unique up to master homotopy within Iacovino’s setup Iacovino 2010, Theorem 2 and Corollary 3, arXiv v2, printed p. 6, eq. (13), Open PDF. This is conditional on a chosen framing; it is not a claim that every perturbative construction canonically selects a two-framing.

Fix this background gravitational normalization as well as the frame, and let $Q_{\mathrm{BV}}$ denote the linearized BV differential. The remaining gauge-field-dependent one-loop lifts can then be written

$$I_1^{\mathrm{lift}}(f) = \frac{\dim G}{48\pi}S_{\mathrm{grav}}^{\mathrm W}(g,f)+J, \qquad Q_{\mathrm{BV}}J=0, \qquad J\sim J+Q_{\mathrm{BV}}K.$$

Their equivalence classes are the $H^0(\operatorname{Def}_{\mathrm{CS}})$ torsor computed above. The gravitational part is unique only up to master homotopy within the stated construction, while for simple $\mathfrak g$ the remaining gauge-field-dependent one-loop freedom is one-dimensional. This is the requested lift classification conditional on the framing trivialization; it does not turn a formal local parameter into an allowed integral level.

A nonlocal primitive does not solve the local problem

The spectral functional $\eta(L_-;g)$ is global: it depends on the spectrum of an elliptic operator on all of $M$. Its variation is local, but $\eta$ itself is not generally the integral of a finite-jet local density. Therefore an equation of unrestricted cochains such as

$$\mathrm d_{\mathrm{Met}}\eta = \text{local anomaly}$$

does not prove that the anomaly vanishes in the local obstruction complex. The admissible trivializing cochain must itself be local. Here the framed gravitational Chern–Simons transgression is the local choice; the bare $\eta$ invariant is not. Ferreiro Pérez formulates this locality test and the Chern–Simons cancellation criterion in Ferreiro Pérez 2018, § 1, printed pp. 2–3, and § 5.1, printed pp. 15–16, Open PDF.

The result of the fixture is now precise:

1. acyclicity removes residual fields and makes the Gaussian determinant nonsingular;
2. the ordinary semisimple gauge-BV obstruction class vanishes, while its lifts retain invariant-pairing freedom;
3. the separate metric anomaly is the local Pontryagin/framing cocycle; and
4. a chosen framing supplies its local transgression, after which the answer is topological but framing-dependent.

Non-acyclic saddles, reducible connections, exact nonperturbative asymptotics, and the theorem that identifies all obstruction and lift groups belong to the later mathematical treatment.

What level quantization does not determine

Passing the global level test establishes a well-defined classical phase in the stated category of manifolds and bundles. It does not determine:

• whether a dynamical path integral exists nonperturbatively;
• the modular tensor category, state spaces, or line-operator spectrum;
• the boundary condition or edge conformal field theory;
• knot invariants and their operator-framing conventions;
• matter-induced parity shifts or the total effective level; or
• whether a perturbative expansion about one flat connection captures the full partition function.

The acyclic fixture is deliberately local in field space and asymptotic in the level. Its one-loop determinant is a controlled calculation, not a proof of the exact TQFT or of convergence of the saddle expansion.

Common pitfalls

Treating the local three-form as the action. On a nontrivial bundle the potential is patchwise and the local Chern–Simons form does not glue to a number. Test the exponentiated phase against large transformations and all closed extensions, or use an intrinsic differential refinement.

Saying “integer level” without naming the lattice. An integer is obtained only after choosing a generator and trace normalization. The global datum is an integral class for the actual gauge group, with spin or torsion refinements when required.

Using spin and framing interchangeably. Spin can enlarge the classical Abelian level lattice. Framing controls a separate quantum gravitational phase; one does not substitute for the other.

Calling every failure an anomaly. A failed large-gauge test, a boundary variation, a gauge-BV obstruction, and a framing anomaly live in different problems. State the field space, cochain complex, and admissible counterterms before comparing their classes.

Check your understanding

1. Why does the $SU(N)$ filling test give integer level in the basic trace normalization?

Answer

Two fillings glue to a closed four-manifold with $Q=(8\pi^2)^{-1}\int\operatorname{tr}_F(F\wedge F)\in\mathbb Z$. Their phase ratio is $e^{2\pi i kQ}$. A unit-charge bundle makes this equal to one for all fillings exactly when $k\in\mathbb Z$. A different global group or trace can change which multiples of the basic class occur.

2. Why is odd compact-$U(1)$ level allowed on spin manifolds but not in the ordinary oriented bosonic theory?

Answer

The filling ambiguity is $e^{i\pi k\langle c_1^2,[X]\rangle}$. On a general oriented $X$, $\langle c_1^2,[X]\rangle$ can be odd, forcing even $k$. On a spin $X$ the intersection form is even, so integer $k$ suffices.

3. For $K_{N,p}=\bigl(\begin{smallmatrix}p&N\\N&0\end{smallmatrix}\bigr)$, what changes when $p$ changes and what does not?

Answer

For an ordinary bosonic theory write $p=2r$ with $r\in\mathbb Z_N$; changing $r$ changes the Dijkgraaf–Witten twist and therefore spins and braiding. The determinant is always $-N^2$, so the torus state-space dimension remains $N^2$. The conclusion assumes both compact fields are integrated with the same global normalization.

4. Does $\eta(L_-)$ trivialize the framing anomaly as a local counterterm?

Answer

No. The $\eta$ invariant is a global spectral functional. Its variation is local, but it is not generally a finite-jet local functional. After choosing a framing, gravitational Chern–Simons supplies the admissible local transgression.

5. What exactly does acyclicity buy in the perturbative fixture?

Answer

$H^0=0$ removes infinitesimal stabilizer and ghost zero modes, while $H^1=0$ removes tangent directions to a flat-connection moduli space; duality then removes the complementary cohomology. The Gaussian determinant is nonsingular. Acyclicity does not cancel the framing anomaly or prove the full saddle expansion converges.

6. Why is the perturbative $H^3_{\mathrm{Lie}}(\mathfrak g)$ lift freedom not the same as the global level lattice?

Answer

The former is a formal local deformation space over the coefficient field and records invariant-pairing counterterms. The latter is the integral global datum $H^4(BG;\mathbb Z)$ that makes the exponentiated phase well defined on all bundles and large transformations. Passing from the local vector space to the integral lattice is an additional global condition.

Continue to exact theories and applications

The Abelian Chern–Simons theory page will construct the exact TQFT, line operators, state spaces, and gluing data. Abelian K-matrix data and fractional quantum Hall fluids will apply the same lattice data to topological order and response.

The obstruction–deformation page will prove the cohomological obstruction and lift statements used in the acyclic fixture. Yang–Mills–Chern–Simons–matter actions will add propagating matter and effective-level shifts, while Chern–Simons gravity and boundary currents will specialize the construction to three-dimensional gravity. These pages are prospective continuations; the global level and one-loop tests above are self-contained.

References

• Belov, Dmitriy M., and Gregory W. Moore. “Classification of Abelian Spin Chern–Simons Theories.” arXiv:hep-th/0505235v1 [hep-th], 2005. Stable record.
• Costello, Kevin, John Francis, and Owen Gwilliam. “Chern–Simons Factorization Algebras and Knot Polynomials.” arXiv:2602.12412v1 [math.QA], 2026. Preliminary version. Stable record.
• Ferreiro Pérez, Roberto. “Locality and Universality in Gravitational Anomaly Cancellation.” arXiv:1805.12068v1 [math-ph], 2018. https://doi.org/10.48550/arXiv.1805.12068.
• Freed, Daniel S. “Classical Chern–Simons Theory, Part 1.” Advances in Mathematics 113, no. 2 (1995): 237–303. https://doi.org/10.1006/aima.1995.1039. Open PDF.
• Freed, Daniel S. “Classical Chern–Simons Theory, Part 2.” Houston Journal of Mathematics 28, no. 2 (2002): 293–310. Journal record. Open PDF.
• Gwilliam, Owen, and Brian R. Williams. “A One-Loop Exact Quantization of Chern–Simons Theory.” arXiv:1910.05230v2 [math-ph], 2020. https://doi.org/10.48550/arXiv.1910.05230.
• Iacovino, Vito. “Master Equation and Perturbative Chern–Simons Theory.” arXiv:0811.2181v2 [math.DG], 2010. https://doi.org/10.48550/arXiv.0811.2181.
• Kapustin, Anton, and Nathan Seiberg. “Coupling a QFT to a TQFT and Duality.” Journal of High Energy Physics 2014, no. 4 (2014): 001. https://doi.org/10.1007/JHEP04(2014)001. arXiv:1401.0740v2.
• Wernli, Konstantin. “Notes on Chern–Simons Perturbation Theory.” Reviews in Mathematical Physics 34, no. 3 (2022): 2230003. https://doi.org/10.1142/S0129055X22300035. Open PDF.
• Witten, Edward. “Quantum Field Theory and the Jones Polynomial.” Communications in Mathematical Physics 121, no. 3 (1989): 351–399. https://doi.org/10.1007/BF01217730. Open PDF.