The Chern-Simons Function and the Quantum Hall Effect

Summary

An expository article by Edward Witten (IAS; hep-th, v1 18 Sep 2026, v2 22 Sep 2026). It is the written version of a lecture given in 2013 at the CUNY Graduate Center conference for Jim Simons’ 75th birthday, and will appear in the Bulletin of the AMS issue in memory of Simons (1938–2024). It is written for mathematicians and has four short sections and 29 references. The article defines the Chern–Simons function as the secondary characteristic class attached to the second Chern class. Using nothing but the effective action of the electromagnetic field, it then explains why the response of a two-dimensional insulator is labeled by an integer, the quantum Hall level $k$ with ${\sigma }_{xy}=k{e}^2/h$. Finally it shows how an emergent Chern–Simons gauge field turns that integer into the rational numbers of the fractional effect.

The chain of reasoning is the part worth keeping:

1. An insulator is a material with no relevant low-energy degrees of freedom. Its electromagnetic response is therefore a local effective action $I_{\mathrm{eff}}[A]$, organized as an expansion in fields and derivatives and constrained by the microscopic symmetries.
2. In $2+1$ dimensions exactly one term in that action cannot be written in terms of $\vec{E}$ and $\vec{B}$. That term is the Chern–Simons function, which is defined only mod $\mathbb{Z}$.
3. Classically an additive ambiguity in the action is harmless. Quantum mechanically $e^{iI}$ must be well defined (Dirac’s lesson), so the coefficient must be $2\pi k$ with $k\in \mathbb{Z}$.
4. The Chern–Simons term is the only one whose variation gives a current proportional to the field rather than to its derivatives. That current is the Hall effect, with an integer coefficient.
5. For $U(1)$, defining the function mod $\mathbb{Z}$ requires a spin structure, because the intersection form of a spin 4-manifold is even. Without one only even $k$ would be allowed, so the experimental observation of odd $k$ is, in Witten’s reading, physical evidence that spacetime carries a spin structure.

For this wiki the paper matters well beyond its expository aim. It states cleanly several global facts that other threads here use but state loosely:

• the integrality of Chern–Simons levels, which underlies the group’s Maxwell–Chern–Simons/self-dual program and Project 18 (non-invertible-symmetries-mcs);
• the spin-structure dependence of θ-periodicity (axionic-electrodynamics; course Semester I Week 12);
• anomaly inflow at a boundary (t-hooft-anomaly);
• the TQFT reading of the fractional effect (topological-order; course Semester II Week 10).

Witten also flags that the emergence of the Chern–Simons gauge field “is more a theoretical interpretation of experimental facts than a theorem about the solutions of the Schrödinger equation”. That is precisely the gap the group’s Julia–Toulouse program addresses: effective gauge fields derived from defect condensation.

Key Results

§1. The Chern–Simons function. Take a $G$-bundle with connection $A$ on a closed oriented 3-manifold $W=\partial X$, for example $G=SU(N)$. Define ${\mathrm{CS}}_X(A)={\int }_X\mathrm{Tr}F\wedge F/8{\pi }^2$. Two extensions $X,X^{\prime }$ glue into a closed $Y=X\cup \overline{X^{\prime }}$, and ${\mathrm{CS}}_X-{\mathrm{CS}}_{X^{\prime }}={\mathcal{I}}_Y\in \mathbb{Z}$ is the instanton number of $Y$. Hence $\mathrm{CS}(A)\in \mathbb{R}/\mathbb{Z}$ depends only on $A{|}_W$. Physicists usually carry a factor $2\pi$, so their $\mathrm{CS}$ takes values in $\mathbb{R}/2\pi \mathbb{Z}$ (footnote 1).

§2. Insulators as effective actions. $I_{\mathrm{eff}}=I_{\mathrm{Maxwell}}+{\int }_{S}f(\vec{E},\vec{B},\partial \vec{E},\dots )$, with $I_{\mathrm{Maxwell}}=\frac{1}{4e^2}\int F\wedge \star F$. The Taylor coefficients are restricted by $O(3)$ and by time reversal $T$:

• $\vec{c}\cdot \vec{E}$: a ferroelectric.
• ${\vec{c}}^{\prime }\cdot \vec{B}$ ($T$-odd): a ferromagnet.
• $ϵ{\vec{E}}^2+\mu {\vec{B}}^2+\gamma \vec{E}\cdot \vec{B}$: a dielectric, with $\gamma =0$ if $T$ holds (footnote 4; see the θ = π caveat in axionic-electrodynamics).
• Derivative terms: dispersion. Anisotropic quadratic forms: birefringence. Cubic and higher terms: nonlinear optics.

The current is $\mathcal{J}=e^2\delta \tilde{I}/\delta A$. Any term built from $\vec{E},\vec{B}$ yields a current proportional to derivatives of the field, so the longitudinal conductivity vanishes. That is the precise sense of “insulator”.

§3. Two-dimensional materials: the integer effect.

• Spin structure is essential. Spacetime is $M=\mathbb{R}\times Y$ with $Y$ a spin 3-manifold, and the film’s worldvolume is $W=\mathbb{R}\times D$. On a spin 4-manifold ${\int }_X{c}_1^2$ is even, so the basic integer is ${\int }_{X}F\wedge F/8{\pi }^2$. For a $U(1)$ connection on a spin 3-manifold $W$ one therefore gets a well-defined $\mathrm{CS}(A)={\int }_{X}F\wedge F/8{\pi }^2\mathrm{mod}\mathbb{Z}$, and it depends on the spin structure. Example: on $T^3=T^2\times S^1$, with $A$ pulled back from a degree-one connection on $T^2$, $\mathrm{CS}(A)=0$ or $\frac{1}{2}$ according to the spin structure.
• Local formulas. $\mathrm{CS}(A)=\frac{1}{8{\pi }^2}{\int }_{W}A\wedge dA$ for a trivialized bundle, and $\delta \mathrm{CS}=\frac{1}{4{\pi }^2}\int \delta A\wedge F$. Critical points are flat connections.
• Quantization. $I_{\mathrm{eff}}\supset 2\pi k\mathrm{CS}(A)=\frac{k}{4\pi }\int AdA$ with $k\in \mathbb{Z}$. Two-dimensional insulators are thus classified by an integer.
• The Hall current. $J_y=\frac{e^{2}k}{2\pi }{E}_x\delta (z)$, i.e. ${\sigma }_{xy}=\frac{k{e}^2}{2\pi \hslash }=k\frac{e^2}{h}$ (von Klitzing 1980; measured to about $10^{-9}$). Odd $k$ is observed, which is the physical imprint of the spin structure.
• Edges. When $\partial W=\Sigma \ne \varnothing$, $\exp (2\pi ik\mathrm{CS}(A))$ is not a number. It is a unit vector in a line $\mathcal{R}$ fixed by $A{|}_{\Sigma }$ (Ramadas–Singer–Weitsman). A boundary QFT whose partition function lives in ${\mathcal{R}}^{-1}$ restores well-definedness. This is anomaly inflow (Callan–Harvey). The edge theory is not fixed by $k$; the simplest choice has $Z=\det (\text{Dirac}{)}^k$, i.e. $k$ chiral fermions.

§4. The fractional effect.

• Topological degrees of freedom. Relax “no relevant degrees of freedom” to “no relevant local degrees of freedom”, which leaves a TQFT. The material carries an emergent $U(1)$ field $a$ with action $2\pi r\mathrm{CS}(a)$. Here $r\in \mathbb{Z}$, a spin structure is needed if $r$ is odd, and a Maxwell term is irrelevant at long distances when $r\ne 0$.
• The coupling to electromagnetism. The mixed invariant $\mathrm{CS}(A,a)$, defined via ${\int }_{X}F\wedge f/4{\pi }^2$, needs no spin structure, since $c_1(\mathcal{L})c_1(\mathcal{M})$ can be odd. It enters with coefficient $2\pi s$, $s\in \mathbb{Z}$:

$$I_{\mathrm{int}}={\int }_W(\frac{k}{4\pi }AdA+\frac{s}{2\pi }Ada+\frac{r}{4\pi }ada),sF+rf=0,{\sigma }_{xy}=(k-\frac{s^2}{r})\frac{e^2}{h}.$$

• Laughlin’s state. $(k,s,r)=(0,1,-3)$ gives $\nu =1/3$ (Tsui–Störmer–Gossard 1982).
• Closing remarks.
  • Almost all known unitary 3d TQFTs are Chern–Simons theories (Moore–Seiberg).
  • Quasiparticle charge and spin, and the ground-state degeneracy on genus $g$, follow from the TQFT.
  • The $\nu =5/2$ state is believed to be non-abelian (Moore–Read), but “decisive experimental proof remains elusive”.
  • Recent zero-field fractional quantum anomalous Hall states have been observed (Park et al. 2023).
  • Non-abelian Chern–Simons theory gives a QFT framework for the Jones polynomial.

Evidence status (labels as in the physics-research methodology):

Claim	Status	Basis
Integer quantization of ${\sigma }_{xy}$	PROVED under gap assumptions	TKNN (1982) and Avron–Seiler–Simon (1983) for free fermions; Hastings–Michalakis, CMP 334 (2015) 433 for interacting gapped systems on a torus. Witten’s argument is the EFT version and assumes a local gapped effective action.
Rational ${\sigma }_{xy}$ with denominator bounded by the torus degeneracy	PROVED under assumptions	Bachmann–Bols–De Roeck–Fraas, J. Math. Phys. 62 (2021) 011901 [arXiv:2001.06458]
FQH described by an emergent abelian Chern–Simons theory	STRONGLY SUPPORTED, not derived	Experiment and numerics; the emergence is interpretation, by Witten’s own caveat
$\nu =5/2$ is non-abelian	CONJECTURED	Moore–Read; experimentally unsettled
Almost all unitary 3d TQFTs are Chern–Simons theories	EXPECTED (folklore)	Moore–Seiberg (1989)

Methods

• Effective field theory as the organizing principle. The effective action is expanded in fields and derivatives and pruned by symmetry. Material properties are read off from its Euler–Lagrange equations and currents.
• Secondary characteristic classes via bordism. A 3-manifold invariant is defined by extending to a bounding 4-manifold and showing independence mod $\mathbb{Z}$ with a gluing argument.
• Well-definedness of $e^{iI}$ (Dirac; Feynman) as the source of quantization.
• Spin geometry. Evenness of the intersection form on spin 4-manifolds (Wu’s formula) is the reason odd levels need a spin structure.
• Boundaries. Determinant lines (Ramadas–Singer–Weitsman) and anomaly inflow (Callan–Harvey).

Relevance

• condensed-matter-connections: a common origin for the line’s three topics.
  • The line studies θ-electrodynamics, Weyl superconductors and monopole operators, and this article is the cleanest statement of the logic behind all three: response = effective action, and quantization = well-definedness of $e^{iI}$.
  • Read backwards, Witten’s eq. (25), ${\mathrm{CS}}_X(A)={\int }_{X}F\wedge F/8{\pi }^2$, says a film’s Chern–Simons level is the $\theta /2\pi$ of a 4d region bounded by it. A θ-wall with $\Delta \theta$ carries level $\Delta \theta /2\pi$, so the $\theta =\pi$ surface of a topological insulator has half-integer level. That is forbidden for a standalone 2d insulator, which is why the surface must be anomalous. See axionic-electrodynamics.
• confinement-duality: the global input the Maxwell–Chern–Simons papers never needed until now.
  • The MCS/self-dual duality (Marcelo’s MSc thesis; PLB 605, 2005) and the Julia–Toulouse papers with a Chern–Simons term (PLB 674, 2009; PRD 88, 2013) were local, perturbative statements.
  • Project 18 (non-invertible-symmetries-mcs) is a question about global symmetry structure. It needs exactly Witten’s §3–4 data: $k\in \mathbb{Z}$, spin structure for odd $k$, $|k{|}^g$ ground states, and the mixed term $\mathrm{CS}(A,a)$ with integer coefficient.
  • Witten’s caveat on emergence is the opening for Project 17: a Julia–Toulouse derivation of the $(k,s,r)$ action with integrality built in.
• bell-inequalities-qft and relative-entropy-qft: the edge as a type III₁ laboratory.
  • By anomaly inflow, the edge of the $(k,s,r)$ fluid carries a chiral $U(1)$ current whose Schwinger term is the Hall conductance: $[\rho (x),\rho (y)]=\frac{i\nu }{2\pi }{\delta }^{\prime }(x-y)$ with $\nu =k-s^2/r$. The Pauli–Jordan distribution of the edge current is ${\sigma }_{xy}$.
  • The edge is a chiral boson whose local algebra contains the electron vertex operator, which is fermionic for odd $r$, while quasiparticle operators with charge $e/|r|$ generate superselection sectors (for a Laughlin state, $k=0$, $s=1$, so $\nu =1/|r|$ with $r<0$ in Witten’s orientation).
  • This is precisely the object of the group’s vertex-operator Tsirelson paper (Caribé–Guimarães–Roditi–Sorella, arXiv:2604.18513) and of the coherent-state Araki–Uhlmann computations. See Questions Raised §1.
• Courses and the immersion program.
  • The article is the natural reading for the “one honest lecture” on the fractional effect in the generalized-symmetries course (Semester II Week 10, wiki/courses/generalized-symmetries-course/). Its §3 gives the spin-structure condition for θ-periodicity, which Semester I Week 12 states in §3.3 and F5.
  • It is Track A reading for the immersion-plan, and a compact specimen for the Witten writing apprenticeship: the question stated in words before any formula (“is it possible for a multiple of $\mathrm{CS}(A)$ to appear…?”), the classical answer set against the quantum one, rigor spent only where it changes the conclusion (the spin structure).

Questions Raised

Discussed with the researcher on ingest; not yet filed as question pages.

1. Edge Bell/relative-entropy laboratory. On the edge of the $(k,s,r)$ fluid, $\nu$ normalizes the current’s Pauli–Jordan distribution.

   • (a) What is the Araki–Uhlmann relative entropy between the vacuum and a quasiparticle-sector state localized in an interval? Longo’s $U(1)$-current formula suggests it scales as $q^2/\nu$ at fixed profile: $\propto 1/|r|$ for the charge-$e/|r|$ Laughlin quasiparticle and $\propto |r|$ for the electron.
   • (b) Does the vertex-operator Tsirelson saturation of arXiv:2604.18513 survive when only the physically local vertex operators are admitted (electron-type, and fermionic for odd $r$, so graded locality applies)?
   • (c) The charge-counting operator $e^{2\pi i{Q}_I}$ acts trivially on integer charge and detects fractional charge; in the bulk it closes into a ${\mathbb{Z}}_{|r|}$ 1-form-symmetry loop. Is it a usable Bell observable?

   Links: bell-inequalities-qft, relative-entropy-qft, weyl-operators.

2. The global form of the MCS/self-dual duality. On $T^2$ the MCS holonomy zero modes form a Landau problem with cyclotron frequency equal to the topological mass $m=k{e}^2/2\pi$ and a $k$-fold degenerate lowest level, the $U(1{)}_k$ TQFT. The self-dual model has no such sector. Is the precise statement MCS $\simeq$ SD $\otimes$ $U(1{)}_k$ (a spin TQFT for odd $k$)? What does this imply for the non-invertible symmetries of Project 18, and for the noncommutative version (PLB 605, 2005) next to Susskind’s noncommutative-CS description of the Laughlin fluid? → non-invertible-symmetries-mcs

3. A Julia–Toulouse derivation of Witten’s $(k,s,r)$ action with exact integrality. Condensing composite-boson vortices (the Zhang–Hansson–Kivelson/Read picture Witten cites) should produce $s=1$ and the level $r$. Can the modified-Villain lattice make $r\in \mathbb{Z}$ and the odd-$r$ spin requirement exact? Jacobson–Sulejmanpasic build lattice $U(1{)}_k$ from a lattice θ-term, which is the lattice version of eq. (25). Is the hierarchy iterated condensation? → julia-toulouse-higher-form-symmetries

4. θ-walls, periodicity and pseudo-axion strings.

   • A θ-wall with $\Delta \theta =\pi$ has half-integer level and must be anomalous.
   • $\theta \cong \theta +2\pi$ holds only with a spin structure; bosonic systems have $4\pi$ periodicity and the statistical Witten effect.
   • For the group’s Weyl-superconductor pseudo-axion (a charge-$2e$ condensate), what are the correct periodicity and the Callan–Harvey modes on pseudo-axion strings?

   → axionic-electrodynamics, condensed-matter-connections

5. Magic along the MCS flow. MCS runs from a local QFT, whose states must carry magic (2026-benedetti-magic-in-qft), to the abelian TQFT $U(1{)}_k$, whose link states are stabilizer states (Salton–Swingle–Walter). Where does the magic go? Long-range-magic work is crowded (2025–26), so the group’s angle would be the modular/type III one. → magic-and-confinement-phases

6. Pedagogical. Footnote 5 notes that the argument can be run in Minkowski space using magnetic monopoles. A monopole crossing a level-$k$ film deposits charge $k$ (Laughlin’s flux argument). Dirac-string invisibility gives $k\in \mathbb{Z}$, and for odd $k$ the resulting dyon’s statistics is the spin-structure refinement. This would make a good course problem for Semester I Week 12 or Semester II Week 10.

Reading Notes

Five-line referee note (the immersion-plan format):

• Claim: two-dimensional insulators are classified electromagnetically by an integer Chern–Simons level, and the fractional effect is an emergent abelian Chern–Simons TQFT coupled through $(k,s,r)$.
• Method: effective action, well-definedness of $e^{iI}$, and spin bordism.
• Weakest step: the existence of a local gapped effective action for $A$ is assumed, and the emergence of $a$ is explicitly not derived.
• What would break it: gapless bulks (e.g. the composite Fermi liquid at $\nu =1/2$), or non-abelian orders beyond abelian Chern–Simons.
• Question for the author: footnote 5 says the argument can be run in Minkowski space using monopoles. Does that version also see the spin-structure refinement, through the statistics of the dyon a monopole becomes after crossing an odd-$k$ film?

Notation ledger.

• Witten’s $\mathrm{CS}\in \mathbb{R}/\mathbb{Z}$; the physics action is $2\pi k\mathrm{CS}=\frac{k}{4\pi }\int AdA$.
• Signature $-+++$ (mostly plus), whereas the playbook’s group convention is $(+,-,-,-)$.
• $A$ is normalized so that charges are integers, with $e$ in front of the Maxwell term.
• $K$-matrix dictionary. In the common convention $\mathcal{L}=-\frac{1}{4\pi }Kada+\frac{t}{2\pi }Ada$, Witten’s $r=-K$ and $s=t$, and ${\sigma }_{xy}=k+t^{\mathsf{T}}{K}^{-1}t$. The generalized-symmetries course (Semester II Week 10) writes $+\frac{1}{4\pi }Kada$, the mirror image; there $r=K$, $s=t$ and ${\sigma }_{xy}=k-t^{\mathsf{T}}{K}^{-1}t$, and Witten’s $\nu =1/3$ example $(k,s,r)=(0,1,-3)$ is $K=-3$, $t=1$.

Minor slips in v1 (18 Sep). The v2 of 22 Sep announces “minor corrections”; whether they cover this list was not checked.

• Eq. (23) integrates over $M$ where $X$ is meant.
• Eq. (17) writes $I(\Phi ,A)$ for $\tilde{I}(\Phi ,A)$.
• Eq. (11) drops the $1/e^2$ of eq. (9).
• Ref. [6] (Laughlin’s 1981 gauge argument, the flux-insertion cousin of footnote 5) is listed but not cited in the text.
• The Chern–Simons 1974 paper itself is not in the bibliography, presumably taken as read in a Simons memorial issue.

INSPIRE walk (parents).

• Belavin–Polyakov–Schwarz–Tyupkin (1975) and ‘t Hooft (1976), on instantons (the latter ★ in Zotero).
• Dirac (1931).
• von Klitzing et al. (1980); Tsui–Störmer–Gossard (1982); Laughlin (1983).
• Witten, CMP 121 (1989).
• Callan–Harvey (1985).
• Zhang–Hansson–Kivelson and Read (1989).

The paper is too new for children.

Related Papers

• E. Witten, Quantum Field Theory and the Jones Polynomial, CMP 121 (1989) 351. Non-abelian Chern–Simons theory, its Hilbert spaces on Riemann surfaces, and knot invariants (ref. [19]; with Wen, ref. [27], for the genus-$g$ degeneracy).
• E. Witten, Fermion Path Integrals and Topological Phases, RMP 88 (2016) 035001 [arXiv:1508.04715], and Three Lectures on Topological Phases of Matter, Riv. Nuovo Cim. 39 (2016) 313 [arXiv:1510.07698]. Both are ★ in Zotero. They carry the story into 3+1d: θ = π, $T$, η-invariants.
• E. Witten, K. Yonekura, Anomaly Inflow and the η-Invariant, arXiv:1909.08775. The modern inflow reference (immersion Track A).
• E. Witten, From Superconductivity and Four-Manifolds to Weak Interactions, Bull. AMS 44 (2007) 361. The companion Bull. AMS article (ref. [20]).
• C. G. Callan, J. A. Harvey, NPB 250 (1985) 427. Anomaly inflow on axion strings and domain walls, the same family as the group’s pseudo-axion work.
• S. C. Zhang, T. H. Hansson, S. Kivelson, PRL 62 (1989) 82; N. Read, PRL 62 (1989) 86. Chern–Simons–Landau–Ginzburg theory and composite bosons, the condensation picture closest to Julia–Toulouse.
• T. Jacobson, T. Sulejmanpasic, Modified Villain formulation of abelian Chern–Simons theory, PRD 107 (2023) 125017 [arXiv:2303.06160]. Lattice $U(1{)}_k$ with level quantization, 1-form anomaly and monopole charge manifest, built from a lattice θ-term. See villain-action.
• G. Salton, B. Swingle, M. Walter, Entanglement from Topology in Chern–Simons Theory, PRD 95 (2017) 105007 [arXiv:1611.01516]. $U(1{)}_k$ link states are stabilizer states.
• R. Longo, Entropy of Coherent Excitations, LMP 109 (2019) 2587. Vacuum relative entropy of coherent states on wedge algebras, including the $U(1)$-current formula: the formula to specialize to the edge.
• S. Hollands, Relative entropy for coherent states in chiral CFT, LMP (2020) [arXiv:1903.07508]. For stress-tensor coherent states the relative entropy is $c$ times a Schwarzian action; this is the $c=1$ edge’s energy sector.
• J. G. A. Caribé, M. S. Guimarães, I. Roditi, S. P. Sorella, Bosonization, vertex operators and maximal violation of the Bell-CHSH inequality in wedge regions, arXiv:2604.18513 (2026). The group’s chiral-boson Tsirelson construction, which the edge program would extend.
• 2026-benedetti-magic-in-qft. Magic separates local QFT from TQFT, relevant to the Maxwell–Chern–Simons → $U(1{)}_k$ flow.