nLab abelian Chern-Simons theory

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∞\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Quantum field theory

Topological physics

Contents

Idea

By abelian Chern-Simons theory one means Chern-Simons theory with abelian gauge group (typically the circle group or a torus-product of copies of that).

One major application of abelian Chern-Simons theory is as an effective field theory of the fractional quantum Hall effect, see below.

Definition

The Lagrangian density and the Level

For the time being we normalize connection forms according to the common convention in theoretical physics, where the gauge curvature/field strength/flux density F A≔dAF_A \coloneqq \mathrm{d}A is such that

(1)12πF A∈Ω int 2 \tfrac{1}{2\pi} F_A \;\in\; \Omega^2_{int}

is an integral form, in that it has integer periods (cf. Dirac charge quantization and ordinary differential cohomology).

The (local) Lagrangian density of abelian Chern-Simons theory is a multiple of the Chern-Simons form of the gauge field

(2)L(A)≔K4πA∧dA L(A) \;\coloneqq\; \tfrac{K}{4\pi} A \wedge \mathrm{d} A \,

for a numerical parameter KK, whose admissible values are to be characterized so that for each choice of domain Σ 3\Sigma^3 (a closed smooth 3-manifold) the action functional

S(Σ 3,A)≔K4π∫ Σ 3A∧dA S(\Sigma^3, A) \;\coloneqq\; \tfrac{K}{4\pi} \textstyle{\int_{\Sigma^3}} A \wedge \mathrm{d}A

has gauge-invariant exponential

(3)exp(iS(A))=exp(iK4π∫ Σ 3A∧dA)∈U(1). \exp\Big( \mathrm{i} S(A) \Big) \;=\; \exp\Big( \mathrm{i} \tfrac{K}{4\pi} \textstyle{\int_{\Sigma^3}} A \wedge \mathrm{d}A \Big) \;\in\; \mathrm{U}(1) \,.

A transparent way to understand this condition is to consider (in the style of the definition of Cheeger-Simons differential characters, popularized by Witten) a compact 4-manifold Σ 4\Sigma^4 with boundary Σ 3=∂Σ 4\Sigma^3 = \partial \Sigma^4 and an extension of the U(1)\mathrm{U}(1)-connection to Σ 4\Sigma^4, where we will denote its curvature/field strength/flux density still by FF.

Then Stokes' theorem gives

exp(iS(A))=exp(iK4π∫ Σ 4F∧F)=exp(Kπi∫ Σ 4F2π∧F2π), \exp\Big( \mathrm{i} S(A) \Big) \;=\; \exp\Big( \mathrm{i} \tfrac{K}{4\pi} \textstyle{\int_{\Sigma^4}} F \wedge F \Big) \;=\; \exp\Big( K \pi \mathrm{i} \textstyle{\int_{\Sigma^4}} \frac{F}{2\pi} \wedge \frac{F}{2\pi} \Big) \,,

which reveals that the expression is well-defined — in that it takes the same value for any other choice of Σ 4\Sigma^4 — if for closed Σ^ 4\widehat{\Sigma}^4 (arising as the gluing of one choice of Σ 4\Sigma^4 with the orientation reversal of any other along their common boundary Σ 3\Sigma^3) we have

12πiKπi∫ Σ^ 4F2π∧F2π⏟c 1 2(Σ 4)∈ℤ. \tfrac{1}{2\pi \mathrm{i}} K \pi \mathrm{i} \underset{ c_1^2(\Sigma^4) }{ \underbrace{ \textstyle{\int_{\widehat{\Sigma}^4}} \frac{F}{2\pi} \wedge \frac{F}{2\pi} } } \;\in\; \mathbb{Z} \,.

If there is no further condition on Σ 3\Sigma^3 and hence on Σ^ 4\widehat{\Sigma}^4, then by (1), c 1 2(Σ 4)∈ℤc_1^2(\Sigma^4) \in \mathbb{Z} and the condition is that

(4)k≔K2∈ℤ k \;\coloneqq\; \tfrac{K}{2} \;\in\; \mathbb{Z}

must be an integer.

This kk is the Chern-Simons level. On the other hand, K=2kK = 2k is the (even, for now) integer that gets promoted to the eponymous matrix (K ij)(K_{i j}) in “K-matrix Chern-Simons theory” ,where several gauge fields couple to each other (see below). But beware that conventions differ and that our “KK” is often denoted “kk” in the literature.

It is also KK that is more fundamental as one restricts attention to spin manifolds Σ 3\Sigma^3, hence to spin Chern-Simons theory (Dijkgraaf & Witten 1990 §5):

from the observation that the cobordism ring for spin manifolds equipped with complex line bundles is trivial in degree 3, and that

Σ^ 4spin manifold⇒∫ Σ^ 4F2π∧F2π∈2ℤ, \widehat{\Sigma}^4 \; \text{spin manifold} \;\; \Rightarrow \;\; \int_{\widehat{\Sigma}^4} \frac{F}{2\pi} \wedge \frac{F}{2\pi} \;\in\; 2\mathbb{Z} \,,

it follows that for spin Chern-Simons theory one may allow also half-integral level kk (4) and hence odd integral KK:

k∈12ℤequivalentlyK≔2k∈ℤ. k \;\in\; \tfrac{1}{2}\mathbb{Z} \;\;\;\;\;\; \text{equivalently} \;\;\;\;\;\; K \coloneqq 2k \;\in\; \mathbb{Z} \,.

Finally, just to note how the resulting expression for the exponentiated action functional (3) changes when one adopts instead of (1) the normalization condition for FF that is more common in mathematics, where it is FF itself that is integral:

physics normalizationmath normalization
integral flux density12πF\tfrac{1}{2\pi} FFF
exponentiated actione iS(A)=exp(iK4π∫ Σ 3A∧dA)e^{\mathrm{i}S(A)} = \exp\Big( \mathrm{i} \tfrac{K}{4\pi} \textstyle{\int_{\Sigma^3}} A \wedge \mathrm{d}A \Big)e iS(A)=exp(2πi∫ Σ 3K2A∧dA)e^{\mathrm{i}S(A)} = \exp\Big( 2\pi\mathrm{i} \textstyle{\int_{\Sigma^3}} \tfrac{K}{2}\, A \wedge \mathrm{d}A \Big)

Some of the following sections stick to one or the other of these conventions.

The equation of motion

The Euler-Lagrange equation of motion associated with the abelian Chern-Simons Lagrangian (2) is immediately found to be

F=0, F = 0 \,,

saying that the on-shell gauge fields are precisely the flat connections with vanishing flux density F=dAF = \mathrm{d}A.

Properties

Space of quantum states

For abelian Chern-Simons theory with NN gauge fields (a (i)) i=1 N(a^{(i)})_{i = 1}^N and Lagrangian density of the form (using Einstein summation convention) K ij12a (i)∧da (j)K_{i j} \tfrac{1}{2} a^{(i)} \wedge \mathrm{d} a^{(j)} (24) , for KK an N×NN \times N even integer symmetric matrix (the diagonal entries are even numbers), the dimension of the Hilbert space of quantum states ℋ g\mathscr{H}_g (obtained by geometric quantization, cf. quantization of D=3 Chern-Simons theory) over an orientable surface of genus gg is the absolute value of the determinant of KK raised to the ggth power:

dim(ℋ g)=|det(K)| g. dim(\mathscr{H}_g) \;=\; \left\vert det(K)\right\vert^g \,.

(for g=1g = 1 this is Wesolowski, Hosotani & Ho 1994 (3.25), for N=1N=1 see Manoliu 1998a p 40, for general NN this goes back to Wen & Zee 1992 (1.2), recalled in Fradkin 2013 (14.23), Belov & Moore 2005 p 26; see also references at theta function).

For the modular group-action on these state spaces see at integer Heisenberg group the section Modular automorphisms (there for g=1g=1).

For a non-orientable surface with kk crosscaps, it is

dim(ℋ k)=|det(K)| k−1. dim(\mathscr{H}_k) \;=\; \left\vert det(K)\right\vert^{k-1} \,.

(e.g. arXiv:1509.03920 (73))


Wilson loop quantum observables

It is physics folklore that the Wilson loop quantum observables of abelian Chern-Simons theory on the 3-sphere evaluate to the exponentiated linking number plus framing number of the given (oriented, framed) link.

The traditional reasoning via “path integral” arguments, followed by a point-splitting renormalization choice via link framing, is as follows (going back to Polyakov 1988, enshrined since Witten 1989 §2.1, reviewed in GMR 1994 §1).

Wilson loops

The gauge field of abelian U ( 1 ) \mathrm{U}(1) Chern-Simons theory is a principal U(1)-bundle with principal connection ∇\nabla.

However, on Σ 3=S 3\Sigma^3 = S^3 the 3-sphere, any such U(1)\mathrm{U}(1)-connection has underlying trivial bundle (since the homotopy classes of maps from S 3S^3 to the classifying space BU(1)B \mathrm{U}(1), being an Eilenberg-MacLane space K(ℤ,2)K(\mathbb{Z},2), are all trivial: π 0Map(S 3,BU(1))≃π 3BU(1)≃1\pi_0 Map\big(S^3, B \mathrm{U}(1)\big) \simeq \pi_3 B \mathrm{U}(1) \simeq 1) and hence is represented by a globally defined differential 1-form

(5)A∈Ω dR 1(S 3). A \in \Omega^1_{dR}\big(S^3\big) \mathrlap{\,.}

Given an oriented link

with nn connected components

its Wilson loop/holonomy with respect to such a connection (5) is the exponentiated integral of AA along the link:

(8)exp(i∫(S 1) nγ *A)∈U(1). \exp\bigg( \mathrm{i} \textstyle{\underset{{(S^1)^n}}{\int}} \gamma^\ast A \bigg) \;\in\; \mathrm{U}(1) \,.

The expectation value ⟨−⟩\langle-\rangle of the γ\gamma-Wilson loop (8), regarded as a quantum observable in abelian Chern-Simons theory,

⟨exp(i∫(S 1) nγ *A)⟩∈ℂ, \Bigg\langle \exp\bigg( \mathrm{i} \textstyle{\underset{{(S^1)^n}}{\int}} \gamma^\ast A \bigg) \Bigg\rangle \;\in\; \mathbb{C} \mathrlap{\,,}

is traditionally meant to be the normalized “path integral ∫DA\int D A” over the space of gauge orbits of AA (5) of the expression (8) weighted by the exponentiated action functional (3) of abelian Chern-Simons theory, suggestively written as:

(9)⟨exp(i∫(S 1) nγ *A)⟩=∫DAexp(iK4π∫S 3A∧dA+i∫(S 1) nγ *A)∫DAexp(iK4π∫S 3A∧dA) " ". \Bigg\langle \exp\bigg( \mathrm{i} \textstyle{\underset{{(S^1)^n}}{\int}} \gamma^\ast A \bigg) \Bigg\rangle \;\; = \;\; \multiscripts{^{\text{"}}}{ \frac{ \displaystyle{\int D A} \; \exp\bigg( \mathrm{i} \tfrac{K}{4\pi} \textstyle{\underset{S^3}{\int}} A \wedge \mathrm{d}A \;+\; \mathrm{i} \textstyle{\underset{{(S^1)^n}}{\int}} \gamma^\ast A \bigg) }{ \displaystyle{\int D A} \; \exp\bigg( \mathrm{i} \tfrac{K}{4\pi} \textstyle{\underset{S^3}{\int}} A \wedge \mathrm{d}A \bigg) } }{^{\text{"}}} \mathrlap{\,.}

As usual for path integrals, the expression on the right of (9) remains undefined, since a suitable measure “DAD A” is not known and may not exist. As also usual for path integrals, one proceeds instead by postulating that the expression on the left of (9) satisfies enough structural properties that are suggested by the symbols on the right as to be uniquely fixed by these conditions — and then take that to be the definition.

We proceed below to discuss the usual such postulates, which proceed by applying expected properties of Gaussian integrals. A slightly different set of postulates, but more explicit than usual, is considered by Guadagnini & Thuillier 2008 §4.3.

The Gaussian path integral

Since the abelian Chern-Simons Lagrangian density K4πA∧dA\tfrac{K}{4\pi} A \wedge \mathrm{d}A (2) is a quadratic form in AA, one envisions that this expression (9) follows the transformation rules of Gaussian integrals over elements ϕ∈ℝ n\phi \in \mathbb{R}^n of finite-dimensional Cartesian spaces, for which

Z(J)≔∫d nϕexp(−12ϕ⋅M⋅ϕ+J⋅ϕ) Z(J) \;\coloneqq\; \int \mathrm{d}^n \phi\; \exp\Big( - \tfrac{1}{2} \phi \cdot M \cdot \phi + J \cdot \phi \Big)

evaluates to

(10)Z(J)=Z(0)exp(12J⋅M −1⋅J). Z(J) \;=\; Z(0) \, \exp\Big( \tfrac{1}{2} \, J \cdot M^{-1} \cdot J \Big) \,.

Systematically following through this analogy (using Fourier transform to turn the derivative in AdAA \mathrm{d} A into a multiplication operation) with some care (such as with attention to gauge fixing) suggests the expression:

⟨exp(i∫(S 1) nγ *A)⟩=exp(−12∫γ∫γ⟨A μ(x),A ν(y)⟩dx μdy ν) " ", \Bigg\langle \exp\bigg( \mathrm{i} \textstyle{\underset{{(S^1)^n}}{\int}} \gamma^\ast A \bigg) \Bigg\rangle \;=\; \multiscripts{^{\text{"}}}{ \exp\bigg( -\tfrac{1}{2} \textstyle{\underset{\gamma}{\int}} \textstyle{\underset{\gamma}{\int}} \big\langle A_\mu(x) , A_\nu(y) \big\rangle \, \mathrm{d} x^\mu\, \mathrm{d} y^\nu \bigg) }{^{\text{"}}} \,,

whereby the Chern-Simons propagator – namely the analog of M −1M^{-1} in (10) – is suggested to be

(11)⟨A μ(x),A ν(y)⟩=i2Kϵ μνρx ρ−y ρ|x−y| 3 " ". \big\langle A_\mu(x) , A_\nu(y) \big\rangle \;=\; \multiscripts{^{\text{"}}}{ \frac{\mathrm{i}}{2K} \epsilon_{\mu \nu \rho} \frac{ x^\rho - y^\rho }{ \left\vert x - y \right\vert^3 } }{^{\text{"}}} \,.

In summary, this suggests that

(12)⟨exp(i∫(S 1) nγ *A)⟩=exp(−i4K∫γ∫γdx μdy νϵ μνρx ρ−y ρ|x−y| 3) " ". \Bigg\langle \exp\bigg( \mathrm{i} \textstyle{\underset{{(S^1)^n}}{\int}} \gamma^\ast A \bigg) \Bigg\rangle \;=\; \multiscripts{^{\text{"}}}{ \exp\bigg( -\tfrac{\mathrm{i}}{4K} \textstyle{\underset{\gamma}{\int}} \textstyle{\underset{\gamma}{\int}} \, \mathrm{d} x^\mu\, \mathrm{d} y^\nu \epsilon_{\mu \nu \rho} \frac{ x^\rho - y^\rho }{ \left\vert x - y \right\vert^3 } \bigg) }{^{\text{"}}} \,.

This is the proposed expression due to Polyakov 1988 (3), popularized by Witten 1989 (2.29). It is almost the time-honored integral expression for the Gauss linking number of the link γ\gamma, except for those contributions where xx and yy run over the same connected component (7), compare the final expression (14) below.

Namely, (12) is of course still not well-defined, since the propagator (11) is ill-defined whenever x=yx = y. In a final step of the traditional argument one applies a renormalization argument to deal with the “divergencies” of the exponent in (12) when x=yx = y.

The renormalization choice

Polyakov 1988 (5) considered one regularization of (11). Another one was by Leukert & Schäfer 1996 §7 (there on a backdrop of a rigorous construction of the path integral measure, which however still leaves the Wilson loop observable ill-defined, see also Hahn 2004a 2004b).

But Witten 1989 p. 363 asserted that it is “clear” that the following “point-splitting regularization” should be used instead, and this has become the commonly accepted choice since (cf. Kaul 1999 (9), Hahn 2004a §9, Hahn 2004b §2.5 & §5 Guadagnini & Thuillier 2008 §3, Mezei, Pufu & Wang 2017 (5.1)):

Choose a framing of the link γ\gamma (6), hence a unit vector field nn in ℝ 3\mathbb{R}^3 defined on γ\gamma, which is normal to the tangent vectors γ˙\dot \gamma to γ\gamma. With that, finally define (12) by replacing all occurences of yy with a shifted version y+sny + s n, for arbitrarily small lengths s∈ℝ +s \in \mathbb{R}_+ of the normal/framing vector sns n:

(13)⟨exp(i∫(S 1) nγ *A)⟩≔lims→0exp(−i4K∬(S 1) ndxdyϵ μνρx ρ−(y ρ+pn ρ)|x−(y+sn)| 3), \Bigg\langle \exp\bigg( \mathrm{i} \textstyle{\underset{{(S^1)^n}}{\int}} \gamma^\ast A \bigg) \Bigg\rangle \;\coloneqq\; \underset{ s \to 0 }{\lim} \exp\bigg( -\tfrac{\mathrm{i}}{4K} \textstyle{\underset{(S^1)^n}{\iint}} \, \mathrm{d} x \, \mathrm{d} y \, \epsilon_{\mu \nu \rho} \frac{ x^\rho - \big(y^\rho + p n^\rho\big) }{ \left\vert x - \big( y + s n \big) \right\vert^3 } \bigg) \,,

where the limit s→0s \to 0 is over normal vector fields that keep the given direction but shrink in length (cf. GMR 1994 (1.5)).

This expression (13) is finally well-defined.

The knot-theoretic expression

The shift along the framing vector in (13) does not change that part of the integral, with respect to the naive (12), where σ\sigma and σ′\sigma' run over distinct connected components (7) of the link, hence that part now does give the sum of the linking numbers among the connected components of the the oriented link. At the same time, the shift makes the contributions where σ\sigma and σ′\sigma' do run over the same connected component become the linking number of that component with its shift by the framing, which is its self-linking or framing number.

The end result of this argument is hence, in the exponent, the sum of the framing numbers frm(γ i)∈ℤfrm(\gamma_i) \in \mathbb{Z} of each link component γ i\gamma_i (7) with the linking numbers lnk(γ i,γ j)∈ℤlnk(\gamma_i, \gamma_j) \in \mathbb{Z} all pairs of distinct link components, which is its writhe wrth(γ)∈ℤwrth(\gamma) \in \mathbb{Z} of the link:

(14)⟨exp(i∫(S 1) nγ *A)⟩=exp(πiK(∑ ifrm(γ i)+∑ i≠jlnk(γ i,γ j)⏟wrth(γ))). \Bigg\langle \exp\bigg( \mathrm{i} \textstyle{\underset{{(S^1)^n}}{\int}} \gamma^\ast A \bigg) \Bigg\rangle \;=\; \exp\bigg( \tfrac{\pi \mathrm{i}}{K} \Big( \underset{ wrth(\gamma) }{ \underbrace{ \textstyle{\sum_{i}} frm(\gamma_i) + \textstyle{\sum_{i \neq j}} lnk(\gamma_i, \gamma_j) } } \Big) \bigg) \mathrlap{\,.}

(Witten 1989 (2.31) ff, cf. Kaul 1999 (9), Guadagnini & Thuillier 2008 (5.2), Mezei, Pufu & Wang 2017 (5.1) 1)

Note though that none of the intermediate steps towards (13) was well-defined, so that this traditional argument — impactful as it historically was and “correct” as it may be in its conclusion (14) — is somewhat unsatisfactory as a derivation of quantum observables from input data.


Abelian Chern-Simons as effective QFT for FQH systems

The following is a streamlined digest of the traditional argument and Ansatz [originally culminating in Zee 1995, Wen 1995, more recently highlighted by Witten 2016 pp 30, Tong 2016 §5] for abelian Chern-Simons theory at level k∈ℕk \in \mathbb{N} as an effective field theory for fractional quantum Hall systems at filling fraction ν=1/2k\nu = 1/2k.


Preliminaries

Consider a spacetime Σ\Sigma of dimension 1+21 + 2, to be thought of as the worldvolume of the conducting sheet that hosts the fractional quantum Hall system.

On this spacetime, the electric current density is a differential 2-form JJ.

Note that the corresponding electric current vector field J→\vec J is characterized by its contraction into the given volume form dvoldvol being equal to the density 2-form

J=dvol(J→,⋯), J \;=\; dvol\big(\, \vec J, \cdots \big) \,,

and that the conservation law for J→\vec J, hence the vanishing of its divergence is equivalent to the density 2-form being closed differential form:

(15)divJ→=0⇔dJ=0. div \vec J \;=\; 0 \;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\; \mathrm{d} J \;=\; 0 \,.

We write AA for the vector potential 1-form on Σ\Sigma which encodes the background electromagnetic field, and F=dAF \,=\, \mathrm{d}A for its field strength, the Faraday tensor. Since we regard this as a fixed background field, we do not (need to) consider the Maxwell Lagrangian density L YM≔F∧⋆FL_{YM} \,\coloneqq\, F \wedge \star F.

But we do (need to) consider the interaction term between the electromagnetic field and the electric current density, which is

(16)L int≔A∧J. L_{int} \,\coloneqq\, A \wedge J \,.

Here we assume that AA is globally defined, in fact we shall assume that Σ\Sigma has vanishing de Rham cohomology in degree=2,

(17)H dR 2(Σ)=0. H^2_{dR}(\Sigma) \;=\; 0 \,.

This means we are focused on the local nature of fields, not on their global properties, as (for better or worse) usual for Lagrangian field theory.


The Ansatz

The auxiliary gauge potential. With the assumption (17) also the conserved current density (15) admits a coboundary, a differential 1-form to be denoted aa:

(18)J=da. J \,=\, \mathrm{d}a \,.

The central Ansatz of the approach is to think of this 1-form as an effective dynamical gauge field for the FQH dynamics.


The Lagrangian density. This in turn suggests that the effective Lagrangian density should be the sum of

  1. the interaction term (16) already mentioned, which thus specializes to

    L int=A∧J=A∧da, L_{int} \;=\; A \wedge J \;=\; A \wedge \mathrm{d}a \,,

  2. a further interaction term for aa, now regarded as a gauge potential, to its own current density jj, to be thought of as the current of FQH quasi-particles (or quasi-holes)

    (19)L quas=a∧j L_{quas} \;=\; a \wedge j

  3. a kinetic term for aa itself, where the only sensible choice is the Chern-Simons form at level k=K/2∈ℤ/2k = K/2 \in \mathbb{Z}/2

    L CS=K12a∧da. L_{CS} \;=\; K \tfrac{1}{2} a \wedge \mathrm{d}a \,.

In total, the traditional Ansatz for the effective Lagrangian density for “single layer” FQH systems at filling fraction 1/K1/K is hence:

(20)L≔K12a∧da−A∧da⏟J−a∧j. L \,\coloneqq\, K \tfrac{1}{2}\, a\wedge \mathrm{d}a \,-\, A \wedge \underset{J}{ \underbrace{ \mathrm{d}a } } \,-\, a \wedge j \,.

[Wen 1995 (2.11), reviewed in Wen 2007 (7.3.10)]


The equation of motion. The Euler-Lagrange equation of motion for the effective gauge field aa induced by (20) is simply

(21)δLδA=0⇔J=1k(F+j). \frac{\delta L}{\delta A} \;=\; 0 \;\;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\;\; J \,=\, \tfrac{1}{k}\Big( F + j \Big) \,.

(This means in particular that if, on topologically non-trivial Σ\Sigma, one were to insist (which is not a logical necessity) to subject the effective gauge field aa to Dirac charge quantization (as AA is), then the ordinary electromagnetic field would have to be assumed to carry magnetic charge being a multiple of kk.)


Reproducing FQH phenomena

Recovering the Hall conductivity. The first plausibility check on the effective model is to observe that its equation of motion (21) implies the correct Hall conductivity relation for a fractional quantum Hall system at filling fraction 1/k1/k:

Namely assuming that the electric field is oriented in the yy-direction, so that the Faraday tensor is of the form (cf there):

F=E ydt∧dy+Bdx∧dy F \;=\; E_y \mathrm{d}t \wedge \mathrm{d}y + B \mathrm{d}x\wedge \mathrm{d}y

and assuming that the quasi-particles are stationary, so that

j∝dvol(∂ 0)∝dx∧dy, j \;\propto\; dvol(\partial_0) \;\propto\; \mathrm{d}x \wedge \mathrm{d}y \mathrlap{\,,}

the temporal component of the equation of motion (21) becomes

J x=1kE y J_x \;=\; \tfrac{1}{k} E_y

which is the correct form of the Hall field E yE_y for given longitudinal current J XJ_X at filling fraction ν=1/k\nu = 1/k – see there.

(Following Zee 1995 (4.5), later authors [Witten 2016 §2.3, Tong 2016 p 161] deduce this in a more roundabout way, by first inserting the equation of motion back into the Lagrangian density and then varying that with respect to AA. It seems that this unnecessary step is brought about by first tacitly renaming “JJ” to “ff” and then forgetting that this was just a renaming.)


Recovering the fractional quasi-particles. Second, the spatial component of the equation of motion (21) says that (1.) the magnetic flux quanta and (2.) the quasi-particles contribute their kkth fraction to the total charge density:

J 0=1K(B+j 0). J_0 \;=\; \tfrac{1}{K}(B + j_0) \,.

Here

  • statement (1) reflects exactly the composite fermion model, where at kkth filling fraction each electron is bound to kk quanta of magnetic flux

  • statement (2) is the desired property of quasi-particles to have kkth fractional charge.


This largely concludes the traditional justification for the effective abelian Chern-Simons theory (20).


Edge modes.

In correspondence of how the effective field theory for fractional quantum Hall systems in the bulk is abelian Chern-Simons theory, so the corresponding effective field theory for the FQH edge modes is Floreanini-Jackiw theory (Wen 1992 §2.5, 1995 §3.3).


Hierarchical K-matrix formalism

There is “hierarchy model” of FQH systems at non-unit filling fractions (due to Haldane 1983 and Halperin 1984, whence “HH-hierachy”), where quasiparticles at unit filling fractions are imagined to themselves exhibit a secondary FQH effect with secondary quasi-particles, and so on. (NB.: this picture faces various experimental challenges, cf. Jain 2007 §12.1, Jain 2014, esp. p 8 and Rem. below. It does however seem to be better fit to describe genuine multi-component systems.)

In elaboration of this hierarchica picture to effective field theory consider abelian Chern Simons theory of multiple gauge fields — U(1) nU(1)^n-Chern-Simons theory — that are hierarchically coupled to each other. Since the coupling matrix here is traditionally denoted “KK” (since it generalizes the single Chern-Simons level “kk” for U(1)U(1)-Chern-Simons theory), this has come to be called the “K-matrix formalism” (Blok & Wen 1990, Wen & Zee 1992, early review is in Wen 1995 §2.1, a textbook account is in Wen 2007 §7.3.3).

The basic idea is to observe that the above step from

(i) introducing a gauge potential (18) for the electron current density JJ

to

(ii) the effective Chern-Simons action (20) with quasi-particle current density jj

may be repeated, by next introducing also a gauge potential for the quasi-particle current (cf. Wen 2007 (7.3.13)), and so on, iteratively.

Concretely, postulating that the quasi-particle current jj (19) is itself the field strength of a secondary auxiliary gauge field a (2)a^{(2)}

j=da (2) j \,=\, \mathrm{d} a^{(2)}

(and renaming the previous auxiliary gauge field to a (1)≡aa^{(1)} \equiv a)

and then adjoining for that secondary gauge field another Chern-Simons kinetic term, turns the effective Lagrangian density (20) into

(22)L 1,2≔K 112a (1)∧da (1)−A∧da (1)⏟J−a (1)∧da (2)⏟j+K 212a (2)∧da (2). L_{1,2} \,\coloneqq\, K_1 \tfrac{1}{2}\, a^{(1)} \wedge \mathrm{d} a^{(1)} \,-\, A \wedge \underset{J}{ \underbrace{ \mathrm{d} a^{(1)} } } \,-\, a^{(1)} \wedge \underset{j}{ \underbrace{ \mathrm{d} a^{(2)} } } \,+\, K_2 \tfrac{1}{2}\, a^{(2)} \wedge \mathrm{d} a^{(2)} \,.

(cf. Wen 1995 (2.15), Wen 2007 (7.3.13))

meant to be an effective theory at filling fraction

(23)ν=1K 1−1K 2. \nu \;=\; \tfrac { 1 } { K_1 - \tfrac{1}{K_2} } \,.

(cf. Wen 1995 (2.18), Wen 2007 (7.3.15))

Now this secondary effective Lagrangian (22) may be rewritten equivalently as (using Einstein summation convention)

(24)L 1,2=K ij12a (i)∧da (j)−Q iA∧da (i) L_{1,2} \;=\; K_{i j} \tfrac{1}{2} a^{(i)} \wedge \mathrm{d} a^{(j)} - Q_i \, A \wedge \mathrm{d} a^{(i)}

with the K-matrix

K≔[K 1 −1 −1 K 2] K \;\coloneqq\; \left[ \begin{matrix} K_1 & -1 \\ -1 & K_2 \end{matrix} \right]

and the charge vector

Q≔[1 0]. Q \;\coloneqq\; \left[ \begin{matrix} 1 \\ 0 \end{matrix} \right] \,.

(cf. Wen 1995 (2.19), Wen 2007 (7.3.15)).

Observing that the inverse K-matrix is

(25)K −1=1K 1K 2−1[K 2 1 1 K 1] K^{-1} \,=\, \tfrac{1}{K_1 K_2 - 1} \left[ \begin{matrix} K_2 & 1 \\ 1 & K_1 \end{matrix} \right]

the second-layer filling fraction (23) is recognized as the top left entry of (25) and hence re-expressed in terms of these matrix quantities as

ν=Q t⋅K −1⋅Q. \nu \,=\, Q^t \cdot K^{-1} \cdot Q \,.

(cf. Wen 2007 (7.3.19))

Remark

The phenomenological (experimental) validity already of the HH-hierarchy underlying this K-matrix formulation, when applied to ordinary single-component FQH systems, has been called into question (Jain 2007 §12.1 and Jain 2014, esp. p 8): The required assumptions on quasi-particle densities necessary for the hierarchy to be physically plausible seem to be drastically violated, and in any case the experimentally observed filling fractions do not reflect the predicted hierarchy.

On the other hand, the K-matrix formalism may be the natural description of multi-component FQH systems, where each column/row of the K-matrix corresponds to one “component” of possible electron modes (e.g. different spin polarizations or different layers in multi-layer materials), cf. Barkeshli & Wen 2017, Sodemann et al 2017, Zeng 2021 Hu, Liu & Zhu 2023, Zeng 2024.


References

General

General discussion of abelian Chern-Simons theory:

Many general reviews of Chern-Simons theory have a section focused on the abelian case, for instance:

  • Gregory Moore, §2 in: Introduction to Chern-Simons Theories, TASI lecture notes (2019) [pdf, pdf]

  • David Grabovsky, §1 in: Chern–Simons Theory in a Knotshell (2022) [pdf, pdf]

On the light-cone quantization of abelian Chern-Simons theory:

Via the Reshetikhin-Turaev construction:

On abelian Chern-Simons theory via (Riemann's) theta functions and the representation theory of the integer Heisenberg group:

On the chiral abelian WZW model (edge modes) at the boundary of abelian Chern-Simons theory:

On boundary conditions and line-defects in abelian Chern-Simons theory and on the corresponding ground state degeneracy (topological order) on surfaces with (“gapped”) boundaries:

and with fermionic boundary 2d CFT:

  • Kohki Kawabata, Tatsuma Nishioka, Takuya Okuda, Shinichiro Yahagi: Fermionic CFTs from topological boundaries in abelian Chern-Simons theories [arXiv:2502.08084]

Discussion via locally covariant algebraic quantum field theory:

On lattice formulation of abelian Chern-Simons theory:

and its canonical quantization:

In relation to the dilogarithm:

On unoriented manifolds:

  • Ippo Orii: On dimensions of (2+1)D(2+1)D abelian bosonic topological systems on unoriented manifolds, Prog. Theor. Exp. Phys. 2025 (2025) 053B017 [arXiv:2502.13532, doi:10.1093/ptep/ptaf056]

  • Ippo Orii: Generalization of anomaly formula for time reversal symmetry in (2+1)(2+1)D abelian bosonic TQFTs, Prog. Theor. Exp. Phys. 2025 123B02 (2025) [arXiv:2508.04990, doi:10.1093/ptep/ptaf155]

  • Ippo Orii: Vanishing of the H^3 obstruction for time-reversal symmetry in (2+1)D abelian bosonic TQFTs [arXiv:2509.07368]

Coupled to a Higgs field:

  • Yoonbai Kim, O-Kab Kwon, Hanwool Song, Chanju Kim: Inhomogeneous Abelian Chern-Simons Higgs Model with New Inhomogeneous BPS Vacuum and Solitons [arXiv:2409.11978]

  • Vaibhav Wasnik: Correlator-Level Verification of Mass and Current Maps in Abelian Chern-Simons Dualities [arXiv:2602.20604]

Wilson loop observables

Discussion of Wilson loop quantum observables in abelian Chern-Simons theory, via path integral arguments followed by renormalization via framing:

Review:

Alternative discussion via proper flux quantization:

Abelian Chern-Simons for fractional quantum Hall effect

The idea of abelian Chern-Simons theory as an effective field theory exhibiting the fractional quantum Hall effect (abelian topological order) goes back to

and was made more explicit in:

Beware that many of these early articles actually consider Maxwell-Chern-Simons theory, which however was later claimed not to actually exhibit the claimed “anyon statistics”:

Early review:

Further review and exposition:

For discussion of the fractional quantum Hall effect via abelian but noncommutative (matrix model-)Chern-Simons theory

More on edge modes via the abelian WZW model/Floreanini-Jackiw theory chiral boson on the boundary of abelian Chern-Simons theory:

and in view of the bulk-edge correspondence:

The symmetry protected situation:

Amplification of the K-matrix formalism as being about (not the usual single-component but) multi-component FQH systems:

Further developments:

Abelian Maxwell-Chern-Simons theory for superconductivity

On 1+2D Maxwell-Chern-Simons theory as an effective description of superconductivity:

(…)

For baryons in N f=1N_f=1 QCD

On baryons in N f = 1 N_f = 1 QCD as quantum Hall droplets effectively described by abelian Chern-Simons theory:

  1. Beware that Witten 1989 drops a factor of 2 in passing from (1.3) to (2.27) there. This amounts to shifting the Chern-Simons level “kk” — our KK (2) — by a factor of 2. His end result for the Wilson loop, equation (2.31) there, is expressed with respect to this shifted level. Transforming back to the original and usual normalization in (1.3) there cancels the factor of 2 in (2.31) there, which thereby agrees with our (14) (under k there=K herek_{there} = K_{here}). Note also that Kaul 1999 follows along with a lost factor of 2 in (1) there and hence gives in (9) there Witten’s formula, while Mezei, Pufu & Wang 2017 (5.1) appear to have noticed the glitch. Unfortunately, this is all the more confusing as there is generally an ambiguity of a factor of 2 in what one may want to mean by the abelian Chern-Simons level in the first place, as discussed above. ↩

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