nLab abelian Chern-Simons theory

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 these).

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

Properties

Space of quantum states

For abelian Chern-Simons theory with $N$ gauge fields $(a^{(i)})_{i = 1}^N$ and Lagrangian density of the form (using Einstein summation convention) $K_{i j} a^{(i)} \wedge \mathrm{d} a^{(j)}$ (10) , for $K$ an $N \times N$ even integer symmetric matrix (the diagonal entries are even numbers), the dimension of the Hilbert space of quantum states $\mathscr{H}_g$ (obtained by geometric quantization, cf. quantization of D=3 Chern-Simons theory) over an orientable surface of genus $g$ is the absolute value of the determinant of $K$ raised to the $g$th power:

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

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

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

For a non-orientable surface with $k$ crosscaps, it is

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

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

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 $k \in \mathbb{N}$ as an effective field theory for fractional quantum Hall systems at filling fraction $\nu = 1/k$.

Preliminaries

Consider a spacetime $\Sigma$ of dimension $1 + 2$, 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 $J$.

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

$$J \;=\; dvol\big(\, \vec J, \cdots \big) \,,$$

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

(1)$$div \vec J \;=\; 0 \;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\; \mathrm{d} J \;=\; 0 \,.$$

We write $A$ for the vector potential 1-form on $\Sigma$ which encodes the background electromagnetic field, and $F \,=\, \mathrm{d}A$ 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 $L_{YM} \,\coloneqq\, F \wedge \star F$.

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

(2)$$L_{int} \,\coloneqq\, A \wedge J \,.$$

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

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

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 (3) also the conserved current density (1) admits a coboundary, a differential 1-form to be denoted $a$:

(4)$$J \,=\, \mathrm{d}a \,.$$

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 (2) already mentioned, which thus specializes to

   $$L_{int} \;=\; A \wedge J \;=\; A \wedge \mathrm{d}a \,,$$

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

   (5)$$L_{quas} \;=\; a \wedge j$$

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

   $$L_{CS} \;=\; K \tfrac{1}{2} a \wedge \mathrm{d}a \,.$$

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

(6)$$L \,\coloneqq\, K \tfrac{1}{2}\, a\wedge \mathrm{d}a \,-\, A \wedge \underset{J}{ \underbrace{ \mathrm{d}a } } \,-\, a \wedge j \,.$$

[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 $a$ induced by (6) is simply

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

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

Reproducing FQH phenomena

Recovering the Hall conductivity. The first plausibility check on the effective model is to observe that its equation of motion (7) implies the correct Hall conductivity relation for a fractional quantum Hall system at filling fraction $1/k$:

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

$$F \;=\; E_y \mathrm{d}t \wedge \mathrm{d}y + B \mathrm{d}x\wedge \mathrm{d}y$$

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

$$j \;\propto\; dvol(\partial_0) \;\propto\; \mathrm{d}x \wedge \mathrm{d}y \mathrlap{\,,}$$

the temporal component of the equation of motion (7) becomes

$$J_x \;=\; \tfrac{1}{k} E_y$$

which is the correct form of the Hall field $E_y$ for given longitudinal current $J_X$ at filling fraction $\nu = 1/k$ – 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 $A$. It seems that this unnecessary step is brought about by first tacitly renaming “$J$” to “$f$” and then forgetting that this was just a renaming.)

Recovering the fractional quasi-particles. Second, the spatial component of the equation of motion (7) says that (1.) the magnetic flux quanta and (2.) the quasi-particles contribute their $k$th fraction to the total charge density:

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

Here

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

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

This largely concludes the traditional justification for the effective abelian Chern-Simons theory (6).

Hierarchical K-matrix formalism

Following the hierarchical picture of FQH systems at non-unit filling fraction (due to Haldane 1983 and Halperin 1984), where quasiparticles at unit filling fractions are imagined to support themselves a secondary FQH effect with secondary quasi-particles, and so on (a picture that faces various experimental challenges, cf. Jain 2007 §12.1, Jain 2014, esp. p 8 and Rem. below), authors consider abelian Chern Simons theory of multiple gauge fields that are hierarchically coupled to each other, called the “K-matrix formalism” (Blok & Wen 1990, Wen & Zee 1992, early review in Wen 1995 §2.1, textbook account in Wen 2007 §7.3.3).

The basic idea is to observe that the above step from (i) introducing a gauge potential (4) for the electron current density $J$ to (ii) the effective Chern-Simons action (6) with quasi-particle current density $j$ may be repeated by next introducing also a gauge potential for the quasi-particle current (cf. Wen 2007 (7.3.13)), and so on.

Concretely, postulating that the quasi-particle current $j$ (5) is itself the field strength of a secondary auxiliary gauge field $a^{(2)}$

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

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

and then adjoining for that secondary gauge field another Chern-Simons kinetic term, turns the effective Lagrangian density (6) into

(8)$$L_{1,2} \,\coloneqq\, K_1 \tfrac{1}{2}\, a^{(1)} \wedge \mathrm{d} a^{(1)} \,-\, A \wedge \underset{J}{ \underbrace{ \mathrm{d} a^{(1)} } } \,-\, a^{(1)} \wedge \underset{j}{ \underbrace{ \mathrm{d} a^{(2)} } } \,+\, K_2 \tfrac{1}{2}\, a^{(2)} \wedge \mathrm{d} a^{(2)} \,.$$

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

meant to be an effective theory at filling fraction

(9)$$\nu \;=\; \tfrac { 1 } { K_1 - \tfrac{1}{K_2} } \,.$$

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

Now this secondary effective Lagrangian (8) may be rewritten equivalently as (using Einstein summation convention)

(10)$$L_{1,2} \;=\; K_{i j} \tfrac{1}{2} a^{(i)} \wedge \mathrm{d} a^{(j)} - Q_i \, A \wedge \mathrm{d} a^{(i)}$$

with the K-matrix

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

and the charge vector

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

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

Observing that the inverse K-matrix is

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

the second-layer filling fraction (9) is recognized as the top left entry of (11) and hence re-expressed in terms of these matrix quantities as

$$\nu \,=\, Q^t \cdot K^{-1} \cdot Q \,.$$

(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.

Related concepts

• Chern-Simons theory

• quantization of D=3 Chern-Simons theory

• fractional quantum Hall effect

• integer Heisenberg group

References

General

• Michiel Bos, V. Parameswaran Nair: $U(1)$ Chern-Simons theory and $c=1$ conformal blocks, Physics Letters B 223 1 (1989) 61-66 [doi:10.1016/0370-2693(89)90920-9]

• Alexios P. Polychronakos: Abelian Chern-Simons theories in $2+1$ dimensions, Annals of Physics 203 2 (1990) 231-254 [doi:10.1016/0003-4916(90)90171-J]

• Sergio Albeverio, J. Schäfer: Rigorous Approach to Abelian Chern-Simons Theory, in Groups and Related Topics, Mathematical Physics Studies 13, Springer (1992) 143-152 [doi:10.1007/978-94-011-2801-8_12]

• D. Wesolowski, Y. Hosotani, C.-L. Ho: Multiple Chern-Simons Fields on a Torus, Int. J. Mod. Phys. A 9 (1994) 969-989 [arXiv:hep-th/9302121, doi:10.1142/S0217751X94000443]

• G. Giavarini, C. P. Martin, F. Ruiz Ruiz: Abelian Chern-Simons theory as the strong large-mass limit of topologically massive abelian gauge theory: the Wilson loop, Nucl.Phys. B 412 (1994) 731-750 [arXiv:hep-th/9309049, doi:10.1016/0550-3213(94)90397-2, pdf]

• Mihaela Manoliu: Abelian Chern-Simons theory, J. Math. Phys. 39 (1998) 170-206 [arXiv:dg-ga/9610001, doi:10.1063/1.532333]

• Mihaela Manoliu: Abelian Chern-Simons theory. II: A functional integral approach, J. Math.Phys. 39 (1998) 207-217 [doi:10.1063/1.532312]

• Dmitriy Belov, Gregory W. Moore: Classification of abelian spin Chern-Simons theories [arXiv:hep-th/0505235]

  (with focus on refinement to Spin Chern-Simons theory)

• Spencer D. Stirling: Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories, PhD thesis, Austin (2008) [arXiv:0807.2857, pdf, ProQuest]

• Lisa Jeffrey, Brendan McLellan: Eta-Invariants and Anomalies in $U(1)$ Chern-Simons Theory [arXiv:1004.2913]

• Diego Delmastro, Jaume Gomis: Symmetries of Abelian Chern-Simons Theories and Arithmetic, J. High Energ. Phys. 2021 6 (2021) [doi:10.1007/JHEP03(2021)006, arXiv:1904.12884]

• Han-Miru Kim, Philippe Mathieu, Michail Tagaris, Frank Thuillier: $U(1)^n$ Chern-Simons theory: partition function, reciprocity formula and CS-duality [arXiv:2409.10734]

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:

• Prem P. Srivastava: Light-Front Dynamics of Chern-Simons Systems [arXiv:hep-th/9412239]

• L. R. U. Manssur: Canonical Quantization of Chern-Simons on the Light-Front, Phys. Lett. B 480 (2000) 229-236 [arXiv:hep-th/9910127v1, doi:10.1016/S0370-2693(00)00380-4]

Via the Reshetikhin-Turaev construction:

• Enore Guadagnini, Francesco Mancarella: Abelian link invariants and homology, J. Math. Phys. 51 (2010) 062301 [arXiv:1004.5211]

• Philippe Mathieu, Frank Thuillier: Abelian Turaev-Virelizier theorem and $U(1)$ BF surgery formulas [arXiv:1706.01845]

On abelian Chern-Simons theory via theta functions and the representation theory of the integer Heisenberg group:

• Răzvan Gelca, Alejandro Uribe: From classical theta functions to topological quantum field theory, in: The Influence of Solomon Lefschetz in Geometry and Topology: 50 Years of Mathematics at CINVESTAV, Contemporary Mathematics 621, AMS (2014) 35-68 [arXiv:1006.3252, doi;10.1090/conm/621, ams:conm-621, slides pdf, pdf]

• Răzvan Gelca, Alastair Hamilton: Classical theta functions from a quantum group perspective, New York J. Math. 21 (2015) 93–127 [arXiv:1209.1135, nyjm:j/2015/21-4]

• Răzvan Gelca, Alastair Hamilton: The topological quantum field theory of Riemann’s theta functions, Journal of Geometry and Physics 98 (2015) 242-261 [doi:10.1016/j.geomphys.2015.08.008, arXiv:1406.4269]

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:

• Anton Kapustin, Natalia Saulina: Topological boundary conditions in abelian Chern-Simons theory, Nucl. Phys. B 845 (2011) 393-435 [arXiv:1008.0654, doi:10.1016/j.nuclphysb.2010.12.017]

• Anton Kapustin, Natalia Saulina, §3 in: Surface operators in 3d TFT and 2d Rational CFT, in: Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics 83, AMS (2011) [arXiv:1012.0911, ams:pspum-83]

• Levin 2013

• Anton Kapustin: Ground-state degeneracy for abelian anyons in the presence of gapped boundaries, Phys. Rev. B 89 (2014) 125307 [arXiv:1306.4254, doi:10.1103/PhysRevB.89.125307]

• Juven C. Wang, Xiao-Gang Wen: Boundary Degeneracy of Topological Order, Phys. Rev. B 91 (2015) 125124 [doi:10.1103/PhysRevB.91.125124, arXiv:1212.4863]

• Jackson R. Fliss, Xueda Wen, Onkar Parrikar, Chang-Tse Hsieh, Bo Han, Taylor L. Hughes, Robert G. Leigh: Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory, JHEP 09 (2017) 056 [arXiv:1705.09611, doi:10.1007/JHEP09(2017)056]

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:

• Claudio Dappiaggi, Simone Murro, Alexander Schenkel: Non-existence of natural states for Abelian Chern-Simons theory, J. Geom. Phys. 116 (2017) 119-123 [arXiv:1612.04080, arXiv:10.1016/j.geomphys.2017.01.015]

On lattice formulation of abelian Chern-Simons theory:

• Kai Sun, Krishna Kumar, Eduardo Fradkin: Discretized Abelian Chern-Simons gauge theory, Phys. Rev. B 92 (2015) 115148 [doi:10.1103/PhysRevB.92.115148]

• Theodore Jacobson, Tin Sulejmanpasic: Modified Villain formulation of abelian Chern-Simons theory, Phys. Rev. D 107 (2023) 125017 [arXiv:2303.06160, doi:10.1103/PhysRevD.107.125017]

and its canonical quantization:

• Theodore Jacobson, Tin Sulejmanpasic: Canonical quantization of lattice Chern-Simons theory, High Energ. Phys. 2024 (2024) 87 [arXiv:2401.09597, doi:10.1007/JHEP11(2024)087]

In relation to the dilogarithm:

• Daniel S. Freed, Andrew Neitzke: The dilogarithm and abelian Chern-Simons, J. Differential Geom. 123 2 (2023) 241-266 [arXiv:2006.12565, doi:10.4310/jdg/1680883577]

On unoriented manifolds:

• Ippo Orii: On dimensions of $(2+1)D$ abelian bosonic topological systems on unoriented manifolds [arXiv:2502.13532]

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]

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

• Steven M. Girvin, around (10.7.15) in: Summary, Omissions and Unanswered Questions, Chapter 10 of: The Quantum Hall Effect, Graduate Texts in Contemporary Physics, Springer (1986, 1990) [doi:10.1007/978-1-4612-3350-3]

• Steven M. Girvin, Allan H. MacDonald, around (10) of: Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect, Phys. Rev. Lett. 58 12 (1987) (1987) 1252-1255 [doi:10.1103/PhysRevLett.58.1252]

• Shou Cheng Zhang, T. H. Hansson S. Kivelson: Effective-Field-Theory Model for the Fractional Quantum Hall Effect, Phys. Rev. Lett. 62 (1989) 82 [doi:10.1103/PhysRevLett.62.82]

and was made more explicit in:

• Xiao-Gang Wen, Anthony Zee: Quantum statistics and superconductivity in two spatial dimensions, Nuclear Physics B – Proceedings Supplements 15 (1990) 135-156 [doi:10.1016/0920-5632(90)90014-L]

• B. Blok, Xiao-Gang Wen: Effective theories of the fractional quantum Hall effect at generic filling fractions, Phys. Rev. B 42 (1990) 8133 [doi:10.1103/PhysRevB.42.8133]

• Xiao-Gang Wen: Topological Orders in Rigid States, Int. J. Mod. Phys. B 4 239 (1990) [doi:10.1142/S0217979290000139]

• Xiao-Gang Wen, Qian Niu: Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces, Phys. Rev. B 41 9377 (1990) [doi:10.1103/PhysRevB.41.9377]

• Jürg Fröhlich, T. Kerler: Universality in quantum Hall systems, Nuclear Physics B 354 2–3 (1991) 369-417 [doi:10.1016/0550-3213(91)90360-A]

• Jürg Fröhlich, Anthony Zee: Large scale physics of the quantum hall fluid, Nuclear Physics B 364 3 (1991) 517-540 [doi:10.1016/0550-3213(91)90275-3]

• Z. F. Ezawa, A. Iwazaki: Chern-Simons gauge theories for the fractional-quantum-Hall-effect hierarchy and anyon superconductivity, Phys. Rev. B 43 (1991) 2637 [doi:10.1103/PhysRevB.43.2637]

• A. P. Balachandran, A. M. Srivastava: Chern-Simons Dynamics and the Quantum Hall Effect [arXiv:hep-th/9111006, spire:319826]

• Ana Lopez, Eduardo Fradkin: Fractional quantum Hall effect and Chern-Simons gauge theories, Phys. Rev. B 44 (1991) 5246 [doi:10.1103/PhysRevB.44.5246, pdf]

• Xiao-Gang Wen, Anthony Zee: Topological structures, universality classes, and statistics screening in the anyon superfluid, Phys. Rev. B 44 (1991) 274 [doi:10.1103/PhysRevB.44.274]

• Roberto Iengo, Kurt Lechner: Anyon quantum mechanics and Chern-Simons theory, Physics Reports 213 4 (1992) 179-269 [doi:10.1016/0370-1573(92)90039-3]

• Xiao-Gang Wen, Anthony Zee: Classification of Abelian quantum Hall states and matrix formulation of topological fluids, Phys. Rev. B 46 (1992) 2290 [doi:10.1103/PhysRevB.46.2290]

• Xiao-Gang Wen, Anthony Zee: Shift and spin vector: New topological quantum numbers for the Hall fluids, Phys. Rev. Lett. 69 (1992) 953, Erratum Phys. Rev. Lett. 69 3000 (1992) [doi:10.1103/PhysRevLett.69.953]

• Xiao-Gang Wen: Theory of Edge States in Fractional Quantum Hall Effects, International Journal of Modern Physics B 06 10 (1992) 1711-1762 [doi:10.1142/S0217979292000840]

• Xiao-Gang Wen, Topological orders and Edge excitations in FQH states, Advances in Physics 44 (1995) 405 [doi:10.1080/00018739500101566, arXiv:cond-mat/9506066]

  (in the context of topological order)

• A. P. Balachandran, L. Chandar, B. Sathiapalan: Chern-Simons Duality and the Quantum Hall Effect, Int. J. Mod. Phys. A11 (1996) 3587-3608 [doi:10.1142/S0217751X96001693, arXiv:hep-th/9509019]

Early review:

• Shou Cheng Zhang: The Chern-Simons-Landau-Ginzburg theory of the fractional quantum Hall effect, International Journal of Modern Physics B 06 01 (1992) 25-58 [doi:10.1142/S0217979292000037, pdf]

• Anthony Zee: Quantum Hall Fluids, in: Field Theory, Topology and Condensed Matter Physics, Lecture Notes in Physics 456, Springer (1995) [doi:10.1007/BFb0113369, arXiv:cond-mat/9501022]

• Xiao-Gang Wen: Topological orders and Edge excitations in FQH states, Advances in Physics 44 5 (1995) 405 [doi:10.1080/00018739500101566, arXiv:cond-mat/9506066]

Further review and exposition:

• Xiao-Gang Wen: Effective theory of fractional quantum Hall liquids, section 7.3 of: Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons, Oxford Academic (2007)[doi:10.1093/acprof:oso/9780199227259.001.0001, pdf]

• Yuan-Ming Lu, Ashvin Vishwanath, part II of: Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach, Phys. Rev. B 86 (2012) 125119, Erratum Phys. Rev. B 89 (2014) 199903 [doi:10.1103/PhysRevB.86.125119, arXiv:1205.3156]

• Eduardo Fradkin, chapter 13.7 of: Field Theories of Condensed Matter Physics, Cambridge University Press (2013) [doi:10.1017/CBO9781139015509, ISBN:9781139015509]

• Edward Witten, pp 30 in: Three Lectures On Topological Phases Of Matter, La Rivista del Nuovo Cimento 39 (2016) 313-370 [doi:10.1393/ncr/i2016-10125-3, arXiv:1510.07698]

• David Tong §5 of: The Quantum Hall Effect, lecture notes (2016) [arXiv:1606.06687, course webpage, pdf, pdf]

• Josef Wilsher: The Chern–Simons Action & Quantum Hall Effect: Effective Theory, Anomalies, and Dualities of a Topological Quantum Fluid, PhD thesis, Imperial College London (2020) [pdf]

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

• see there.

More on edge modes:

• Michael Levin: Protected edge modes without symmetry, Phys. Rev. X 3 021009 (2013) [doi:10.1103/PhysRevX.3.021009, arXiv:1301.7355]

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

• Maissam Barkeshli, Xiao-Gang Wen: Bilayer quantum Hall phase transitions and the orbifold non-Abelian fractional quantum Hall states, Phys. Rev. B 95 (2017) 085135 [doi:10.1103/PhysRevB.95.085135, arXiv:1010.4270]

• Inti Sodemann, Itamar Kimchi, Chong Wang, T. Senthil: Composite fermion duality for half-filled multicomponent Landau levels, Phys. Rev. B 95 (2017) 085135 [doi:10.1103/PhysRevB.95.085135, doi:1609.08616]

• Tian-Sheng Zeng: Fractional quantum Hall effect of Bose-Fermi mixtures, Phys. Rev. B 103 (2021) L201118 [doi:10.1103/PhysRevB.103.L201118, arXiv:2012.08203]

• Liangdong Hu, Zhao Liu, W. Zhu: Modular transformation and anyonic statistics of multi-component fractional quantum Hall states, Phys. Rev. B 108 (2023) 235121 [arXiv:2301.06427, doi:10.1103/PhysRevB.108.235121]

• Tian-Sheng Zeng: Three-component fractional quantum Hall effect in topological flat bands, Phys. Rev. B 110 (2024) 165126 [doi:10.1103/PhysRevB.110.165126, arXiv:2407.08568]

Further developments:

• Dmitriy Belov, Gregory W. Moore, §7 of: Classification of abelian spin Chern-Simons theories [arXiv:hep-th/0505235]

• Christian Fräßdorf: Abelian Chern-Simons Theory for the Fractional Quantum Hall Effect in Graphene, Phys. Rev. B 97 115123 (2018) [doi:10.1103/PhysRevB.97.115123, arXiv:1712.03595]

• Kristan Jensen, Amir Raz: The Fractional Hall hierarchy from duality [arXiv:2412.17761]

• Abhishek Agarwal, Dimitra Karabali, V. Parameswaran Nair: Fractional quantum Hall effect in higher dimensions, Phys. Rev. D 111 (2025) 025002 [doi:10.1103/PhysRevD.111.025002, arXiv:2410.14036]

  (higher dimensional Chern-Simons theory)

Abelian Maxwell-Chern-Simons theory for superconductivity

Coupling the Chern-Simons term to 1+2D Maxwell theory for an effective description of superconductivity:

• M. Cristina Diamantini, P. Sodano, Carlo A. Trugenberger: Gauge Theories of Josephson Junction Arrays, Nucl. Phys. B 474 (1996) 641-677 [doi:10.1016/0550-3213(96)00309-4, arXiv:hep-th/9511168]

(…)

For baryons in $N_f=1$ QCD

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

• Zohar Komargodski: Baryons as Quantum Hall Droplets [arXiv:1812.09253]

• Yong-Liang Ma, Maciej A. Nowak, Mannque Rho, Ismail Zahed, Baryon as a Quantum Hall Droplet and the Quark-Hadron Duality, Phys. Rev. Lett. 123 (2019) 172301 [doi:10.1103/PhysRevLett.123.172301]

• Kentaro Nishimura, Naoki Yamamoto, Ryo Yokokura: Quantum Hall liquids in high-density QCD [arXiv:2410.07665]