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 $K^{i j} A_i \wedge \mathrm{d} A_j$ (for $K$ an $N \times N$ symmetric matrix and using Einstein summation convention), 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 a 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 $N=1$ see Manoliu 1998a p 40, for general $N$ cf. Belov & Moore 2005 p 26)

References

General

• 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]

• 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]

• 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]

• 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]

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]

On boundary conditions and line-defects in abelian Chern-Simons theory:

• 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]

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]

Abelian Chern-Simons for frational 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, A. 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]

• S. C. 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]

• 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]

• 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]

• 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]

• 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]

Early review:

• 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:

• 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]

• Edward Witten: 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.

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]

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]