nLab W algebra

Idea

For $N \in \mathbb{N}$, $N \geq 2$, W-algebras $W_N$ are associative algebras which are higher conformal spin extensions of the Virasoro algebra (which is the case $N=2$) — originally discovered as extended symmetry algebras of 2D conformal field theories.

For $N \to \infty$ the commutator in these algebras becomes linear in the standard generators (traditionally denoted “$W$”), whence the limiting case $W_\infty$ is the universal enveloping algebra of a Lie algebra. These $W_\infty$-algebras (with capital “W”) are deformation quantizations of $w_\infty$-algebras (with lower case “w”), which in turn are central extensions of Lie algebras of area-preserving diffeomorphisms.

In the fractional quantum Hall effect, $W_\infty$-algebras describe the symmetries of collective excitations of the 2D electron gas, notably of the GMP magneto-roton mode (Girvin, MacDonald & Platzman 1986, see at Laughlin wavefunction – GMP exitations), while their classical limit $w_\infty$-algebras describes the symmetries of the corresponding long-wavelength limit, known as the “chiral graviton”-excitation. In fact, the supersymmetric form of these $W_\infty$-symmetries is asymptotically realized in FQH systems (see at effective supersymmetry of FQH systems), with the superpartner of the GMP mode being the “neutral fermion” excitation whose long-wavelength $w_\infty$ limit is the corresponding “gravitino”.

Related concepts

• area-preserving diffeomorphisms

• Virasoro algebra

• quantum Hall effect via noncommutative geometry

References

General

Original discussion of the $W_3$-algebra:

• Alexander B. Zamolodchikov: Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor Math Phys 65 (1985) 1205–1213 [doi:10.1007/BF01036128, mathnet:5141]

and the original generalization to $W_N$ algebras:

• Vladimir A. Fateev, Alexander B. Zamolodchikov: Conformal quantum field theory models in two dimensions having $\mathbb{Z}_3$ symmetry, Nuclear Physics B 280 (1987) 644-660 [doi:10.1016/0550-3213(87)90166-0]

• Vladimir A. Fateev, Sergei L. Lukyanov: The Models of Two-Dimensional Conformal Quantum Field Theory with $Z_n$ Symmetry, Int. J. Mod. Phys. A 3 (1988) 507 [doi:10.1142/S0217751X88000205]

Early consideration (not using that terminology, though) of the $w_\infty$-algebra of the torus:

• Vladimir Arnold; equation (109) in: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Annales de l’Institut Fourier 16 1 (1966) 319–361 [numdam:AIF_1966__16_1_319_0/]

and of the $W_\infty$-algebra of the noncommutative torus:

• Girvin, MacDonald & Platzman 1986 (4.13)

• David B. Fairlie, Paul Fletcher, Cosmas K. Zachos; equation (6) in: Trigonometric structure constants for new infinite-dimensional algebras, Physics Letters B 218 2 (1989) 203-206 [doi:10.1016/0370-2693(89)91418-4]

Discussion of $W_\infty$ as the limiting case of $W_N$:

• Adel Bilal: A remark on the $N \to \infty$ limit of $W_N$-algebras, Physics Letters B 227 3–4 (1989) 406-410 [doi:10.1016/0370-2693(89)90951-9]

• Christopher N. Pope, Larry J. Romans, X. Shen: The complete structure of $W_\infty$, Physics Letters B 236 2 (1990) 173-178 [doi:10.1016/0370-2693(90)90822-N]

Survey and review:

• Ioannis Bakas, Elias B. Kiritsis: Structure and representations of the $W_\infty$ algebra, Prog. Theor. Phys. Suppl. 102 (1990) 15-38 [inSpire:298496]

• Christopher N. Pope: Lectures on W algebras and W gravity, Summer School in High-energy Physics and Cosmology (1991) 827-867 [arXiv:hep-th/9112076, inSpire:321766]

• X. Shen: W-infinity and String Theory, Int. J. Mod. Phys. 7 A (1992) 6953 [arXiv:hep-th/9202072, doi:10.1142/S0217751X92003203]

• Peter Bouwknegt, Kareljan Schoutens: W symmetry in conformal field theory, Phys. Rep. 223 4 (1993) 183–276 [doi:10.1016/0370-1573(93)90111-P]

• Peter Bouwknegt, Kareljan Schoutens: $W$-Symmetry, Advanced Series in Mathematical Physics 22, World Scientific (1995) [doi:10.1142/2354, inSpire:406843]

• Peter Bouwknegt, Jim McCarthy, Krzysztof Pilch: The $W_3$ algebra: modules, semi-infinite cohomology, and BV algebras, Lec. Notes in Phys. 42, Springer (1996) [doi:10.1007/978-3-540-68719-1]

• D. V. Artamonov: Introduction to finite $W$-algebras, Boletín de Matemáticas, 23 2 (2016) 165-219 [arXiv:1607.01697]

• Wikipedia: W-algebra

Relation of $w_\infty$-algebras to area-preserving diffeomorphisms:

• Eric Bergshoeff, Miles P. Blencowe, Kellogg S. Stelle: Area-preserving diffeomorphisms and higher-spin algebras, Commun. Math. Phys. 128 (1990) 213–230 [doi:10.1007/BF02108779]

• Ergin Sezgin: Area-Preserving Diffeomorphisms, $w_\infty$ Algebras and $w_\infty$ Gravity [arXiv:hep-th/9202086]

• Ian I. Kogan: Area Preserving Diffeomorphisms and Symmetry in a Chern-Simons Theory [arXiv:hep-th/9208028]

Relation to Jordan algebra:

• L. J. Romans: Realisations of classical and quantum $W_3$ symmetry, Nuclear Physics B 352 3 (1991) 829–848 [doi:10.1016/0550-3213(91)90108-A]

Relation to $L_\infty$-algebra:

• Ralph Blumenhagen, Michael Fuchs, Matthias Traube, $\mathcal{W}$-Algebras are $L_\infty$-algebras [arXiv:1705.00736]

Representation theory

On the representation theory:

• M. Golenishcheva-Kutuzova, D. Lebedev, Vertex operator representation of some quantum tori Lie algebras, Commun. Math. Phys. 148 (1992) 403–416 [doi:10.1007/BF02100868]

• H. Awata, M. Fukuma, Y. Matsuo, S. Odake: Representation Theory of The $W_{1+\infty}$ Algebra, Prog. Theor. Phys. Suppl. 118 (1995) 343-374 [doi:10.1143/PTPS.118.343, arXiv:hep-th/9408158]

W algebra CFT

On 2d CFT with extended W algebra symmetry:

• Federico Ambrosino, Tomáš Procházka: RG flows of minimal $\mathcal{W}$-algebra CFTs via non-invertible symmetries [arXiv:2601.18667]

On conformal blocks in $\mathcal{W}_3$-Algebra CFT:

• V. Belavin, Mikhail Pavlov: Towards $\mathcal{W}_3$ classical blocks with semi-degenerate operators [arXiv:2512.23868]

See also:

• Harshal Kulkarni, Christopher Beem: Towards a classification of graded unitary $\mathcal{W}_3$ algebras [arXiv:2602.15944]

• Thomas Creutzig, Niklas Garner, Byeonggi Go, Heeyeon Kim: W-algebras of the Deligne-Cvitanović Exceptional series and the minimal 3d $\mathcal{N}=4$ SCFT [arXiv:2603.17394]

In Fractional Quantum Hall systems

On $W_\infty$-algebra symmetry in fractional quantum Hall systems:

The $W_\infty$-algebra of the noncommutative torus (FFZ 1989) first appears (not named or recognized as such) in discussion of FQH systems as the Lie algebra of projected density operators $\overline{\rho}_{\mathbf{k}}$ in:

• Steven M. Girvin, A. H. MacDonald, P. M. Platzman; equation (4.13) in: Magneto-roton theory of collective excitations in the fractional quantum Hall effect, Phys. Rev. B 33 (1986) 2481 [doi:10.1103/PhysRevB.33.2481, inSpire:244382, pdf]

whence often also called here the GMP algebra.

Further discussion:

• Satoshi Iso, Dimitra Karabali, B. Sakita: Fermions in the lowest Landau level. Bosonization, $W_\infty$ algebra, droplets, chiral bosons, Physics Letters B 296 1–2 (1992) 143-150 [doi:10.1016/0370-2693(92)90816-M]

• Andrea Cappelli, Carlo A. Trugenberger, G. R. Zemba: Infinite Symmetry in the Quantum Hall Effect, Nucl. Phys. B 396 (1993) 465-490 [doi:10.1016/0550-3213(93)90660-H, arXiv:hep-th/9206027]

• Andrea Cappelli, Gerald V. Dunne, Carlo A. Trugenberger, G. R. Zemba: Conformal Symmetry and Universal Properties of Quantum Hall States, Nucl. Phys. B 398 (1993) 531-567 [doi:10.1016/0550-3213(93)90603-M, arXiv:hep-th/9211071]

• Andrea Cappelli, Carlo A. Trugenberger, G. R. Zemba: Classification of Quantum Hall Universality Classes by $W_{1+\infty}$ symmetry, Phys. Rev. Lett. 72 (1994) 1902-1905 [doi:10.1103/PhysRevLett.72.1902, arXiv:hep-th/9310181]

• Andrea Cappelli: $W_\infty$ Symmetry in the Quantum Hall Effect, talk at Geometric and analytic aspects of the Quantum Hall effect (May 2023) [video, pdf]

• Wang Yuzhu: $W_\infty$-algebra and GMP algebra, appendix B in: Graviton Modes in Fractional Quantum Hall Liquids, PhD thesis, Nanyang TU (2023) [hndl:10356/165156, pdf]

• Yi-Hsien Du: GMP/$W_\infty$ algebra, section IV.B in: Chiral Graviton Theory of Fractional Quantum Hall States [arXiv:2509.04408]

On $W_\infty$-algebra symmetry in fractional Chern insulators (fractional quantum anomalous Hall systems):

• Siddharth A. Parameswaran, Rahul Roy, Shivaji L. Sondhi: Fractional Chern Insulators and the $W_\infty$ Algebra, Phys. Rev. B 85 (2012) 241308(R) [arXiv:1106.4025, doi:10.1103/PhysRevB.85.241308]

On the sphere:

• Rakesh K. Dora, Ajit C. Balram: Dispersion of collective modes in spinful fractional quantum Hall states on the sphere [arXiv:2509.13100]

• Rakesh K. Dora, Ajit C. Balram: Static structure factor and the dispersion of the Girvin-MacDonald-Platzman density mode for fractional quantum Hall fluids on the Haldane sphere, Phys. Rev. B 111 (2025) 115132 [arXiv:2410.00165, doi:10.1103/PhysRevB.111.115132]

and in relation to the fuzzy sphere:

• Yin-Chen He: Free real scalar CFT on fuzzy sphere: spectrum, algebra and wavefunction ansatz [arXiv:2506.14904]

• Luisa Eck, Zhenghan Wang: 3d Conformal Field Theories via Fuzzy Sphere Algebra [arXiv:2602.15025]

In String/M-theory

Relation to M-brane intersections:

• Davide Gaiotto, Miroslav Rapčák, Yehao Zhou: Deformed Double Current Algebras, Matrix Extended $W_{\infty}$ Algebras, Coproducts, and Intertwiners from the M2-M5 Intersection [arXiv:2309.16929]

category: physics, algebra