nLab volume preserving diffeomorphism

Idea

A diffeomorphism $\phi \colon X \longrightarrow X$ of a (pseudo-) Riemannian manifold $X$ is volume preserving (VPD) if its pullback of differential forms fixes the volume form: $\phi^\ast vol = vol$.

If the dimension is $dim(X) = 2$, then one also speaks of area-preserving diffeomorphisms (APD).

Properties

Volume preserving diffeomorphisms form a topological subgroup

$$SDiff(X) \subset Diff(X)^+$$

of the general (orientation-preserving) diffeomorphism group of the underlying smooth manifold of $X$.

Proposition

This inclusion is a weak homotopy equivalence, in fact a deformation retract.

This follows by Moser's theorem (cf. Banyaga 1997 Cor. 1.5.4).

For $X$ compact, $SDiff$ is an infinite-dimensional Fréchet Lie group.

Its Lie algebra then is the algebra of incompressible (i.e. zero-divergence) vector fields on $X$. For the case $dim(X) = 2$, these Lie algebras (or their central extensions) are known as $w_\infty$-algebras (with lower case “w”) and certain deformation quantizations of these as $W_\infty$-algebras (with capital “W”).

In physics

In physics (field theory), volume-preserving diffeomorphisms appear:

• as enhanced symmetries of 2D Yang-Mills theory (cf. Witten 1991, Kogan 1992, Pickrell 1996),

• as the gauge symmetries of unimodular gravity,

• as residual gauge symmetries of relativistic brane sigma models after light cone gauge-fixing (cf. below),

• as (gauge) symmetries of effective descriptions of fractional quantum Hall systems (cf. below).

But beware that most of the physics literature considers this at the level of Lie algebras, which can be misleading as the relation of infinite-dimensional Lie groups to their Lie algebras is more loose. For instance:

• the relation between $Lie(SDiff(\Sigma^2))$ and $\mathfrak{su}(\infty)$ (cf. references below) fails drastically at the level of groups (already for basic topological reasons, as highlighted by Swain 2004);

• the canonical group action of $SDiff(\Sigma^2)$ on quantum states of 3D Maxwell-Chern-Simons theory is (continuous but) not differentiable, hence has no reflection on the level of Lie algebras (Pickrell 2000).

References

General

Monograph:

• Augustin Banyaga; Volume Preserving Diffeomorphisms, chapter 5 of: The Structure of Classical Diffeomorphism Groups, Mathematics and Its Applications 400, Springer (1997) [doi:10.1007/978-1-4757-6800-8]

Original discussion 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/]

Proof that the volume-preserving diffeomorphism group of Cartesian space $\mathbb{R}^n$ is a perfect group when $n \geq 3$:

• Dusa McDuff: On the Group of Volume-Preserving Diffeomorphisms of $\mathbf{R}^n$, Transactions of the AMS 261 1 (1980) [doi:10.1090/S0002-9947-1980-0576866-3]

Discussion in relation to $w_\infty$-algebras:

• 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]

  (in 3D Maxwell-Chern-Simons theory)

Linear representations

There is no general representation theory of VPDs, but (cf. Pickrell 2000 p. 179) the following examples of linear representations of $SDiff$ have been constructed.

Reps of $SDiff(\mathbb{R}^3)$

In the context of discussion of vortices:

• Mario Rasetti, Tullio Regge: Vortices in He II, current algebras and quantum knots, Physica A: Statistical Mechanics and its Applications 80 3 (1975) 217-233 [doi:10.1016/0378-4371(75)90105-3]

• Mario Rasetti, Tullio Regge: Quantum Vortices, preprint TH.3643-CERN (1983) [pdf]

• Mario Rasetti, Tullio Regge: Quantization of Extended Vortices and $SDiff(\mathbb{R}^3)$, in: Algebraic Analysis,

  2, Academic Press (1988) 727-734 [doi:10.1016/B978-0-12-400466-5.50022-6]

• Mauro Spera: Moment map and gauge geometric aspects of the Schrödinger and Pauli equations, International Journal of Geometric Methods in Modern Physics 13 04 (2016) 1630004 [doi:10.1142/S021988781630004X]

Reps of $SDiff(S^2)$

Representation on sections of determinant line bundles of (tacitly) 3D Maxwell-Chern-Simons theory:

• Doug Pickrell: On the Action of the Group of Diffeomorphisms of a Surface on Sections of the Determinant Line Bundle, Pacific Journal of Mathematics 193 1 (2000) 177-199 [doi:10.2140/pjm.2000.193.177, pdf]

  (via methods of constructive field theory, also discusses $SDiff(T^2)$)

using a rigorous APD-invariant path integral measure for 2D Yang-Mills theory, due to:

• Doug Pickrell: On $YM_2$ measures and area-preserving diffeomorphisms, Journal of Geometry and Physics 19 4 (1996) 315-367 [doi:10.1016/0393-0440(95)00034-8]

In the context of the Plebański formulation of gravity:

• Boris Kruglikov, Oleg Morozov: $SDiff(2)$ and uniqueness of the Plebański equation, J. Math. Phys. 53 083506 (2012) [doi:10.1063/1.4739749, arXiv:1204.3577]

Via the orbit method:

• Robert F. Penna: $SDiff(S^2)$ and the orbit method, J. Math. Phys. 61 012301 (2020) [arXiv:1806.05235, doi:10.1063/1.5140475]

  (though no real results obtained here)

Relation between $\mathfrak{su}(\infty)$ and $Lie(SDiff(\Sigma))$

On the identification of the special unitary Lie algebra $\mathfrak{su}(n)$, as $n \to \infty$, with the Lie algebra of area-preserving diffeomorphisms of surfaces $\Sigma$ (cf. also at Quantization of the M2-brane to the BFSS matrix model):

For the 2-sphere:

• Jens Hoppe; section B.II of: Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, PhD thesis, MIT (1982) [hdl:1721.1/15717, pdf]

• Bernard de Wit, Jens Hoppe, Hermann Nicolai; pp. 563 of: On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 (1988) 545-581 [doi:10.1016/0550-3213(88)90116-2, spire:261702, pdf, pdf]

• Emmanuel G. Floratos, John Iliopoulos, G. Tiktopoulos: A note on $SU(\infty)$ classical Yang-Mills theories, Physics Letters B 217 3 (1989) 285-288 [doi:10.1016/0370-2693(89)90867-8]

For the 2-torus:

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

• Christopher N. Pope, Kellogg S. Stelle: $SU(\infty)$, $SU_+(\infty)$ and area-preserving algebras, Physics Letters B 226 3–4 (1989) 257-263 [doi:10.1016/0370-2693(89)91191-X]

• Hermann Nicolai, Robert C. Helling; p. 20 of: Supermembranes and M(atrix) Theory, In: Trieste 1998, Nonperturbative aspects of strings, branes and supersymmetry (1998) 29-74 [arXiv:hep-th/9809103, spire:476366]

Warning that the analogous statements for the Lie groups (as opposed to their Lie algebras) fail dramatically, for basic topological reasons:

• John Swain: On the limiting procedure by which $SDiff(T^2)$ and $SU(\infty)$ are associated [arXiv:hep-th/0405002]

• John Swain: The Topology of $SU(\infty)$ and the Group of Area-Preserving Diffeomorphisms of a Compact 2-manifold [arXiv:hep-th/0405003]

• John Swain: The Majorana representation of spins and the relation between $SU(\infty)$ and $SDiff(S^2)$ [arXiv:hep-th/0405004]

Analogous discussions:

for the tetrahedron:

• A. Wolski, J. S. Dowker: Area‐preserving diffeomorphisms of the tetrahedron, J. Math. Phys. 32 (1991) 857–863 [doi:10.1063/1.529343]

for $\mathfrak{sl}(n;\mathbb{N})$ and $\mathfrak{su}(n,n)$:

• William Donnelly, Laurent Freidel, Seyed Faroogh Moosavian, Antony J. Speranza: Matrix quantization of gravitational edge modes, J. High Energ. Phys. 2023, 163 (2023) [doi:10.1007/JHEP05(2023)163]

and in relation to the KP hierarchy:

• Kanehisa Takasaki, Takashi Takebe: $SDiff(2)$ KP hierarchy, Int. J. Mod. Phys. A 7 supp01B (1992) 889-922 [doi:10.1142/S0217751X92004099, arXiv:hep-th/9112046]

As symmetries of brane dynamics

On volume-preserving diffeomorphisms as residual gauge symmetries of brane sigma-models in light cone gauge:

On APD symmetry of the M2-brane sigma-model (cf. M2-brane – Light-cone quantization to the BFSS matrix model):

• Emmanuel G. Floratos, John Iliopoulos: A note on the classical symmetries of the closed bosonic membranes, Physics Letters B 201 2 (1988) 237-240 [doi:10.1016/0370-2693(88)90220-1]

• Bernard de Wit, Jens Hoppe, Hermann Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 (1988) 545-581 [doi:10.1016/0550-3213(88)90116-2, spire:261702, pdf, pdf]

  (gauging of APD symmetry on p. 553)

• Bernard de Wit, U. Marquard, Hermann Nicolai: Area-preserving diffeomorphisms and supermembrane Lorentz invariance, Commun. Math. Phys. 128 (1990) 39–62 [doi:10.1007/BF02097044, inSpire:277216]

and on VPD symmetry in the M2’s covariant formulation:

• So Katagiri: A Lorentz Covariant Matrix Model for Bosonic M2-Branes: Nambu Brackets and Restricted Volume-Preserving Deformations [arXiv2504.05940]

• So Katagiri: Quantum Stability at One Loop for BPS Membranes in a Lorentz-Covariant RVPD Matrix Model [arXiv:2509.23853]

Generalization to light cone gauge VPD symmetry of the higher-dimensional super p-branes without gauge fields on their worldvolume:

• Eric Bergshoeff, Ergin Sezgin, Y. Tanii, Paul K. Townsend: Super $p$-branes as gauge theories of volume preserving diffeomorphisms, Annals of Physics 199 2 (1990) 340-365 [doi:10.1016/0003-4916(90)90381-W, pdf]

See also:

• Y. Matsuo, Y. Shibusa: Volume Preserving Diffeomorphism and Noncommutative Branes, JHEP 0102:006 (2001) [arXiv:hep-th/0010040]

On VPD symmetry in the Nambu-Poisson M5-brane model:

• Pei-Ming Ho, Yosuke Imamura, Yutaka Matsuo, Shotaro Shiba; section 4 of: M5-brane in three-form flux and multiple M2-branes, JHEP 0808:014 (2008) [arXiv:0805.2898, doi:10.1088/1126-6708/2008/08/014]

• Pei-Ming Ho, Chi-Hsien Yeh; section 2 of: D-brane in R-R Field Background, JHEP 1103:143 (2011) [arXiv:1101.4054, doi:10.1007/JHEP03(2011)143]

• Pei-Ming Ho, Chen-Te Ma, Chi-Hsien Yeh; section 2.2.1 of: BPS States on M5-brane in Large $C$-field Background, J. High Energ. Phys. 2012 76 (2012) [arXiv:1206.1467, doi:10.1007/JHEP08(2012)076]

Review:

• Pei-Ming Ho; section 3.2.1 of: A Concise Review on M5-brane in Large $C$-Field Background, Chin. J. Phys. 48 1 (2010) [arXiv:0912.0445]

• Pei-Ming Ho, Yutaka Matsuo; section 4.3 of: Nambu bracket and M-theory, Progress of Theoretical and Experimental Physics 2016 6 (2016) 06A104 [arXiv:1603.09534, doi:10.1093/ptep/ptw075]

On VPD symmetry of the actual M5-brane sigma-model:

• Igor A. Bandos, Paul K. Townsend: Light-cone M5 and multiple M2-branes, Class. Quant. Grav. 25 245003 (2008) [doi:10.1088/0264-9381/25/24/245003, arXiv:0806.4777]

See also:

• Pei-Ming Ho, Hikaru Kawai, Henry Liao: General Actions of Extended Objects and Volume-Preserving Diffeomorphism [arXiv:2602.23443]

As symmetries of fractional quantum Hall systems

As (gauge) symmetries of fractional quantum Hall systems (cf. supersymmetry in FQH systems and for more see at $W_\infty$-algebra):

• A. 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]

• Dimitra Karabali, V. Parameswaran Nair: The effective action for edge states in higher dimensional quantum Hall systems, Nucl. Phys. B 679 (2004) 427-446 [doi:10.1016/j.nuclphysb.2003.11.020, arXiv:hep-th/0307281]

• Yi-Hsien Du, Umang Mehta, Dung Xuan Nguyen, Dam Thanh Son: Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry, SciPost Phys. 12 050 (2022) [arXiv:2103.09826, doi:10.21468/SciPostPhys.12.2.050]

• Yi-Hsien Du, Umang Mehta, Dam Thanh Son: Noncommutative gauge symmetry in the fractional quantum Hall effect, J. High Energ. Phys. 2024 125 (2024) [doi:10.1007/JHEP08(2024)125, arXiv:2110.13875]

• Yi-Hsien Du: Chiral Graviton Theory of Fractional Quantum Hall States [arXiv:2509.04408]

• Hisham Sati, Urs Schreiber: Non-Perturbative SDiff Covariance of Fractional Quantum Hall Excitations, European Physics Letters 154 (2026) 16002 [arXiv:2602.02292 cond-mat.str-el, doi:10.1209/0295-5075/ae57d7]

Analog for 3d VPDs:

• Giandomenico Palumbo: Generalized GMP Algebra for Three-Dimensional Quantum Hall Fluids of Extended Objects [arXiv:2602.15664]