L-infinity algebras in physics


The concept of L-∞ algebras as graded vector spaces equipped with nn-ary brackets satisfying a generalized Jacobi identity is due to:

At least Stasheff 92 was following Zwiebach 92, who had observed that the n-point functions in closed string field theory equip the BRST complex of the closed bosonic string with L L_\infty-algebra structure (see further reference there). Zwiebach, in turn, was following the BV-formalism due to Batalin-Vilkovisky 83, Batakin-Fradkin 83.

See also at L-infinity algebra – History.

Discussion in terms of cofibrant resolutions of the Lie operad:

A historical survey is

See also

  • Marilyn Daily, L L_\infty-structures, PhD thesis, 2004 (web)

  • Klaus Bering, Tom Lada, Examples of Homotopy Lie Algebras Archivum Mathematicum (arXiv:0903.5433)

A detailed reference for Lie 2-algebras is:

As models for rational homotopy types

That L L_\infty-algebras are models for rational homotopy theory is implicit in Quillen 69 (via their equivalence with dg-Lie algebras) and was made explicit in Hinich 98. Exposition is in

and genralization to non-connected rational spaces is discussed in

L L_\infty-algebras in physics

The following lists, mainly in chronological order of their discovery, L-∞ algebra structures appearing in physics, notably in supergravity, BV-BRST formalism, deformation quantization, string theory, higher Chern-Simons theory/AKSZ sigma-models and local field theory.

For more see also at higher category theory and physics.

In supergravity

In their equivalent formal dual guise of Chevalley-Eilenberg algebras (see above), L L_\infty-algebras of finite type – in fact super L-∞ algebras – appear in pivotal role in the D'Auria-Fré formulation of supergravity at least since

In the supergravity literature these CE-algebras are referred to as “FDA”s. This is short for “free differential algebra”, which is a slight misnomer for what in mathematics are called semifree dgas (or sometimes “quasi-free” dga-s).

The translation of D'Auria-Fré formalism to explicit (super) L L_\infty-algebra language is made in

connecting them to the higher WZW terms of the Green-Schwarz sigma models of fundamental super p-branes (The brane bouquet).

Further exposition of this includes

See also at supergravity Lie 3-algebra, and supergravity Lie 6-algebra.

Notice that there is a different concept of “Filipov n-Lie algebra” suggested in (Bagger-Lambert 06) to play a role in the description of the conformal field theory in the near horizon limit of black p-branes, notably the BLG model for the conformal worldvolume theory on the M2-brane .

A realization of thse “Filippov 33-Lie algebras” as 2-term L L_\infty-algebras (Lie 2-algebras) equipped with a binary invariant polynomial (“metric Lie 2-algebras”) is in

based on

See also

  • José Figueroa-O'Farrill, section Triple systems and Lie superalgebras in M2-branes, ADE and Lie superalgebras, talk at IPMU 2009 (pdf)

In BV-BRST formalism

The introduction of BV-BRST complexes as a model for the derived critical locus of the action functionals of gauge theories is due to

  • Igor Batalin, Grigori Vilkovisky, Gauge Algebra and Quantization, Phys. Lett. B 102 (1981) 27–31. doi:10.1016/0370-2693(81)90205-7

  • Igor Batalin, Grigori Vilkovisky, Feynman rules for reducible gauge theories, Phys. Lett. B 120 (1983) 166-170.


  • Igor Batalin, Efim Fradkin, A generalized canonical formalism and quantization of reducible gauge theories, Phys. Lett. B122 (1983) 157-164.

  • Igor Batalin, Grigori Vilkovisky, Quantization of Gauge Theories with Linearly Dependent Generators, Phys. Rev. D 28 (10): 2567–258 (1983) doi:10.1103/PhysRevD.28.2567. Erratum-ibid. 30 (1984) 508 doi:10.1103/PhysRevD.30.508

as reviewed in

The understanding that these BV-BRST complexes mathematically are the formal dual Chevalley-Eilenberg algebra of a derived L-∞ algebroid originates around

Discussion in terms of homotopy Lie-Rinehart pairs is due to

  • Lars Kjeseth, Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs, Homology Homotopy Appl. Volume 3, Number 1 (2001), 139-163. (Euclid)

The L-∞ algebroid-structure is also made explicit in (def. 4.1 of v1) of (Sati-Schreiber-Stasheff 09).

In string field theory

The first explicit appearance of L L_\infty-algebras in theoretical physics is the L L_\infty-algebra structure on the BRST complex of the closed bosonic string found in the context of closed bosonic string field theory in

  • Barton Zwiebach, Closed string field theory: Quantum action and the B-V master equation , Nucl.Phys. B390 (1993) 33 (arXiv:hep-th/9206084)

  • Jim Stasheff, Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space Talk given at the Conference on Topics in Geometry and Physics (1992) (arXiv:hep-th/9304061)

Generalization to open-closed bosonic string field theory yields L-∞ algebra interacting with A-∞ algebra:

See also

  • Jim Stasheff, Higher homotopy algebras: String field theory and Drinfeld’s quasiHopf algebras, proceedings of International Conference on Differential Geometric Methods in Theoretical Physics, 1991 (spire)

For more see at string field theory – References – Relation to A-infinity and L-infinity algebras.

In deformation quantization

The general solution of the deformation quantization problem of Poisson manifolds due to

makes crucial use of L-∞ algebra. Later it was understood that indeed L-∞ algebras are equivalently the universal model for infinitesimal deformation theory (of anything), also called formal moduli problems:

In heterotic string theory

Next it was again L L_\infty-algebras of finite type that drew attention. It was eventually understood that the string structures which embody a refinement of the Green-Schwarz anomaly cancellation mechanism in heterotic string theory have a further smooth refinement as G-structures for the string 2-group, which is the Lie integration of a Lie 2-algebra called the string Lie 2-algebra. This is due to

and the relation to the Green-Schwarz mechanism is made explicit in

This article also observes that an analogous situation appears in dual heterotic string theory with the fivebrane Lie 6-algebra in place of the string Lie 2-algebra.

Higher Chern-Simons field theory and AKSZ sigma-models

Ordinary Chern-Simons theory for a simple gauge group is all controled by a Lie algebra 3-cocycle. The generalization of Chern-Simons theory to AKSZ-sigma models was understood to be encoded by symplectic Lie n-algebroids (later re-popularized as “shifted symplectic structures”) in

The globally defined AKSZ action functionals obtained this way were shown in

to be a special case of the higher Lie integration process of

Further exmaples of non-symplectic L L_\infty-Chern-Simons theory obtained this way include 7-dimensional Chern-Simons theory on string 2-connections:

In local prequantum field theory

Infinite-dimensional L L_\infty-algebras that behaved similar to Poisson bracket Lie algebrasPoisson bracket Lie n-algebras – were noticed


these were shown to be the infinitesimal version of the symmetries of prequantum n-bundles as they appear in local prequantum field theory, in higher generalization of how the Poisson bracket is the Lie algebra of the quantomorphism group.

These also encode a homotopy refinement of the Dickey bracket on Noether conserved currents which for Green-Schwarz sigma models reduces to Lie nn-algebras of BPS charges which refine super Lie algebras such as the M-theory super Lie algebra:

This makes concrete the suggestion that there should be L L_\infty-algebra refinements of the Dickey bracket of conserved currents in local field theory that was made in

Comprehesive survey and exposition of this situation is in

In perturbative quantum field theory

Further identification of L-∞ algebras-structure in the Feynman amplitudes/S-matrix of Lagrangian perturbative quantum field theory:

In double field theory

In double field theory:

Last revised on July 23, 2021 at 08:01:38. See the history of this page for a list of all contributions to it.