# nLab L-infinity algebras in physics

### General

The concept of L-∞ algebras as graded vector spaces equipped with $n$-ary brackets satisfying a generalized Jacobi identity was introduced in generality in

These authors were following Zwiebach 92, who had found in his work on closed string field theory that the n-point functions equip the BRST complex of the closed bosonic string with such a structure. Zwiebach in turn was following the BV-formalism due to Batalin-Vilkovisky 83, Batakin-Fradkin 83

A discussion in terms of resolutions of the Lie operad is for instance in

A historical survey is

• Marilyn Daily, $L_\infty$-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 $L_\infty$-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

• Urtzi Buijs, Yves Félix, Aniceto Murillo, section 2 of $L_\infty$-rational homotopy of mapping spaces (arXiv:1209.4756), published as $L_\infty$-models of based mapping spaces, J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524.

and genralization to non-connected rational spaces is discussed in

• Urtzi Buijs, Aniceto Murillo, Algebraic models of non-connected spaces and homotopy theory of $L_\infty$-algebras, Advances in Mathematics 236 (2013): 60-91. (arXiv:1204.4999)

### $L_\infty$-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.

#### In supergravity

In their equivalent formal dual guise of Chevalley-Eilenberg algebras (see above), $L_\infty$-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) $L_\infty$-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

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 $3$-Lie algebras” as 2-term $L_\infty$-algebras (Lie 2-algebras) equipped with a binary invariant polynomial (“metric Lie 2-algebras”) is in

based on

• 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.

doi:10.1016/0370-2693(83)90645-7

• 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 $L_\infty$-algebras in theoretical physics is the $L_\infty$-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:

• 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)

#### 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 $L_\infty$-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 $L_\infty$-Chern-Simons theory obtained this way include 7-dimensional Chern-Simons theory on string 2-connections:

#### In local prequantum field theory

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

In

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 $n$-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 $L_\infty$-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

Last revised on December 6, 2019 at 10:10:01. See the history of this page for a list of all contributions to it.