The concept of L-∞ algebras as graded vector spaces equipped with $n$-ary brackets satisfying a generalized Jacobi identity was introduced in generality in
Jim Stasheff, Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras, in Quantum groups Number 1510 in Lecture Notes in Math. Springer, Berlin, 1992
Tom Lada, Jim Stasheff, Introduction to sh Lie algebras for physicists, Int. J. Theo. Phys. 32 (1993), 1087–1103. (arXiv:hep-th/9209099)
Tom Lada, Martin Markl, Strongly homotopy Lie algebras Communications in Algebra Volume 23, Issue 6, (1995) (arXiv:hep-th/9406095)
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
See also at L-infinity algebra – History.
A discussion in terms of resolutions of the Lie operad is for instance in
A historical survey is
See also
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:
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
and genralization to non-connected rational spaces is discussed in
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 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
Hisham Sati, Urs Schreiber, Jim Stasheff, example 5 in section 6.5.1, p. 54 of L-infinity algebra connections and applications to String- and Chern-Simons n-transport, in Quantum Field Theory, Birkhäuser (2009) 303-424 (arXiv:0801.3480)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018 (arXiv:1308.5264)
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 $3$-Lie algebras” as 2-term $L_\infty$-algebras (Lie 2-algebras) equipped with a binary invariant polynomial (“metric Lie 2-algebras”) is in
Sam Palmer, Christian Saemann, section 2 of M-brane Models from Non-Abelian Gerbes, JHEP 1207:010, 2012 (arXiv:1203.5757)
Patricia Ritter, Christian Saemann, section 2.5 of Lie 2-algebra models, JHEP 04 (2014) 066 (arXiv:1308.4892)
based on
See also
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
Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems, Princeton University Press 1992. xxviii+520 pp.
Joaquim Gomis, J. Paris, S. Samuel, Antibrackets, Antifields and Gauge Theory Quantization (arXiv:hep-th/9412228)
The understanding that these BV-BRST complexes mathematically are the formal dual Chevalley-Eilenberg algebra of a derived L-∞ algebroid originates around
Jim Stasheff, Homological Reduction of Constrained Poisson Algebras, J. Differential Geom. Volume 45, Number 1 (1997), 221-240 (arXiv:q-alg/9603021, Euclid)
Jim Stasheff, The (secret?) homological algebra of the Batalin-Vilkovisky approach (arXiv:hep-th/9712157)
Discussion in terms of homotopy Lie-Rinehart pairs is due to
The L-∞ algebroid-structure is also made explicit in (def. 4.1 of v1) of (Sati-Schreiber-Stasheff 09).
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:
Hiroshige Kajiura, Homotopy Algebra Morphism and Geometry of Classical String Field Theory (2001) (arXiv:hep-th/0112228)
Hiroshige Kajiura, Jim Stasheff, Homotopy algebras inspired by classical open-closed string field theory, Comm. Math. Phys. 263 (2006) 553–581 (2004) (arXiv:math/0410291)
Martin Markl, Loop Homotopy Algebras in Closed String Field Theory (1997) (arXiv:hep-th/9711045)
See also
For more see at string field theory – References – Relation to A-infinity and L-infinity algebras.
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:
Vladimir Hinich, DG coalgebras as formal stacks (arXiv:9812034)
Jonathan Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no.3, 772-826 (arXiv:0705.0344)
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
John Baez, Alissa Crans, Urs Schreiber, Danny Stevenson, From loop groups to 2-groups, Homotopy, Homology and Applications 9 (2007), 101-135. (arXiv:math.QA/0504123)
André Henriques, Integrating $L_\infty$ algebras, Compos. Math. 144 (2008), no. 4, 1017–1045 (doi,math.AT/0603563)
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.
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
Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis (arXiv:9910078)
Pavol Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one, based on a talk at “Poisson 2000”, CIRM Marseille, June 2000; (arXiv:0105080)
Dmitry Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids in Quantization, Poisson Brackets and Beyond , Theodore Voronov (ed.), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002 (arXiv)
Dmitry Roytenberg, AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories Lett.Math.Phys.79:143-159,2007 (arXiv:hep-th/0608150).
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:
Infinite-dimensional $L_\infty$-algebras that behaved similar to Poisson bracket Lie algebras – Poisson bracket Lie n-algebras – were noticed
Chris Rogers, $L_\infty$ algebras from multisymplectic geometry , Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (arXiv:1005.2230, journal).
Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068)
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:
Hisham Sati, Urs Schreiber, Lie n-algebras of BPS charges (arXiv:1507.08692)
Igor Khavkine, Urs Schreiber, Lie n-algebras of higher Noether currents
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
Further identification of L-∞ algebras-structure in the Feynman amplitudes/S-matrix of Lagrangian perturbative quantum field theory:
Markus Fröb, Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories (arXiv:1803.10235)
Alex Arvanitakis, The $L_\infty$-algebra of the S-matrix (arXiv:1903.05643)
Andreas Deser, Jim Stasheff, Even symplectic supermanifolds and double field theory, Communications in Mathematical Physics November 2015, Volume 339, Issue 3, pp 1003-1020 (arXiv:1406.3601)
Olaf Hohm, Barton Zwiebach, $L_\infty$ Algebras and Field Theory (arXiv:1701.08824)
Last revised on June 12, 2019 at 05:55:50. See the history of this page for a list of all contributions to it.