The concept of L-∞ algebras as graded vector spaces equipped with $n$-ary brackets satisfying a generalized Jacobi identity is due to:
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 (doi:10.1007/BFb0101184).
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)
Maxim Kontsevich, Section 4.3 of: Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66 (2003) 157-216 (arXiv:q-alg/9709040, doi:10.1023/B:MATH.0000027508.00421.bf)
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 $L_\infty$-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:
Igor Kriz, Peter May, p. 28 of: Operads, algebras, modules and motives, Astérisque 233, Société Mathématique de France (1995) (pdf, numdam:AST_1995__233__1_0)
Jean-Louis Loday, Bruno Vallette, Sec. 3.2.12 and onwards in: Algebraic Operads, Grundlehren der mathematischen Wissenschaften 346, Springer 2012 (ISBN 978-3-642-30362-3, pdf)
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, doi:10.1007/978-3-7643-8736-5_17)
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 Communications in Mathematical Physics 2019 (online first) (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 July 23, 2021 at 08:01:38. See the history of this page for a list of all contributions to it.