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 [doi:10.1007/BF00671791, arXiv:hep-th/9209099]
Tom Lada, Martin Markl, Strongly homotopy Lie algebras, Communications in Algebra 23 6 (1995) [doi:10.1080/00927879508825335, 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)
Comprehensive survey with emphasis on $L_\infty$-algebra cohomology:
Review for the special case of Lie 2-algebras with emphasis on the perspective of categorification:
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.
Implicitly, in their equivalent formal dual guise of Chevalley-Eilenberg algebras (see above), $L_\infty$-algebras of finite type – in fact super L-∞ algebras – play a pivotal role in the D'Auria-Fré formulation of supergravity at least since
Peter van Nieuwenhuizen, Free Graded Differential Superalgebras, in: Group Theoretical Methods in Physics, Lecture Notes in Physics 180, Springer (1983) 228–247 [doi:10.1007/3-540-12291-5_29, spire:182644]
Riccardo D'Auria, Pietro Fré, Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B 201 (1982) 101-140 [doi:10.1016/0550-3213(82)90376-5, errata]
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Ch III.6 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, ch III.6: pdf]
Pietro Fré, §6.3 in: Gravity, a Geometrical Course, Volume 2: Black Holes, Cosmology and Introduction to Supergravity, Springer (2013) [doi:10.1007/978-94-007-5443-0]
Leonardo Castellani, §6 in: Supergravity in the group-geometric framework: a primer, Fortschr. Phys. 66 4 (2018) [doi:10.1002/prop.201800014, arXiv:1802.03407]
where they are called “free differential algebras” (“FDA”s, apparently following can Nieuwenhuizen 1982), which is a misnomer for what in mathematics are called semifree dgas (since it is only the underlying graded-commutative algebra that is required to be free, the differential is crucially not free in general, otherwise one has just a Weil algebra).
The translation of D'Auria-Fré formalism (“FDA”s) to explicit (super) $L_\infty$-algebra language was made in:
Hisham Sati, Urs Schreiber, Jim Stasheff, example 5 in section 6.5.1, p. 54 of: $L_\infty$ 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, Int. J. of Geometric Methods in Modern Physics 12 02 (2015) 1550018 [arXiv:1308.5264, doi:10.1142/S0219887815500188]
connecting them to the higher WZW terms of the Green-Schwarz sigma models of fundamental super p-branes (The brane bouquet).
See also at supergravity Lie 3-algebra, and supergravity Lie 6-algebra.
Further exposition and review of the (dual) identification of supergravity “FDAs” with super $L_\infty$-algebras:
Urs Schreiber, Homotopy Lie n-algebras in Supergravity, PhysicsForums-Insights (2015)
Branislav Jurčo, Christian Saemann, Urs Schreiber, Martin Wolf: Higher Structures in M-Theory, Introduction to Higher Structures in M-Theory 2018, Fortsch. d. Phys. 67 8-9 (2019) 1910001 [arXiv:1903.02807, doi:10.1002/prop.201910001]
Domenico Fiorenza, Hisham Sati, Urs Schreiber: The rational higher structure of M-theory, in Higher Structures in M-Theory 2018, Fortschr. der Physik 67 8-9 (2019) 1910017 [arXiv:1903.02834, doi:10.1002/prop.201910017]
Notice that there is a different concept of “Filipov n-Lie algebra” suggested by Bagger& Lambert 2006 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 these “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
Identifying the super-graded gauge algebra of the C-field in D=11 supergravity (with non-trivial super Lie bracket $[v_3, v_3] = -v_6$):
Eugene Cremmer, Bernard Julia, H. Lu, Christopher Pope, Equation (2.6) of Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities, Nucl.Phys. B 535 (1998) 242-292 [doi:10.1016/S0550-3213(98)00552-5, arXiv:hep-th/9806106]
I. V. Lavrinenko, H. Lu, Christopher N. Pope, Kellogg S. Stelle, (3.4) in: Superdualities, Brane Tensions and Massive IIA/IIB Duality, Nucl. Phys. B 555 (1999) 201-227 [doi:10.1016/S0550-3213(99)00307-7, arXiv:hep-th/9903057]
Jussi Kalkkinen, Kellogg S. Stelle, (75) of: Large Gauge Transformations in M-theory, J. Geom. Phys. 48 (2003) 100-132 [doi:10.1016/S0393-0440(03)00027-5, arXiv:hep-th/0212081]
Igor A. Bandos, Alexei J. Nurmagambetov, Dmitri P. Sorokin, (86) in: Various Faces of Type IIA Supergravity, Nucl.Phys. B 676 (2004) 189-228 [doi:10.1016/j.nuclphysb.2003.10.036, arXiv:hep-th/0307153]
Identification as an $L_\infty$-algebra (a dg-Lie algebra, in this case):
and identificatoin with the rational Whitehead $L_\infty$-algebra (the rational Quillen model) of the 4-sphere (cf. Hypothesis H):
Hisham Sati, Alexander Voronov, (13) in: Mysterious Triality and M-Theory [arXiv:2212.13968]
Hisham Sati, Urs Schreiber, (22) in: Flux Quantization on Phase Space [arXiv:2312.12517]
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 extraction of $L_\infty$-algebras from the formal neighbourhood of a derived critical locus is maybe first made explicit in:
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)
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