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 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

Jim Stasheff, Higher homotopy structures, then and now, talk at Opening workshop of Higher Structures in Geometry and Physics, MPI Bonn 2016 (pdf, arXiv:1809.02526)

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.

For more see also at higher category theory and physics.

In supergravity

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:

Urs Schreiber, SuGra 3-Connection Reloaded (2006)
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

Paul de Medeiros, José Figueroa-O'Farrill, Elena Méndez-Escobar, Patricia Ritter, On the Lie-algebraic origin of metric 3-algebras, Commun.Math.Phys.290:871-902,2009 (arXiv:0809.1086)

Supergravity C-Field gauge algebra

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):

Hisham Sati, (4.9) in: Geometric and topological structures related to M-branes, in Superstrings, Geometry, Topology, and $C^\ast$ -algebras, Proc. Symp. Pure Math. 81 (2010) 181-236 [ams:pspum/081, arXiv:1001.5020]

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]

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

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

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).

The extraction of $L_\infty$ -algebras from the formal neighbourhood of a derived critical locus is maybe first made explicit in:

Maxim Grigoriev, Dmitry Rudinsky, Notes on the $L_\infty$ -approach to local gauge field theories $[$ arXiv2303.08990 $]$

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:

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)

In deformation quantization

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

Kontsevich 97

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)
Jacob Lurie, Formal Moduli Problems
Jonathan Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no.3, 772-826 (arXiv:0705.0344)

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

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

Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics, 2012, Volume 315, Issue 1, pp 169-213 (arXiv:0910.4001)

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

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

Domenico Fiorenza, Chris Rogers, Urs Schreiber, AKSZ Sigma-Models in Higher Chern-Weil Theory, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 (arXiv:1108.4378)

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

Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (arXiv:1011.4735)

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

Domenico Fiorenza, Hisham Sati, Urs Schreiber, Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory, Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014) p. 229–321

In local prequantum field theory

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)

Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 (arXiv:1304.6292)

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

Glenn Barnich, Ronald Fulp, Tom Lada, Jim Stasheff, The sh Lie structure of Poisson brackets in field theory (arXiv:hep-th/9702176)

Comprehesive survey and exposition of this situation is in

Urs Schreiber, Higher Prequantum Geometry, in Gabriel Catren, Mathieu Anel (eds.), New Spaces for Mathematics and Physics, 2016

In perturbative quantum field theory

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)

In double field theory

In double field theory:

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)

Related expositions

Last revised on September 27, 2024 at 08:23:26. See the history of this page for a list of all contributions to it.