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\section*{L-infinity algebras in physics}

\hypertarget{general}{}\subsubsection*{{General}}\label{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

\begin{itemize}%
\item [[Jim Stasheff]], \emph{Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras}, in \emph{Quantum groups} Number 1510 in Lecture Notes in Math. Springer, Berlin, 1992


\item [[Tom Lada]], [[Jim Stasheff]], \emph{Introduction to sh Lie algebras for physicists}, Int. J. Theo. Phys. 32 (1993), 1087--1103. (\href{http://arxiv.org/abs/hep-th/9209099}{arXiv:hep-th/9209099})


\item [[Tom Lada]], [[Martin Markl]], \emph{Strongly homotopy Lie algebras} Communications in Algebra Volume 23, Issue 6, (1995) (\href{http://arxiv.org/abs/hep-th/9406095}{arXiv:hep-th/9406095})



\end{itemize}
These authors were following \hyperlink{Zwiebach92}{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 string|closed]] [[bosonic string]] with such a structure. Zwiebach in turn was following the [[BV-formalism]] due to \hyperlink{BatalinVilkovisky83}{Batalin-Vilkovisky 83}, \hyperlink{BatakinFradkin83}{Batakin-Fradkin 83}

See also at \emph{\href{L-infinity-algebra#History}{L-infinity algebra -- History}}.

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

\begin{itemize}%
\item [[Igor Kriz]], [[Peter May]], \emph{Operads, algebras, modules and motives} (\href{http://www.math.uchicago.edu/~may/PAPERS/kmbooklatex.pdf}{pdf})

\end{itemize}
A historical survey is

\begin{itemize}%
\item [[Jim Stasheff]], \emph{Higher homotopy structures, then and now}, talk at \emph{\href{https://www.mpim-bonn.mpg.de/node/6356}{Opening workshop}} of \emph{\href{https://www.mpim-bonn.mpg.de/node/5883}{Higher Structures in Geometry and Physics}}, MPI Bonn 2016 ([[StasheffHomotopyStructuresReview.pdf:file]])

\end{itemize}
See also

\begin{itemize}%
\item Marilyn Daily, \emph{$L_\infty$-structures}, PhD thesis, 2004 (\href{http://www.lib.ncsu.edu/resolver/1840.16/5282}{web})


\item Klaus Bering, [[Tom Lada]], \emph{Examples of Homotopy Lie Algebras} Archivum Mathematicum (\href{http://arxiv.org/abs/0903.5433}{arXiv:0903.5433})



\end{itemize}
A detailed reference for [[Lie 2-algebras]] is:

\begin{itemize}%
\item [[John Baez]] and Alissa Crans, \emph{Higher-dimensional algebra VI: Lie 2-algebras}, \href{http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html}{TAC} 12, (2004), 492--528. (\href{http://arxiv.org/abs/math/0307263}{arXiv:math/0307263})

\end{itemize}
\hypertarget{as_models_for_rational_homotopy_types}{}\subsubsection*{{As models for rational homotopy types}}\label{as_models_for_rational_homotopy_types}

That $L_\infty$-algebras are models for [[rational homotopy theory]] is implicit in \href{rational+homotopy+theory#Quillen69}{Quillen 69} (via their \href{model+structure+on+dg-Lie+algebras#RectificationResolution}{equivalence with dg-Lie algebras}) and was made explicit in \hyperlink{Hinich98}{Hinich 98}. Exposition is in

\begin{itemize}%
\item [[Urtzi Buijs]], [[Yves Félix]], [[Aniceto Murillo]], section 2 of \emph{$L_\infty$-rational homotopy of mapping spaces} (\href{https://arxiv.org/abs/1209.4756}{arXiv:1209.4756}), published as \emph{$L_\infty$-models of based mapping spaces}, J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524.

\end{itemize}
and genralization to non-[[connected topological space|connected]] rational spaces is discussed in

\begin{itemize}%
\item [[Urtzi Buijs]], [[Aniceto Murillo]], \emph{Algebraic models of non-connected spaces and homotopy theory of $L_\infty$-algebras}, Advances in Mathematics 236 (2013): 60-91. (\href{https://arxiv.org/abs/1204.4999}{arXiv:1204.4999})

\end{itemize}
\hypertarget{ReferencesInPhysics}{}\subsubsection*{{$L_\infty$-algebras in physics}}\label{ReferencesInPhysics}

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 \emph{[[higher category theory and physics]]}.

\hypertarget{in_supergravity}{}\paragraph*{{In supergravity}}\label{in_supergravity}

In their equivalent [[formal dual]] guise of [[Chevalley-Eilenberg algebras]] (see \hyperlink{ReformulationInTermsOfSemifreeDGAlgebra}{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

\begin{itemize}%
\item [[Riccardo D'Auria]], [[Pietro Fré]], \emph{[[GeometricSupergravity.pdf:file]]}, Nuclear Physics B201 (1982) 101-140

\end{itemize}
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 dga]]s (or sometimes ``quasi-free'' dga-s).

The translation of [[D'Auria-Fré formulation of supergravity|D'Auria-Fré formalism]] to explicit ([[super L-∞ algebra|super]]) $L_\infty$-algebra language is made in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], example 5 in section 6.5.1, p. 54 of \emph{L-infinity algebra connections and applications to String- and Chern-Simons n-transport}, in Quantum Field Theory, Birkh\"a{}user (2009) 303-424 (\href{http://arxiv.org/abs/0801.3480}{arXiv:0801.3480})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|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 (\href{http://arxiv.org/abs/1308.5264}{arXiv:1308.5264})



\end{itemize}
connecting them to the [[higher WZW terms]] of the [[Green-Schwarz sigma models]] of fundamental [[super p-branes]] ([[schreiber:The brane bouquet]]).

Further exposition of this includes

\begin{itemize}%
\item [[Urs Schreiber]], \emph{\href{https://www.physicsforums.com/insights/homotopy-lie-n-algebras-supergravity/}{Homotopy Lie n-algebras in Supergravity}}, PhysicsForums-Insights 2015

\end{itemize}
See also at \emph{[[supergravity Lie 3-algebra]]}, and \emph{[[supergravity Lie 6-algebra]]}.

Notice that there is a \emph{different} concept of ``Filipov [[n-Lie algebra]]'' suggested in (\href{BLG+model#BaggerLambert06}{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

\begin{itemize}%
\item Sam Palmer, [[Christian Saemann]], section 2 of \emph{M-brane Models from Non-Abelian Gerbes}, JHEP 1207:010, 2012 (\href{http://arxiv.org/abs/1203.5757}{arXiv:1203.5757})


\item [[Patricia Ritter]], [[Christian Saemann]], section 2.5 of \emph{Lie 2-algebra models}, JHEP 04 (2014) 066 (\href{http://arxiv.org/abs/1308.4892}{arXiv:1308.4892})



\end{itemize}
based on

\begin{itemize}%
\item Paul de Medeiros, [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], [[Patricia Ritter]], \emph{On the Lie-algebraic origin of metric 3-algebras}, Commun.Math.Phys.290:871-902,2009 (\href{http://arxiv.org/abs/0809.1086}{arXiv:0809.1086})

\end{itemize}
See also

\begin{itemize}%
\item [[José Figueroa-O'Farrill]], section \emph{Triple systems and Lie superalgebras} in \emph{M2-branes, ADE and Lie superalgebras}, talk at IPMU 2009 (\href{http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Hongo.pdf}{pdf})

\end{itemize}
\hypertarget{ReferencesBVBRSTFormalism}{}\paragraph*{{In BV-BRST formalism}}\label{ReferencesBVBRSTFormalism}

The introduction of [[BV-BRST complexes]] as a model for the [[derived critical locus]] of the [[action functionals]] of [[gauge theories]] is due to

\begin{itemize}%
\item [[Igor Batalin]], [[Grigori Vilkovisky]], \emph{Gauge Algebra and Quantization}, Phys. Lett. B 102 (1981) 27--31. doi:10.1016/0370-2693(81)90205-7


\item [[Igor Batalin]], [[Grigori Vilkovisky]], \emph{Feynman rules for reducible gauge theories}, Phys. Lett. B 120 (1983) 166-170. doi:10.1016/0370-2693(83)90645-7


\item [[Igor Batalin]], [[Efim Fradkin]], \emph{A generalized canonical formalism and quantization of reducible gauge theories}, Phys. Lett. B122 (1983) 157-164.


\item [[Igor Batalin]], [[Grigori Vilkovisky]], \emph{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



\end{itemize}
as reviewed in

\begin{itemize}%
\item [[Marc Henneaux]], [[Claudio Teitelboim]], \emph{[[Quantization of Gauge Systems]]}, Princeton University Press 1992. xxviii+520 pp.


\item [[Joaquim Gomis]], J. Paris, S. Samuel, \emph{Antibrackets, Antifields and Gauge Theory Quantization} (\href{http://arxiv.org/abs/hep-th/9412228}{arXiv:hep-th/9412228})



\end{itemize}
The understanding that these [[BV-BRST complexes]] mathematically are the [[formal dual]] [[Chevalley-Eilenberg algebra]] of a [[derived L-∞ algebroid]] originates around

\begin{itemize}%
\item [[Jim Stasheff]], \emph{Homological Reduction of Constrained Poisson Algebras}, J. Differential Geom. Volume 45, Number 1 (1997), 221-240 (\href{http://arxiv.org/abs/q-alg/9603021}{arXiv:q-alg/9603021}, \href{https://projecteuclid.org/euclid.jdg/1214459757}{Euclid})


\item [[Jim Stasheff]], \emph{The (secret?) homological algebra of the Batalin-Vilkovisky approach} (\href{http://arxiv.org/abs/hep-th/9712157}{arXiv:hep-th/9712157})



\end{itemize}
Discussion in terms of homotopy [[Lie-Rinehart pairs]] is due to

\begin{itemize}%
\item [[Lars Kjeseth]], \emph{Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs}, Homology Homotopy Appl. Volume 3, Number 1 (2001), 139-163. (\href{https://projecteuclid.org/euclid.hha/1140370269}{Euclid})

\end{itemize}
The [[L-∞ algebroid]]-structure is also made explicit in (\href{http://arxiv.org/abs/0910.4001v1}{def. 4.1 of v1}) of (\hyperlink{SatiSchreiberStasheff09}{Sati-Schreiber-Stasheff 09}).

\hypertarget{in_string_field_theory}{}\paragraph*{{In string field theory}}\label{in_string_field_theory}

The first \emph{explicit} appearance of $L_\infty$-algebras in theoretical physics is the $L_\infty$-algebra structure on the [[BRST complex]] of the [[closed string|closed]] [[bosonic string]] found in the context of closed bosonic [[string field theory]] in

\begin{itemize}%
\item [[Barton Zwiebach]], \emph{Closed string field theory: Quantum action and the B-V master equation} , Nucl.Phys. B390 (1993) 33 (\href{http://arxiv.org/abs/hep-th/9206084}{arXiv:hep-th/9206084})


\item [[Jim Stasheff]], \emph{Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space} Talk given at the \emph{Conference on Topics in Geometry and Physics} (1992) (\href{http://arxiv.org/abs/hep-th/9304061}{arXiv:hep-th/9304061})



\end{itemize}
Generalization to open-closed bosonic string field theory yields [[L-∞ algebra]] interacting with [[A-∞ algebra]]:

\begin{itemize}%
\item [[Hiroshige Kajiura]], \emph{Homotopy Algebra Morphism and Geometry of Classical String Field Theory} (2001) (\href{http://arxiv.org/abs/hep-th/0112228}{arXiv:hep-th/0112228})


\item [[Hiroshige Kajiura]], [[Jim Stasheff]], \emph{Homotopy algebras inspired by classical open-closed string field theory}, Comm. Math. Phys. 263 (2006) 553--581 (2004) (\href{http://arxiv.org/abs/math/0410291}{arXiv:math/0410291})


\item [[Martin Markl]], \emph{Loop Homotopy Algebras in Closed String Field Theory} (1997) (\href{http://arxiv.org/abs/hep-th/9711045}{arXiv:hep-th/9711045})



\end{itemize}
See also

\begin{itemize}%
\item [[Jim Stasheff]], \emph{Higher homotopy algebras: String field theory and Drinfeld's quasiHopf algebras}, proceedings of \emph{International Conference on Differential Geometric Methods in Theoretical Physics}, 1991 (\href{https://inspirehep.net/record/327712}{spire})

\end{itemize}
For more see at \emph{\href{string%20field%20theory#ReferencesHomotopyAlgebra}{string field theory -- References -- Relation to A-infinity and L-infinity algebras}}.

\hypertarget{in_deformation_quantization}{}\paragraph*{{In deformation quantization}}\label{in_deformation_quantization}

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

\begin{itemize}%
\item [[Maxim Kontsevich]], \emph{Deformation quantization of Poisson manifolds}, Lett. Math. Phys. \textbf{66} (2003), no. 3, 157--216, (\href{http://arxiv.org/abs/q-alg/9709040}{arXiv:q-alg/9709040}).

\end{itemize}
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]]:

\begin{itemize}%
\item [[Vladimir Hinich]], \emph{DG coalgebras as formal stacks} (\href{http://arxiv.org/abs/math/9812034}{arXiv:9812034})


\item [[Jacob Lurie]], \emph{[[Formal Moduli Problems]]}


\item [[Jonathan Pridham]], \emph{Unifying derived deformation theories}, Adv. Math. 224 (2010), no.3, 772-826 (\href{http://arxiv.org/abs/0705.0344}{arXiv:0705.0344})



\end{itemize}
\hypertarget{in_heterotic_string_theory}{}\paragraph*{{In heterotic string theory}}\label{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 \emph{[[string Lie 2-algebra]]}. This is due to

\begin{itemize}%
\item [[John Baez]], [[Alissa Crans]], [[Urs Schreiber]], [[Danny Stevenson]], \emph{From loop groups to 2-groups}, \emph{Homotopy, Homology and Applications} \textbf{9} (2007), 101-135. (\href{http://arxiv.org/abs/math.QA/0504123}{arXiv:math.QA/0504123})


\item [[André Henriques]], \emph{Integrating $L_\infty$ algebras}, Compos. Math. \textbf{144} (2008), no. 4, 1017--1045 (\href{http://dx.doi.org/10.1112/S0010437X07003405}{doi},\href{http://arxiv.org/abs/math.AT/0603563}{math.AT/0603563})



\end{itemize}
and the relation to the Green-Schwarz mechanism is made explicit in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Twisted Differential String and Fivebrane Structures]]}, Communications in Mathematical Physics, 2012, Volume 315, Issue 1, pp 169-213 (\href{http://arxiv.org/abs/0910.4001}{arXiv:0910.4001})

\end{itemize}
This article also observes that an analogous situation appears in [[dual heterotic string theory]] with the \emph{[[fivebrane Lie 6-algebra]]} in place of the string Lie 2-algebra.

\hypertarget{higher_chernsimons_field_theory_and_aksz_sigmamodels}{}\paragraph*{{Higher Chern-Simons field theory and AKSZ sigma-models}}\label{higher_chernsimons_field_theory_and_aksz_sigmamodels}

Ordinary [[Chern-Simons theory]] for a simple gauge group is all controled by a [[Lie algebra cohomology|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

\begin{itemize}%
\item [[Dmitry Roytenberg]], \emph{Courant algebroids, derived brackets and even symplectic supermanifolds} PhD thesis (\href{http://arxiv.org/abs/math/9910078}{arXiv:9910078})


\item [[Pavol Ševera]], \emph{[[Some title containing the words ``homotopy'' and ``symplectic'', e.g. this one]]}, based on a talk at ``Poisson 2000'', CIRM Marseille, June 2000; (\href{http://arxiv.org/abs/math/0105080}{arXiv:0105080})


\item [[Dmitry Roytenberg]], \emph{On the structure of graded symplectic supermanifolds and Courant algebroids} in \emph{Quantization, Poisson Brackets and Beyond} , [[Theodore Voronov]] (ed.), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002 (\href{http://arxiv.org/abs/math/0203110}{arXiv})


\item [[Dmitry Roytenberg]], \emph{AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories} Lett.Math.Phys.79:143-159,2007 (\href{http://arxiv.org/abs/hep-th/0608150}{arXiv:hep-th/0608150}).



\end{itemize}
The globally defined AKSZ action functionals obtained this way were shown in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]}, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 (\href{http://arxiv.org/abs/1108.4378}{arXiv:1108.4378})

\end{itemize}
to be a special case of the higher Lie integration process of

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech cocycles for differential characteristic classes]]}, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (\href{http://arxiv.org/abs/1011.4735}{arXiv:1011.4735})

\end{itemize}
Further exmaples of non-symplectic $L_\infty$-Chern-Simons theory obtained this way include [[7-dimensional Chern-Simons theory]] on [[differential string structure|string 2-connections]]:

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:7d Chern-Simons theory and the 5-brane|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

\end{itemize}
\hypertarget{in_local_prequantum_field_theory}{}\paragraph*{{In local prequantum field theory}}\label{in_local_prequantum_field_theory}

Infinite-dimensional $L_\infty$-algebras that behaved similar to [[Poisson bracket]] [[Lie algebras]] -- \emph{[[Poisson bracket Lie n-algebras]]} -- were noticed

\begin{itemize}%
\item [[Chris Rogers]], \emph{$L_\infty$ algebras from multisymplectic geometry} , Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (\href{http://arxiv.org/abs/1005.2230}{arXiv:1005.2230}, \href{http://link.springer.com/article/10.1007%2Fs11005-011-0493-x}{journal}).


\item [[Chris Rogers]], \emph{Higher symplectic geometry} PhD thesis (2011) (\href{http://arxiv.org/abs/1106.4068}{arXiv:1106.4068})



\end{itemize}
In

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:L-∞ algebras of local observables from higher prequantum bundles ]]}, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 -- 142 (\href{http://arxiv.org/abs/1304.6292}{arXiv:1304.6292})

\end{itemize}
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 theorem|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]]:

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Lie n-algebras of BPS charges]]} (\href{http://arxiv.org/abs/1507.08692}{arXiv:1507.08692})


\item [[Igor Khavkine]], [[Urs Schreiber]], \emph{[[schreiber:Lie n-algebras of higher Noether currents ]]}



\end{itemize}
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

\begin{itemize}%
\item [[Glenn Barnich]], [[Ronald Fulp]], [[Tom Lada]], [[Jim Stasheff]], \emph{The sh Lie structure of Poisson brackets in field theory} (\href{http://arxiv.org/abs/hep-th/9702176}{arXiv:hep-th/9702176})

\end{itemize}
Comprehesive survey and exposition of this situation is in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:Higher Prequantum Geometry]]}, in [[Gabriel Catren]], [[Mathieu Anel]] (eds.), \emph{\href{https://ncatlab.org/nlab/show/New+Spaces+for+Mathematics+and+Physics}{New Spaces for Mathematics and Physics}}, 2016

\end{itemize}
\hypertarget{in_perturbative_quantum_field_theory}{}\paragraph*{{In perturbative quantum field theory}}\label{in_perturbative_quantum_field_theory}

Further identification of [[L-∞ algebras]]-[[structure]] in the [[Feynman amplitudes]]/[[S-matrix]] of [[Lagrangian field theory|Lagrangian]] [[perturbative quantum field theory]]:

\begin{itemize}%
\item [[Markus Fröb]], \emph{Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories} \href{https://dx.doi.org/10.1007/s00220-019-03558-6}{Communications in Mathematical Physics} 2019 (online first) (\href{https://arxiv.org/abs/1803.10235}{arXiv:1803.10235})


\item [[Alex Arvanitakis]], \emph{The $L_\infty$-algebra of the S-matrix} (\href{https://arxiv.org/abs/1903.05643}{arXiv:1903.05643})



\end{itemize}
\hypertarget{in_double_field_theory}{}\paragraph*{{In double field theory}}\label{in_double_field_theory}

In [[double field theory]]:

\begin{itemize}%
\item [[Andreas Deser]], [[Jim Stasheff]], \emph{Even symplectic supermanifolds and double field theory}, Communications in Mathematical Physics November 2015, Volume 339, Issue 3, pp 1003-1020 (\href{http://arxiv.org/abs/1406.3601}{arXiv:1406.3601})


\item [[Olaf Hohm]], [[Barton Zwiebach]], \emph{$L_\infty$ Algebras and Field Theory} (\href{https://arxiv.org/abs/1701.08824}{arXiv:1701.08824})



\end{itemize}
\hypertarget{related_expositions}{}\subsubsection*{{Related expositions}}\label{related_expositions}

\begin{itemize}%
\item [[fiber bundles in physics]]


\item [[higher category theory and physics]]


\item [[string theory FAQ]]


\item [[motives in physics]]


\item [[Hilbert's sixth problem]]


\item [[model theory and physics]]


\item [[motivation for sheaves, cohomology and higher stacks]]


\item [[applications of (higher) category theory]]


\item [[motivation for higher differential geometry]]


\item [[motivation for cohesion]]



\end{itemize}


\end{document}