L-infinity-algebra in nLab
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Context
-Lie theory
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
Higher algebra
Algebraic theories
Algebras and modules
Higher algebras
-
symmetric monoidal (∞,1)-category of spectra
Model category presentations
Geometry on formal duals of algebras
Theorems
Rational homotopy theory
and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
dg-Algebra
Rational spaces
PL de Rham complex
Sullivan models
Contents
- Idea
- History
- Definition
- In terms of algebras over an operad
- In terms of higher brackets
- In terms of semifree differential coalgebra
- In terms of semifree differential algebra
- In terms of algebras over an operad
- Examples
- Properties
- Related concepts
- References
- General
- As models for rational homotopy types
- -algebras in physics
- In supergravity
- Supergravity C-Field gauge algebra
- In BV-BRST formalism
- In string field theory
- In deformation quantization
- In heterotic string theory
- Higher Chern-Simons field theory and AKSZ sigma-models
- In local prequantum field theory
- In perturbative quantum field theory
- In double field theory
- Related expositions
Idea
-algebras (or strong homotopy Lie algebras) are a higher generalization (a “vertical categorification”) of Lie algebras: in an -algebra the Jacobi identity is allowed to hold (only) up to higher coherent homotopy.
An -algebra that is concentrated in lowest degree is an ordinary Lie algebra. If it is concentrated in the lowest two degrees is is a Lie 2-algebra, etc.
From another perspective: an -algebra is a Lie ∞-algebroid with a single object.
-algebras are infinitesimal approximations of smooth ∞-groups in analogy to how an ordinary Lie algebra is an infinitesimal approximation of a Lie group. Under Lie integration every -algebra “exponentiates” to a smooth ∞-group .
History
Definition
In terms of algebras over an operad
An -algebra is an algebra over an operad in the category of chain complexes over the L-∞ operad.
In the following we spell out in detail what this means in components.
In terms of higher brackets
We now state the definition of -algebras that is most directly related to the traditional definition of ordinary Lie algebras, namely as -graded vector space equipped with -ary multilinear and graded-skew symmetric maps – the “brackets” – that satisfy a generalization of the Jacobi identity.
To that end, we here choose grading conventions such that the following definition of -algebras reduces to that of ordinary Lie algebras when is concentrated in degree zero. Moreover we take the differential of the underlying chain complex of the -algebra to have degree (“homological grading”). Together this means in particular that is a Lie n-algebra for , , if it is concentrated in degrees 0 to .
Beware that there are also other conventions possible, and there are other conventions in use, for both these choices, leading to different signs in the following formulas.
Definition
(graded signature of a permuation)
Let be a -graded vector space, and for let
be an n-tuple of elements of of homogeneous degree , i.e. such that .
For a permutation of elements, write for the signature of the permutation, which is by definition equal to if is the composite of permutations that each exchange precisely one pair of neighboring elements.
We say that the -graded signature of
is the product of the signature of the permutation with a factor of for each interchange of neighbours to involved in the decomposition of the permuation as a sequence of swapping neighbour pairs.
Definition
An -algebra is
-
a -graded vector space ;
-
for each , a multilinear map, called the -ary bracket, of the form
and of degree
(if one includes here then one speaks of a curved L-infinity algebra)
such that the following conditions hold:
-
(graded skew symmetry) each is graded antisymmetric, in that for every permutation of elements and for every n-tuple of homogeneously graded elements then
where is the -graded signature of the permuation , according to def. ;
-
(strong homotopy Jacobi identity) for all , and for all -tuples of homogeneously graded elements the following equation holds
(1)
where the inner sum runs over all -unshuffles and where is the graded signature sign from def. .
Example
In lowest degrees the generalized Jacobi identity says that
-
for : the unary map squares to 0:
-
for : the unary map is a graded derivation of the binary map
hence
Example
When all higher brackets vanish, then for :
this is the graded Jacobi identity. So in this case the -algebra is equivalently a dg-Lie algebra.
Example
When is possibly non-vanishing, then on elements on which vanishes, the generalized Jacobi identity for gives
This shows that the Jacobi identity holds up to an “exact” term, hence up to homotopy.
In terms of semifree differential coalgebra
In (Lada-Stasheff 92) it was pointed out that the higher brackets of an -algebra (def. ) induce on the graded-co-commutative cofree coalgebra over the underlying graded vector space the structure of a differential graded coalgebra, with differential the sum of the higher brackets, extended as graded coderivations. The higher Jacobi identity is equivalently the condition that . In (Lada-Markl 94) it was observed that conversely, such “semifree” differential graded coalgebras are an equivalent incarnation of -algebras.
(If one uses unital dg-co-algebras then the -algbras encoded with way are generally curved L-infinity algebras. To restrict to the non-curved one one either considers co-augmented unital dg-co-algebras or non-unital coalgebras.)
Notice that this immediately imples that if is degreewise finite dimensional, then passing to dual vector spaces turns semifree differential graded coalgebra into semifree differential graded algebras, which hence are opposite-equivalent to -algebras of finite type. For an ordinary finite dimensional Lie algebra, then this dg-algebras is its Chevalley-Eilenberg algebra, hence we may generally speak of Chevalley-Eilenberg algebras of -algebras of finite type (and also more generally, if one invokes pro-objects, see at model structure for L-infinity algebras – Use of pro-dg-algebras ).
In term of the operadic definition of -algebras above this equivalence is an incarnation of the Koszul duality between the Lie operad and the commutative operad.
We now spell out this dg-coalgebraic incarnation of -algebras.
A (connected) -algebra is
-
an -graded vector space ;
-
equipped with a differential of degree on the free graded co-commutative coalgebra over that squares to 0
Here the free graded co-commutative co-algebra is, as a vector space, the same as the graded Grassmann algebra whose elements we write as
etc (where the is just a funny way to write the wedge , in order to remind us that:…)
but thought of as equipped with the standard coproduct
(work out or see the references for the signs and prefacors).
Since this is a free graded co-commutative coalgebra, one can see that any differential
on it is fixed by its value “on cogenerators” (a statement that is maybe unfamiliar, but simply the straightforward dual of the more familar statement to which we come below, that differentials on free graded algebras are fixed by their action on generators) which means that we can decompose as
where each acts as when evaluated on a homogeneous element of the form and is then uniquely extended to all of by extending it as a coderivation on a coalgebra.
For instance acts on homogeneous elements of word lenght 3 as
exercise for the reader: spell this all out more in detail with all the signs and everyrthing. Possibly by looking it up in the references given below.
Using this, one checks that the simple condition that squares to 0 is precisely equivalent to the infinite tower of generalized Jacobi identities:
So in conclusion we have:
An -algebra is a dg-coalgebra whose underlying coalgebra is cofree and concentrated in negative degree.
In terms of semifree differential algebra
The reformulation of an -algebra as simply a semi-co-free graded-co-commutative coalgebra is a useful repackaging of the original definition, but the coalgebraic aspect tends to be not only unfamiliar, but also a bit inconvenient. At least when the graded vector space is degreewise finite dimensional, we may simply pass to its degreewise dual graded vector space .
(Fully generally the following works when using not just dg-algebras but pro-objects in dg-algebras, see at model structure for L-infinity algebras – Use of pro-dg-algebras).
Its Grassmann algebra is then naturally equipped with an ordinary differential which acts on as
When the grading-dust has settled one finds that with
with the ground field in degree 0, the degree 1-elements of in degree 1, etc, that is of degree +1 and of course squares to 0
This means that we have a semifree dga
In the case that happens to be an ordinary Lie algebra, this is the ordinary Chevalley-Eilenberg algebra of this Lie algebra. Hence we should generally call the Chevalley-Eilenberg algebra of the -algebra .
One observes that this construction is bijective: every (degreewise finite dimensional) cochain semifree dga generated in positive degree comes from a (degreewise finite dimensional) -algebra this way.
This means that we may just as well define a (degreewise finite dimensional) -algebra as an object in the opposite category of (degreewise finite dimensional) commutative dg-algebras that are semifree dgas and generated in positive degree.
(In general this corresponds to curved L-infinity algebra. The flat -algebras dually correspond to the dg-algebras which are augmented over , i.e for which the canonical projection is a homomorphism of dg-algebras.)
And this turns out to be one of the most useful perspectives on -algebras.
In particular, if we simply drop the condition that the dg-algebra be generated in positive degree and allow it to be generated in non-negative degree over the algebra in degree 0, then we have the notion of the (Chevalley-Eilenberg algebra of) an L-infinity-algebroid.
Details
We discuss in explit detail the computation that shows that an -algebra structure on is equivalently a dg-algebra-structure on .
Let be a degreewise finite-dimensional graded vector space equipped with multilinear graded-symmetric maps
of degree -1, for each .
Let be a basis of and a dual basis of the degreewise dual . Equip the Grassmann algebra with a derivation
defined on generators by
Here we take to be of the same degree as . Therefore this derivation has degree +1.
We compute the square :
Here the wedge product on the right projects the nested bracket onto its graded-symmetric components. This is produced by summing over all permutations weighted by the Koszul-signature of the permutation:
The sum over all permutations decomposes into a sum over the -unshuffles and a sum over permutations that act inside the first and the last indices. By the graded-symmetry of the bracket, the latter do not change the value of the nested bracket. Since there are many of them, we get
Therefore the condition is equivalent to the condition
for all and all . This is equation (1) which says that is an -algebra.
In terms of algebras over an operad
-algebras are precisely the algebras over an operad of the cofibrant resolution of the Lie operad.
Examples
Special cases
-
An -algebra for which is concentrated in the first degree is a Lie -algebra (sometimes also: “-algebra”).
-
An -algebra for which only the unary operation and the binary bracket are non-trivial is a dg-Lie algebra: a Lie algebra internal to the category of dg-algebras. From the point of view of higher Lie theory this is a strict -algebra: one for which the Jacobi identity does happen to hold “on the nose”, not just up to nontrivial coherent isomorphisms.
-
So in particular
-
an -algebra generated just in degree 1 is an ordinary Lie algebra ;
-
an -algebra generated just in degree 1 and 2 is a Lie 2-algebra ;
- an -algebra generated just in degree 1 and 2 and with at most binary brackets is a strict Lie 2-algebra , equivalently encoded in a differential crossed module.
-
an -algebra generated just in degree 1, 2 and 3 is a Lie 3-algebra ;
-
-
if is a Lie algebra over , and is the complex consisting of the field in degree , then an -algebra morphism from to is precisely a degree Lie algebra cocycle.
-
The skew-symmetry of the Lie bracket is retained strictly in -algebras. It is expected that weakening this, too, yields a more general vertical categorification of Lie algebras. For this has been worked out by Dmitry Roytenberg: On weak Lie 2-algebras.
-
The horizontal categorification of -algebras are -algebroids.
-
An -algebra with only non-vanishing is called an n-Lie algebra – to be distinguished from a Lie -algebra ! However, in large parts of the literature -Lie algebras are considered for which is not of the required homogeneous degree in the grading, or in which no grading is considered in the first place. Such -Lie algebras are not special examples of -algebras, then. For more see n-Lie algebra.
-
An -algebra internal to super vector spaces is a super L-∞ algebra.
Classes of examples
-
automorphism Lie 2-algebra?
-
For every -Lie algebra there is its automorphism ∞-Lie algebra. In terms of rational homotopy theory this models the rational automorphism group of the rational space corresponding to .
-
Heisenberg Lie n-algebra of an n-plectic manifold or more generally of an n-plectic smooth infinity-groupoid
-
some classes of W-algebras are claimed to induce -algebras in Blumenhagen-Fuchs-Traube 17
Specific examples
Properties
Ind-Conilpotency
Model category structure
See model structure for L-∞ algebras.
Relation to dg-Lie algebras
Every dg-Lie algebra is in an evident way an -algebra. Dg-Lie algebras are precisely those -algebras for which all -ary brackets for are trivial. These may be thought of as the strict -algebras: those for which the Jacobi identity holds on the nose and all its possible higher coherences are trivial.
Theorem
Let be a field of characteristic 0 and write for the category of -algebras over .
Then every object of is quasi-isomorphic to a dg-Lie algebra.
Moreover, one can find a functorial replacement: there is a functor
such that for each
-
is a dg-Lie algebra;
-
there is a quasi-isomorphism
This appears for instance as (Kriz & May 1995, Cor. 1.6).
For more see at relation between L-∞ algebras and dg-Lie algebras.
Relation to -Lie groupoids
In generalization to how a Lie algebra integrates to a Lie group, -algebras integrate to ∞-Lie groups.
See
and
Lie integrated ∞-Lie groupoids.
-
-algebra
References
General
The concept of L-∞ algebras as graded vector spaces equipped with -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 -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)
See also
-
Marilyn Daily, -structures, PhD thesis, 2004 (web)
-
Klaus Bering, Tom Lada, Examples of Homotopy Lie Algebras Archivum Mathematicum (arXiv:0903.5433)
Comprehensive survey with emphasis on -algebra cohomology:
- Ben Reinhold, -algebras and their cohomology, Emergent Scientist 3 4 (2019) [doi:10.1051/emsci/2019003]
Review for the special case of Lie 2-algebras with emphasis on the perspective of categorification:
- John Baez, Alissa Crans, Higher-dimensional algebra VI: Lie 2-algebras, TAC 12, (2004), 492–528. (arXiv:math/0307263)
As models for rational homotopy types
That -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 -rational homotopy of mapping spaces (arXiv:1209.4756), published as -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 -algebras, Advances in Mathematics 236 (2013): 60-91. (arXiv:1204.4999)
-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), -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]
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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) -algebra language was made in:
-
Hisham Sati, Urs Schreiber, Jim Stasheff, example 5 in section 6.5.1, p. 54 of: 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 -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 -Lie algebras” as 2-term -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)
See also
- José Figueroa-O'Farrill, section Triple systems and Lie superalgebras in M2-branes, ADE and Lie superalgebras, talk at IPMU 2009 (pdf)
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 ):
-
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 -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 -algebras, Proc. Symp. Pure Math. 81 (2010) 181-236 [ams:pspum/081, arXiv:1001.5020]
and identificatoin with the rational Whitehead -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 -algebras from the formal neighbourhood of a derived critical locus is maybe first made explicit in:
- Maxim Grigoriev, Dmitry Rudinsky, Notes on the -approach to local gauge field theories arXiv2303.08990
In string field theory
The first explicit appearance of -algebras in theoretical physics is the -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)
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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)
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Martin Markl, Loop Homotopy Algebras in Closed String Field Theory (1997) (arXiv:hep-th/9711045)
See also
- Jim Stasheff, Higher homotopy algebras: String field theory and Drinfeld’s quasiHopf algebras, proceedings of International Conference on Differential Geometric Methods in Theoretical Physics, 1991 (spire)
For more see at string field theory – References – Relation to A-infinity and L-infinity algebras.
In deformation quantization
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)
In heterotic string theory
Next it was again -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 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)
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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)
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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)
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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 -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 -algebras that behaved similar to Poisson bracket Lie algebras – Poisson bracket Lie n-algebras – were noticed
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Chris Rogers, algebras from multisymplectic geometry , Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (arXiv:1005.2230, journal).
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Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068)
In
- 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 -algebras of BPS charges which refine super Lie algebras such as the M-theory super Lie algebra:
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Hisham Sati, Urs Schreiber, Lie n-algebras of BPS charges (arXiv:1507.08692)
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Igor Khavkine, Urs Schreiber, Lie n-algebras of higher Noether currents
This makes concrete the suggestion that there should be -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:
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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)
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Alex Arvanitakis, The -algebra of the S-matrix (arXiv:1903.05643)
In double field theory
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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)
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Olaf Hohm, Barton Zwiebach, Algebras and Field Theory (arXiv:1701.08824)
Last revised on September 27, 2024 at 08:24:38. See the history of this page for a list of all contributions to it.