January 2017 MIT-CTP-4875 L Algebras and Field Theory ∞ 7 1 0 2 r a M Olaf Hohm1 and Barton Zwiebach2 6 2 ] h 1Simons Center for Geometry and Physics, t - Stony Brook University, p e Stony Brook, NY 11794-3636, USA h [ 2Center for Theoretical Physics, 3 v Massachusetts Institute of Technology 4 Cambridge, MA 02139, USA 2 8 8 [email protected], [email protected] 0 . 1 0 7 1 : v i X Abstract r a We review and develop the general properties of L algebras focusing on the gauge ∞ structure of the associated field theories. Motivated by the L homotopy Lie algebra ∞ of closed string field theory and the work of Roytenberg and Weinstein describing the Courant bracket in this language we investigate the L structure of general gauge in- ∞ variant perturbative field theories. We sketch such formulations for non-abelian gauge theories, Einstein gravity, and for double field theory. We find that there is an L alge- ∞ brafor thegauge structureandalarger one forthe fullinteracting field theory. Theories where the gauge structure is a strict Lie algebra often require the full L algebra for ∞ the interacting theory. The analysis suggests that L algebras provide a classification ∞ of perturbative gauge invariant classical field theories. Contents 1 Introduction and summary 2 2 L algebra and gauge Jacobiator 6 ∞ 2.1 The multilinear products and main identity . . . . . . . . . . . . . . . . . . . . . 6 2.2 A family of identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Gauge transformations and algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Gauge Jacobiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 L algebra in ℓ picture and field theory 14 ∞ 3.1 L algebra identities; ℓ-picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 ∞ 3.2 From b-picture to ℓ-picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 General remarks on the L algebra of field theories . . . . . . . . . . . . . . . . 21 ∞ 4 Non-abelian gauge theories and L algebras 24 ∞ 4.1 Generalities on Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 Chern-Simons Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3 Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5 Double field theory and L algebras 33 ∞ 5.1 DFT C-bracket algebra as an L algebra . . . . . . . . . . . . . . . . . . . . . . . 33 3 5.2 Off-shell DFT as extended L algebra . . . . . . . . . . . . . . . . . . . . . . . . 37 3 5.3 Perturbative DFT as L algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ∞ 5.4 Comments on Einstein gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6 A algebras and revisiting Chern-Simons 47 ∞ 7 Conclusions and outlook 49 1 1 Introduction and summary In bosonic open string field theory [1] the interaction of strings is defined by a multiplication rule, a star product of string fields that happens to be associative. While this formulation is advantageous for finding classical solutions of the theory, associativity is not strictly necessary for the formulation of the string field theory; homotopy associativity is. This more intricate structure has been investigated in [2] and has appeared when one incorporates closed strings explicitly in the open string theory [3–5]. The homotopy associative A algebra of Stasheff [6] ∞ is the mathematical structure underlying the general versions of the classical open string field theory sector. The product of closed string fields is analogous to Lie brackets in that they are graded commutative. But a strict Lie algebra does not appear to allow for the formulation of closed string field theory. Instead one requires a collection of higher products satisfying generalized versions of Jacobi identities. The classical string field theory is thus organized by a homotopy Lie algebra, an L algebra whose axioms and Jacobi-like identities were given explicitly in [7]. ∞ The axioms and identities were later given in different but equivalent conventions in [8]. These two formulations are related by a “suspension”, an operation where the degree of all vector spaces is shifted by one unit. An earlier mathematical motivation for homotopy Lie algebras is found in [9]. The L algebra describes the structure of classical string field theory: the ∞ collection of string field products is all one needs to write gauge transformations and field equations. Equipped with a suitable inner product, one can also write an action. The interplay of A and L algebras feature in the study of open-closed string field theory [3,5,10]. L ∞ ∞ ∞ algebras have recently featured in a study of massive two-dimensional field theory [11]. The relevance of L to closed string field theory, which is a field theory for an infinite ∞ number of component fields, suggests that it should also be relevant to arbitrary field theories, and this provided motivation for the present study. In particular, in a recent work Sen has shown how to define consistent truncations for a set of closed string modes [12]. For these degrees of freedom one has an effective field theory organized by an L algebra that can be ∞ derived from the full algebra of the closed string field theory. This again suggests the general relevance of L to field theories. ∞ A first look into the problem of identifying the L gauge structure of some field theories ∞ was given by Barnich et.al. [13]. The early investigation of non-linear higher-spin symmetries in Berends et.al. [14] eventually led to an analysis by Fulp et.al. [15] of gauge structures that under some assumptions define L algebras. More recently, Yang-Mills-type gauge theories ∞ were fully formulated as L algebras by Zeitlin using the BRST complex of open string field ∞ theory [16–18]. Inaninteresting paper,RoytenbergandWeinstein [19]systematically analyzed theCourant algebroid in the language of L algebras. The authors explicitly identified the relevant vector ∞ spaces, the products, and proceeded to show that the Jacobi-like homotopy identities are all satisfied. As it turns out, in this algebra there is a bracket [·,·] that applied to two gauge parameters coincides with the Courant bracket, and a triple product [·,·, ·] that applied to three gauge parameters gives a function whose gradient is the Jacobiator. No higher products 2 exist in this particular L algebra. TheL formulation of the C-bracket, the duality-covariant ∞ ∞ extension of theCourantbracket, was consideredbyDeser andSaemann[20]. Theauthorsused ‘derived brackets’ in their construction, making use of the results of [21–23]. Since the Courant bracket underlies the gauge structure of double field theory [24–28], we were motivated by the above results to try to understand what would be the L algebra of double field theory. ∞ ForthestudyofL infieldtheoryafactaboutL inclosedstringfieldtheorywaspuzzling. ∞ ∞ In this theory there is a triple product [·,·, ·] and it controls several aspects of the theory. It enters in the quartic interactions of fields, as the inner product of Ψ with [Ψ,Ψ,Ψ]. It appears in the gauge transformations as a nonlinear contribution δ Ψ ∼ ···+[Λ,Ψ, Ψ]. It Λ makes the commutator of two gauge transformations δ and δ a gauge transformation with Λ1 Λ2 a gauge parameter that includes a field-dependent term [Λ ,Λ ,Ψ]. Finally, the triple product 1 2 implies that the gauge transformations only close on shell, the extra term being [Λ ,Λ ,F], 1 2 where F is the string field equation. In closed string field theory all these peculiarities happen simultaneouslybecausethetripleproductisnon-vanishing;thesefactsarecorrelated. Moreover, the definition of the product is universal, valid for arbitrary input string fields. These facts, however, are not correlated in ordinary field theories. Yang-Mills, for example, has a quartic interaction in the Lagrangian but the commutator of gauge transformations is a gauge transformation with a field-independent gauge parameter. The same is true for Einstein gravity in a perturbative expansion around a background. The gauge algebra of double field theory is field independent but there is a triple bracket associated to the failure of the Jacobi identity for the Courant bracket. The lack of correlation is quickly demystified by exploring such examples. Gauge parameters, fields, and field equations appear in different vector spaces, according to some relevant grading. In defining a triple product one must state its value for all possible gradings of the various inputs. While in the string field theory one has a universal definition controlled by some data about four-punctured Riemann spheres, the definition of productsin a given field theory has to bedonein a case by case approach as we vary the inputs. The product is non-vanishing for inputs with certain gradings and can vanish for other sets of inputs. The correlation observed in string field theory is not discernible in the various separate field theories. Another important feature is revealed by the explicit analysis: there are at least two L ∞ gauge algebras associated to a field theory, one a subalgebra of the other. There is an L for the ∞ full gaugestructurewhichisasubalgebraofthelargerL thatincludestheinteractionsassociated ∞ with the action and the field equations: gauge full L ⊂ L . (1.1) ∞ ∞ Any algebra for the gauge structure must include a vector space for gauge parameters. If the algebra is field dependent, it must include a vector space for fields. This algebra does not include interactions. When including interactions, the gauge algebra is supplemented by a new vector space for field equations and a set of new products, including some defined on fields and gauge full field equations. One can easily have a theory where L is a Lie algebra but L has higher ∞ ∞ products, because the theories have quartic or higher-order interactions. This is the case for Yang-Mills theory and for Einstein gravity. For a discussion of L algebras associated with ∞ some class of gauge algebras with field-dependent structure constants see also [15]. 3 gauge If L is field independent, there is a third, intermediate algebra ∞ gauge+fields L , (1.2) ∞ that includes the off-shell realization of the gauge transformations on fields but that does not includedynamics. RoytenbergandWeinstein [19], forexample, computedthefield-independent gauge algebraL fortheCourantbracket. Wewillreviewthatwork,translatedinO(d,d)covariant ∞ gauge+fields full language, and extend it to consider the algebra L and the algebra L for the ∞ ∞ interacting double field theory. Our work here points to an intriguing possibility: any gauge invariant perturbative field theory may berepresented by an L algebra that encodes all the information about thetheory, ∞ namely, the gauge algebra and its interactions. It is then possible that gauge invariant pertur- bative field theories are classified by L algebras. A few comments are needed here. Associated ∞ toany gaugestructure, fieldtheories can differby theinteractions. Sincetheinteractions define full someoftheproductsinL ,thataspectofthetheoryisproperlyincorporated. Ofcourse,field ∞ redefinitions establish equivalences between theories, and such equivalences must correspond to suitable isomorphisms of L algebras, presumablyin the way discussed for A in [2]. We seem ∞ ∞ to be constrained to theories formulated in perturbative form, that is theories in which one can identify unambiguously terms with definite powers of the fields in the Lagrangian, although there may beexceptions. For Einstein gravity, which in the standard formulation contains both the metric and its inverse, one must expand around a background to obtain a perturbative expansion in terms of the fluctuating field. This formulation of gravity as L is completely ∞ straightforward, as will be clear to the reader of this paper, but real insight would come only if some elegant explicit definition of the products could be given. Constrained fields, such as the generalized metric of double field theory, are also problematic as the power of the field in any expression may be altered by use of the constraint. All in all, we do not attempt to show that any gauge invariant field theory has a description as an L algebra, although we suspect ∞ that the result is true for unconstrained fields. Perhaps a proof could be built using a different approach. Consider a perturbative theory that can be formulated in the Batalin-Vilkovisky formalism with a master action S satisfying the classical Batalin-Vilkovisky master equation {S,S} = 0; see [29] for a review of these techniques. General arguments indicate that an L ∞ structure can be systematically extracted from the master action [30]. SinceA isthealgebraicsetupforopenstringfieldtheory,onecanaskwhyisL ,thesetup ∞ ∞ for closed string field theory, chosen for perturbative field theories. We have no general answer butitappearsthattheL setupis ratherflexible. Weshowthatfor Chern-Simonstheory both ∞ an A and a L formulation exists. The first formulation requires describing the Lie algebra ∞ ∞ in terms of an associative algebra of matrix multiplication. The second formulation requires the use of a background metric in the definition of the products. We have not investigated if other field theories have both formulations. See, however, the general discussion in [31] giving an A setup to Yang-Mills theory. The formulation of gauge theories as A algebras will also ∞ ∞ be investigated in [32]. Afraction of theworkheredeals withthestructureofL algebras. Following [7]wediscuss ∞ the axioms and main identities, but develop a bit further the analysis. We show explicitly that given an L algebra with multilinear products with n ≥ 1 inputs, one can construct consistent ∞ 4 modified products with n≥ 0 inputs. A product [·]′ without an input is just a special vector F in the algebra and that vector is in fact the field equation for a field Ψ in the theory defined by the original products. The modified product with one input, [B]′ = Q′B, defines a linear operator Q′, built from Ψ and a Q operator that squares to zero and defines the one-input original product. We establish the L identity ∞ Q′F = 0, (1.3) which can be viewed as the Bianchi identity of the original theory, and Q′ may be thought of as a covariant derivative. Indeed we also have Q′2 ∼ F. The modified products simplify the analysis of the gauge structure of the theory. The gauge transformations take the form δ Ψ = Q′Λ, (1.4) Λ and the computation of the gauge algebra [δ ,δ ] can be simplified considerably. We also Λ2 Λ1 compute the ‘gauge Jacobiator’ J(Λ ,Λ ,Λ ) ≡ δ , [δ , δ ] . (1.5) 1 2 3 Λ3 Λ2 Λ1 Xcyc(cid:2) (cid:3) The right hand side is trivially zero for any theory with well-defined gauge transformations; this is clear by expansion of the commutators. On the other hand, this vanishing is a nontrivial constraintontheformofthegaugealgebra. Thisconstraintissatisfiedbyvirtueoftheidentities satisfied by the higher products in the L algebra. ∞ Intheaboveapproach, calledtheb-pictureoftheL algebra, thesignsinthefieldequation, ∞ gauge transformations, action, and gauge algebra are known. There is another picture, the ℓ- picture of the algebra [8] in which the signs of the Jacobi-like identities are more familiar. The two pictures are related by suspension, a shift in the degree of the various spaces involved. We use this suspension to derive the form of field equations, gauge transformation, action, and gauge algebra in the ℓ picture. Here is a brief summary of this paper. We begin in section 2 with a description of the L algebra in the conventions of the original closed string field theory and discuss the gauge ∞ structure, particularly the closure of the algebra and the triviality of the Jacobiator. In sec. 3 we begin by defining the axioms of L algebras in two conventions, one (the ℓ-picture) that is ∞ conventional in themathematics literature andone(the b-picture) thatis conventional in string field theory and hence directly related to sec. 2. In sec. 3.3 we make some general remarks how to identify for agiven field theory thecorrespondingstructuresof an L algebra. Moreover, we ∞ explain how gauge covariance of the field equations and closure of the gauge transformations imply that large classes of L identities hold. (Readers mainly interested in the applications ∞ to field theory can skip sec. 2 and sec. 3.2.) These results will be applied in sec. 4 in order to describe Yang-Mills-like gauge theories, both for Chern-Simons actions in 3D and for general Yang-Mills actions. In sec. 5 wediscuss the L description of doublefield theory, which in turn ∞ is an extension of theconstruction by Roytenberg andWeinstein for the Courantalgebroid. We finally compare these results with A algebras by giving the A description of Chern-Simons ∞ ∞ theory in sec. 6. We close with a summary and an outlook in sec. 7. 5 2 L algebra and gauge Jacobiator ∞ In this section we review the definition of an L algebra, state the main identities,1 and ∞ introduce the field equation and the action. We then turn to a family of identities for modified products, giving the details of a result anticipated in [7]. They correspond to the products that would arise after the expansion of the string field theory action around a background that does not solve the string field theory equations of motion.2 With these products, the Bianchi identities of string field theory become clear and the modified BRST operator functions as a covariant derivative. We elaborate on the types of gauge transformations, and the modified products simplify the calculation of the gauge algebra. We are also able to verify that the gauge Jacobiator for a general field theory described with an L algebra vanishes, as required ∞ by consistency. 2.1 The multilinear products and main identity In an L algebra we have a vector space V graded by a degree, which is an integer. We will ∞ typically work with elements B ∈ V of fixed degree.3 The degree enters in sign factors where, i for convenience, we omit the ‘deg’ label. Thus, for example: (−1)B1B2 ≡ (−1)deg(B1)·deg(B2). (2.1) In exponents, the degrees are relevant only mod 2. In an L algebra we have multilinear ∞ products. In the notation used for string field theory the multilinear products are denoted by brackets [B ,...,B ] and are graded commutative 1 n [···B ,B ,···]= (−1)BiBj[···B ,B ,···]. (2.2) i j j i All products are defined to be of intrinsic degree −1, meaning that the degree of a product of a given number of inputs is given by n deg([B , ... ,B ]) = −1+ deg(B ). (2.3) 1 n i i=1 X The product with one input is sometimes called the Q operator (for BRST) [B] ≡ QB. (2.4) We also have a product [·] with no input whose value is just some special vector in the vector space. The L relations can be written in the form [7]: ∞ σ(i ,j ) B , ... ,B [B , ... ,B ] = 0, n ≥ 0. (2.5) l k i1 il j1 jk ll+X,kk≥=0nXσs (cid:2) (cid:3) 1The identities for gauge invariance of the classical theory first appeared in [35] and were re-cast as L∞ identities in [7]. Note that the structureof quantum closed string field theory goes beyond L∞ algebras. 2After expansion of the string field theory around a background that satisfies the equations of motion, the typeof algebraic structureis not changed [34]. 3In closed string field theory degree ‘deg’ is related to ghost number‘gh’ as deg =2−gh. 6 Herenisthenumberofinputs(ifn = 0westillgetanontrivialidentity). TheinputsB ,...,B 1 n are split into two sets: a first set {B ...B } with l elements and a second set {B ...B } i1 il j1 jk with k elements, where l+k = n. The first set is empty if l = 0 and the second set is empty if k = 0. The two sets do not enter the identity symmetrically: the second set has the inputs for a product nested inside a product that involves the first set of elements. The set of numbers {i ,...,i ,j ,...,j } is a permutation of the list {1,...,n}. 1 l 1 k The sums are over inequivalent splittings. Sets with different values of l and k are inequiv- alent, so we must sum over all possible values of k and l. Two splittings with the same values of l and k are equivalent if the first set {B ...B } contains the same elements, regardless i1 il of order. The factor σ(i ,j ) is the sign needed to rearrange the list {B ,B ,...,B } into l k ∗ 1 n {B ,...B ,B , B ,...B }: i1 il ∗ j1 jk {B ,B ,...,B } → {B ,...B ,B , B ,...B }, (2.6) ∗ 1 n i1 il ∗ j1 jk using the degrees to commute the B’s according to (2.2) and thinking of B as an element of ∗ odd degree. The element B is needed to take into account that the products are odd. ∗ For classical string field theory, or for any field theory expanded around a classical solution, the value of the zeroth product [·] will be set equal to the zero vector: [·] ≡ 0. (2.7) Usingtheabove rulesforsign factors, wecan writeouttheL identities (2.5). Note thatin the ∞ absence of a zeroth product k > 0 and thus n > 0 to get a nontrivial identity. For n = 1,2,3 one gets: 0 = Q(QB), 0 = Q[B ,B ]+[QB ,B ]+(−1)B1[B ,QB ], 1 2 1 2 1 2 0 = Q[B ,B ,B ] 1 2 3 (2.8) +[QB ,B ,B ]+(−1)B1[B ,QB ,B ]+(−1)B1+B2[B ,B ,QB ] 1 2 3 1 2 3 1 2 3 +(−1)B1[B ,[B ,B ]] +(−1)B2(1+B1)[B ,[B ,B ]] 1 2 3 2 1 3 +(−1)B3(1+B1+B2)[B ,[B ,B ]]. 3 1 2 We will now discuss how to define in this language equations of motion and actions for a field theory. To this end and for brevity, we write products with repeated inputs as powers. When there is no possible confusion we also omit the commas between the inputs: [Ψ3] ≡ [Ψ,Ψ,Ψ], [BΨ3] ≡ [B,Ψ,Ψ,Ψ]. (2.9) Here Ψ, called the field, is an element of degree zero: degΨ = 0. (2.10) If Ψ had been of odd degree, the above products would vanish by the graded commutativity property. 7 Given a set of products satisfying the L conditions and a Grassmann even field Ψ we ∞ introduce a field equation F of degree minus one: ∞ 1 F = [Ψn] = QΨ+ 1[Ψ2]+ 1[Ψ3]+... = QΨ+ 1[Ψ,Ψ]+ 1[Ψ,Ψ,Ψ]+... . (2.11) n! 2 3! 2 3! n=0 X Again, we used that the term with n = 0 vanishes, as it involves a product with no input. The field equation F is of degree minus one because Ψ is of degree zero and all products are of degree minus one. Certain infinite sums appear often when dealing with gauge transformations and make it convenient to define modified, primed products: ∞ 1 [A ...A ]′ ≡ [A ...A Ψp], n≥ 1. (2.12) 1 n 1 n p! p=0 X Thus, for example, [A]′ ≡ Q′A = QA+[AΨ]+ 1[AΨ2]+... , 2 (2.13) [A ...A ]′ = [A ...A ]+[A ...A Ψ]+ 1[A ...A Ψ2] + ... . 1 n 1 n 1 n 2 1 n The variation of those products is rather simple: δ[A ...A ]′ = [δA ...A ]′+...+[A ...δA ]′+[A ...A δΨ]′. (2.14) 1 n 1 n 1 n 1 n The identification of [A]′ with Q′A is natural given (2.4). The variation of the field equation takes the form of a modified product. We have δF = Q′(δΨ), (2.15) which is readily established: ∞ ∞ ∞ 1 k 1 δF = δ [Ψk] = [Ψk−1δΨ] = [ΨkδΨ] = [δΨ]′. (2.16) k! k! k! k=0 k=1 k=0 X X X Inner product and action: The action exists if there is a suitable inner product h·,·i. One requires that hA,Bi = (−1)(A+1)(B+1)hB,Ai, (2.17) and that the expression hB ,[B ,...,B ]i, (2.18) 1 2 n for n ≥ 1 is a multilinear graded-commutative function of all the arguments. From the above one can show, for example, that hQA,Bi = (−1)AhA,QBi. (2.19) The action is given by ∞ 1 S = hΨ, [Ψn]i. (2.20) (n+1)! n=1 X A short calculation shows that that under a variation δΨ one has δS = hδΨ,Fi, (2.21) confirming that F = 0 is the field equation corresponding to the action. 8 2.2 A family of identities In this subsection we will establish a number of identities that will be useful below when computing the Jacobiator. The products [...]′ can in fact be viewed as a set of products satisfying a simple extension of the L identities. To see this and to get the general picture we ∞ consider a few examples. Consider (2.5) when all B’s are of even degree. The sign factor is then always equal to +1 and we have B ...B [B ...B ] = 0, n ≥ 0 B even ∀k. (2.22) i1 il j1 jk k ll+X,kk≥=0nXσs (cid:2) (cid:3) For l and k fixed there are n!/(l!k!) inequivalent splittings of the inputs; this is the number of terms in the sum . Assume now that all the B’s are the same: B = B = ... = B, so that σs 1 2 all those terms are equal. We then have P 1 Bl[Bk] = 0, n ≥ 0, B even, (2.23) l!k! l,k≥0 l+Xk=n (cid:2) (cid:3) where we have taken out an overall factor of n! from the numerator. If we now take B = Ψ and sum over n this identity becomes 1 Ψl[Ψk] = 0. (2.24) l!k! n≥0 l,k≥0 X l+Xk=n (cid:2) (cid:3) Reordering the double sum we have 1 1 Ψl[Ψk] = ΨlF = 0, (2.25) l!k! l! l,k≥0 l≥0 X (cid:2) (cid:3) X (cid:2) (cid:3) where we summed over k in the second step and used (2.11). Recalling (2.13), the sum over l finally gives Q′F = 0. (2.26) If we view Q′ as the analogue of the covariant derivative D and F as the analogue of the non- abelian field strength F in Yang-Mills theory, then this identity is the analogue of the Bianchi identity DF = 0. Let us consider a second L identity again based on (2.5) but with n+1 inputs ∞ B = A, B = ... = B = B, B even. (2.27) 1 2 n+1 There are two possible classes of splittings, both of which involve separating the n copies of B into a set with l elements and a set with k elements, with l+k = n. These are {ABl}, {Bk} and {Bl}, {ABk}. (2.28) The sign factors arise from reordering B ABlBk into ABlB Bk for the first sequence, giving a ∗ ∗ sign (−1)A, and into BlB ABk for the second sequence, giving no sign. We thus have ∗ n! (−1)A[ABl[Bk]]+[Bl[ABk]] = 0. (2.29) l!k! l,k≥0 l+Xk=n (cid:0) (cid:1) 9