L ∞ algebras for extended geometry from Borcherds superalgebras
aa r X i v : . [ h e p - t h ] A p r Gothenburg preprintApril, 2018 L algebras for extended geometryfrom Borcherds superalgebras Martin Cederwall and Jakob Palmkvist
Division for Theoretical Physics, Department of Physics,Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
Abstract:
We examine the structure of gauge transformations in extendedgeometry, the framework unifying double geometry, exceptional geometry,etc. This is done by giving the variations of the ghosts in a Batalin–Vilkoviskyframework, or equivalently, an L algebra. The L brackets are given asderived brackets constructed using an underlying Borcherds superalgebra B p g r ` q , which is a double extension of the structure algebra g r . The con-struction includes a set of “ancillary” ghosts. All brackets involving the in-finite sequence of ghosts are given explicitly. All even brackets above the -brackets vanish, and the coefficients appearing in the brackets are given byBernoulli numbers. The results are valid in the absence of ancillary transfor-mations at ghost number . We present evidence that in order to go further,the underlying algebra should be the corresponding tensor hierarchy algebra. email: [email protected], [email protected] ontents L algebras 216 The L structure, ignoring ancillary ghosts 27 The full L structure Examples
Conclusions Introduction
The ghosts in exceptional field theory [1], and generally in extended field theory with anextended structure algebra g r [2], are known to fall into B ` p g r q , the positive levels of aBorcherds superalgebra B p g r q [3, 4]. We use the concept of ghosts, including ghosts forghosts etc., as a convenient tool to encode the structure of the gauge symmetry (structureconstants, reducibility and so on) in a classical field theory using the (classical) Batalin–Vilkovisky framework.It was shown in ref. [3] how generalised diffeomorphisms for E r have a natural formu-lation in terms of the structure constants of the Borcherds superalgebra B p E r ` q . Thisgeneralises to extended geometry in general [2]. The more precise rôle of the Borcherdssuperalgebra has not been spelt out, and one of the purposes of the present paper isto fill this gap. The gauge structure of extended geometry will be described as an L algebra, governed by an underlying Borcherds superalgebra B p g r ` q . The superalgebra B p g r ` q generalises B p E r ` q in ref. [3], and is obtained from the structure algebra g r by adding two more nodes to the Dynkin diagram, as will be explained in Section 2.In cases where the superalgebra is finite-dimensional, such as double field theory [5–19],the structure simplifies to an L n ă8 algebra [20–22], and the reducibility becomes finite.It is likely that a consistent treatment of quantum extended geometry will requirea full Batalin–Vilkovisky treatment of the ghost sector, which is part of the motivationbehind our work. Another, equally strong motivation is the belief that the underlyingsuperalgebras carry much information about the models — also concerning physicalfields and their dynamics — and that this can assist us in the future when investigatingextended geometries bases on infinite-dimensional structure algebras.The first ´ r levels in B p E r q consist of E r -modules for form fields in exceptional fieldtheory [1,23–40], locally describing eleven-dimensional supergravity. Inside this window,there is a connection-free but covariant derivative, taking an element in R p at level p to R p ´ at level p ´ [31]. Above the window, the modules, when decomposed as gl p r q modules with respect to a local choice of section, start to contain mixed tensors, andcovariance is lost. For E , the window closes, not even the generalised diffeomorphismsare covariant [39] and there are additional restricted local E transformations [38].Such transformations were named “ancillary” in ref. [2]. In the present paper, we willnot treat the situation where ancillary transformations arise in the commutator of twogeneralised diffeomorphisms, but we will extend the concept of ancillary ghosts to higher3host number. It will become clear from the structure of the doubly extended Borcherdssuperalgebra B p g r ` q why and when such extra restricted ghosts appear, and what theirprecise connection to e.g. the loss of covariance is.A by-product of our construction is that all identities previously derived on a case-by-case basis, relating to the “form-like” properties of the elements in the tensor hierarchies[31, 41], are derived in a completely general manner.Although the exceptional geometries are the most interesting cases where the struc-ture has not yet been formulated, we will perform all our calculations in the generalsetting with arbitrary structure group (which for simplicity will be taken to be simplylaced, although non-simply laced groups present no principal problem). The general for-mulation of ref. [2] introduces no additional difficulty compared to any special case, andin fact provides the best unifying formalism also for the different exceptional groups. Wenote that the gauge symmetries of exceptional generalised geometry have been dealt within the L algebra framework earlier [42]. However, this was done in terms of a formalismwhere ghosts are not collected into modules of E r , but consist of the diffeomorphismparameter together with forms for the ghosts of the tensor gauge transformations ( i.e. ,in generalised geometry, not in extended geometry).In Section 2, details about the Borcherds superalgebra B p g r ` q are given. Especially,the double grading relevant for our purposes is introduced, and the (anti-)commutatorsare given in this basis. Section 3 introduces the generalised Lie derivative and the sec-tion constraint in terms of the Borcherds superalgebra bracket. In Section 4 we showhow the generalised Lie derivative arises naturally from a nilpotent derivative on the B p g r q subalgebra, and how ancillary terms/ghosts fit into the algebraic structure. Somefurther operators related to ancillary terms are introduced, and identities between theoperators are derived. Section 5 is an interlude concerning L algebras and Batalin–Vilkovisky ghosts. The non-ancillary part of the L brackets, i.e. , the part where ghostsand brackets belong to the B ` p g r q subalgebra, is derived in Section 6. The completenon-ancillary variation p S, C q “ ř n “ rr C n ss can formally be written as p S, C q “ dC ` g p ad C q L C C , (1.1)where g is the function g p x q “ ´ e ´ x ´ x , (1.2)4ontaining Bernoulli numbers in its Maclaurin series. Ancillary ghosts are introducedin Section 7, and the complete structure of the L brackets is presented in Section8. Some examples, including ordinary diffeomorphisms (the algebra of vector fields),double diffeomorphisms and exceptional diffeomorphisms, are given in Section 9. Weconclude with a discussion, with focus on the extension of the present construction tosituations where ancillary transformations are present already in the commutator of twogeneralised diffeomorphisms. For simplicity we assume the structure algebra g r to be simply laced, and we normalisethe inner product in the real root space by p α i , α i q “ . We let the coordinate module,which we denote R “ R p´ λ q , be a lowest weight module with lowest weight ´ λ .Then the derivative module is a highest weight module R p λ q with highest weight λ , and R p´ λ q “ R p λ q .As explained in ref. [3] we can extend g r to a Lie algebra g r ` or to a Lie superalgebra B p g r q by adding a node to the Dynkin diagram. In the first case, the additional node is anordinary “white” node, the corresponding simple root α satisfies p α , α q “ , and theresulting Lie algebra g r ` is a Kac–Moody algebra like g r itself. In the second case, theadditional node is “grey”, corresponding to a simple root β . It satisfies p β , β q “ , andis furthermore a fermionic ( i.e. , odd) root, which means that the associated Chevalleygenerators e and f belong to the fermionic subspace of the resulting Lie superalgebra B p g r q . In both cases, the inner product of the additional simple root with those of g r isgiven by the Dynkin labels of λ , with a minus sign, ´ λ i “ ´p λ, α i q “ p α , α i q “ p β , β i q , (2.1)where we have set α i “ β i ( i “ , , . . . , r ).We can extend g r ` and B p g r q further to a Lie superalgebra B p g r ` q by adding onemore node to the Dynkin diagrams. We will then get two different Dynkin diagrams In refs. [2,40], the coordinate module was taken to be a highest weight module. We prefer to reversethese conventions (in agreement with ref. [3]). With the standard basis of simple roots in the superalge-bra, its positive levels consists of lowest weight g r -modules. In the present paper the distinction is notessential, since the cases treated all concern finite-dimensional g r and finite-dimensional g r -modules. In ref. [2], the algebras g r ` , B p g r q and B p g r ` q were called A , B and C , respectively. ´ γ γ γ r ´ γ r ´ γ r ´ γ r ´ γ r β ´ β β β r ´ β r ´ β r ´ β r ´ β r Figure 1 : Dynkin diagrams of B p E r ` q together with our notation for the simple rootsrepresented by the nodes. (two different sets of simple roots) corresponding to the same Lie superalgebra B p g r ` q .These are shown in Figure in the case when g “ E r and λ is the highest weight of thederivative module in exceptional geometry. The line between the two grey nodes in thesecond diagram indicate that the inner product of the two corresponding simple rootsis p β ´ , β q “
1, not ´ γ ´ “ ´ β ´ , γ “ β ´ ` β , γ i “ β i . ( . )This corresponds to a “generalised Weyl transformation” or “odd Weyl reflection” [ ],which provides a map between the two sets of Chevalley generators mapping the definingrelations to each other, thus inducing an isomorphism.In spite of the notation B p g r ` q we choose to consider this algebra as constructedfrom the second Dynkin diagram in Figure , which means that we let e , f and h beassociated to β rather than γ . For β ´ , we drop the subscript and write the associatedgenerators simply as e , f and h . They satisfy the (anti-)commutation relations r h, e s “ r h, f s “ r e, f s “ h . ( . ) cting with h on e and f we have r h, e s “ e , r h, f s “ ´ f . ( . )Throughout the paper the notation r¨ , ¨s is used for the Lie super-bracket of the super-algebra, disregarding the statistics of the generators. Thus, we do not use a separatenotation ( e.g. t¨ , ¨u , common in the physics literature) for brackets between a pair offermionic elements.Let k be an element in the Cartan subalgebra of B p g r q that commutes with g r andsatisfies r k, e s “ e and r k, f s “ ´ f when we extend B p g r q to B p g r ` q . In the Cartansubalgebra of B p g r ` q , set r k “ k ` h , so that r e, f s “ h “ r k ´ k . We then have r k, e s “ ´p λ, λ q e , r k, e s “ e , r k, f s “ p λ, λ q f , r k, f s “ ´ f , ( . ) r r k, e s “ p ´ p λ, λ qq e , r r k, e s “ e , r r k, f s “ pp λ, λ q ´ q f , r r k, f s “ ´ f . ( . )The Lie superalgebra B p g r ` q can be given a p Z ˆ Z q -grading with respect to β and β ´ . It is then decomposed into a direct sum of g r modules B p g r ` q “ à p p,q qP Z ˆ Z R p p,q q , ( . )where R p p,q q is spanned by root vectors (together with the Cartan generators if p “ q “ p and q for β and β ´ , respectively,when expressed as linear combinations of the simple roots. We will refer to the degrees p and q as level and height , respectively. They are the eigenvalues of the adjoint actionof h “ r k ´ k and the Cartan element q “ p ´ p λ, λ qq k ` p λ, λ q r k “ k ` p λ, λ q h , ( . )respectively. Thus r q, e s “ r q, f s “ , r q, e s “ e , r q, f s “ ´ f . ( . ) n the same way as the Lie superalgebra B p g r ` q can be decomposed with respect to β and β ´ , it can also be decomposed with respect to γ and γ ´ . Then the degrees m and n , corresponding to γ and γ ´ , respectively, are related to the level and height by m “ p and n “ p ´ q . The L structure on B p g r ` q that we are going to introduce isbased on yet another Z -grading, B p g r ` q “ à ℓ P Z L ℓ , ( . )where the degree ℓ of an element in R p p,q q is given by ℓ “ p ` q . The L structure isthen defined on (a part of) the subalgebra of B p g r ` q corresponding to positive levels ℓ ,and all the brackets have level ℓ “ ´
1. It is important, however, to note that the subsetof B p g r ` q on which the ghosts live is not closed under the superalgebra bracket, so thespace on which the L algebra is defined will not support a Lie superalgebra structure.The subset in question consists of the positive levels of the subalgebra B p g r q at p ą q “
0, together with a subset of the elements at p ą q “
1. See further Sections and . The ghost number is identified with the level ℓ “ p ` q in Table . ¨ ¨ ¨ p “ ´ p “ p “ p “ p “ ¨ ¨ ¨¨ ¨ ¨ n “ q “ rr R ❦❦❦❦❦❦❦❦ n “ q “ r R ❦❦❦❦❦❦❦❦❦❦ r R ‘ rr R ❦❦❦❦❦❦ n “ ❦❦❦❦❦❦ q “ ❦❦❦❦❦❦❦❦❦❦❦ R ❦❦❦❦❦❦❦❦❦❦ R ‘ r R ❦❦❦❦❦ R ‘ r R ❙❙❙❙❙❙❙ ❙❙❙❙❙❙❙❙ n “ ❦❦❦❦❦❦❦❦ q “ R ‘ adj ‘ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ R ❙❙❙❙❙❙❙❙❙❙❙❦❦❦❦❦❦❦❦❦❦❦ ❦❦❦❦❦❦❦❦ R ❙❙❙❙❙❙❙❙❙❙ ❦❦❦❦❦❦❦❦ R ❙❙❙❙❙❙❙❙ ¨ ¨ ¨¨ ¨ ¨ ❦❦❦❦❦❦❦❦❦❦ R ℓ “ ❙❙❙❙❙❙❙❙❙ ℓ “ ❙❙❙❙❙❙❙❙❙ ℓ “ ❙❙❙❙❙❙❙❙❙ Table : The general structure of the superalgebra B p g r ` q . The blue lines are the L -levels, given by ℓ “ p ` q . We also have m “ p . Red lines are the usual levels in thelevel decomposition of B p g r ` q , and form g r ` modules. Tables with specific examplesare given in Section 9, and use the same gradings as this table. Following ref. [ ], we let E M and F M be fermionic basis elements of R p , q “ R and p´ , q “ R , respectively, in the subalgebra B p g r q , while r E M and r F M are bosonic basiselements of R p , q “ R and R p´ , ´ q “ R in the subalgebra g r ` . Furthermore, we let T α be generators of g r , and p t α q M N representation matrices in the R representation.Adjoint indices will be raised and lowered with the Killing metric η αβ and its inverse.Then the remaining (anti-)commutation relations of generators at levels ´
1, 0 and 1 inthe “local superalgebra” ( i.e. , where also the right hand side belongs to level ´
1, 0 or1) that follow from the Chevalley–Serre relations are r T α , E M s “ ´p t α q M N E N , r T α , r E M s “ ´p t α q M N r E N , r k, E M s “ ´p λ, λ q E M , r r k, r E M s “ p ´ p λ, λ qq r E M , r r k, E N s “ p ´ p λ, λ qq E N , r k, r E N s “ p ´ p λ, λ qq r E N , r e, E N s “ r E N , r e, r E N s “ , r f, E N s “ , r f, r E N s “ E N , ( . ) r T α , F N s “ p t α q M N F M , r T α , r F N s “ p t α q M N r F M , r k, F N s “ p λ, λ q F N , r r k, r F N s “ pp λ, λ q ´ q r F N , r r k, F N s “ pp λ, λ q ´ q F N , r k, r F N s “ pp λ, λ q ´ q r F N , r e, F N s “ , r e, r F N s “ F N , r f, F N s “ ´ r F N , r f, r F N s “ , ( . ) r E M , F N s “ ´p t α q M N T α ` δ M N k , r r E M , r F N s “ ´p t α q M N T α ` δ M N r k , r E M , r F N s “ δ M N f , r r E M , F N s “ ´ δ M N e . ( . )From this we get rr E M , F N s , E P s “ f M N P Q E Q , rr r E M , r F N s , r E P s “ r f M N P Q r E Q , rr E M , F N s , r E P s “ δ M N r E P ` f M N P Q r E Q , rr r E M , r F N s , E P s “ δ M N E P ` f M N P Q E Q , rr E M , r F N s , E P s “ , rr r E M , F N s , r E P s “ , rr E M , r F N s , r E P s “ δ M N E P , rr r E M , F N s , E P s “ ´ δ M N r E P , ( . ) here f M N P Q “ p t α q M N p t α q P Q ´ p λ, λ q δ M N δ P Q , ( . )and r f M N P Q “ p t α q M N p t α q P Q ` ` ´ p λ, λ q ˘ δ M N δ P Q . ( . )In particular we have the identities rr E M , F N s , E P s “ rr r E M , r F N s , E P s ` rr E M , r F N s , r E P s , rr r E M , r F N s , r E P s “ rr E M , F N s , r E P s ´ rr r E M , F N s , E P s , ( . )which follow from acting with e and f on rr E M , r F N s , E P s “ rr r E M , F N s , r E P s “ E M and r E M fulfil certain “covariantised Serrerelations”, following from the Serre relations for e and r e, e s , the generators correspond-ing to the roots β and γ , respectively. The Serre relation in the B p g r q subalgebra statesthat r E M , E N s only spans a submodule R of the symmetric product of two R ’s. Thecomplement of R in the symmetric product is R p´ λ q , the only module appearing inthe square of an object in a minimal orbit. Similarly, the Serre relation in the g r ` subalgebra states that r r E M , r E N s only spans r R , the complement of which is the highestmodule in the antisymmetric product of two R ’s. The bracket r E M , r E N s spans R ‘ r R .The conjugate relations apply to F M and r F M . We thus have r E M , E N s P R , r F M , F N s P R , r E M , r E N s P R ‘ r R , r F M , r F N s P R ‘ r R , r r E M , r E N s P r R , r r F M , r F N s P r R . ( . )The modules R and r R are precisely the ones appearing in the symmetric and anti-symmetric parts of the section constraint in Section . For more details, e.g. on theconnection to minimal orbits and to a denominator formula for the Borcherds superal-gebra, we refer to refs. [ – ]. The (anti-)commutation relations with generators at level ˘ . ) at level ¯ . ) by the Jacobi identity.An important property of B p g r ` q is that any non-zero level decomposes into dou- lets of the Heisenberg superalgebra spanned by e , f and h . This follows from eqs. ( . ).An element at positive level and height 0 is annihilated by ad f . It can be “raised” toheight 1 by ad e and lowered back by ad f . We define, for any element at a non-zerolevel p , A “ p r A, e s , ( . ) A “ ´r A, f s . ( . )Then A “ A ` A . Occasionally, for convenience, we will write raising and loweringoperators acting on algebra elements. We then use the same symbols for the operators: A “ A and A “ A .As explained above B p g r ` q decomposes into g r modules, where we denote the oneat level p and height q by R p p,q q . Every g r -module R p “ R p p, q at level p ą R p p, q “ R p ‘ r R p . Sometimes, r R p may vanish. The occurrence of non-zero modules r R p is responsible for the appearance of “ancillary ghosts”. Let A and B be elements at positive level and height 0 (or more generally, annihilatedby ad f ), and denote the total statistics of an element A by | A | . The notation is such that | A | takes the value 0 for a totally bosonic element A and 1 for a totally fermionic one.“Totally” means statistics of generators and components together, so that a ghost C al-ways has | C | “
0, while its derivative (to be defined in eq. ( . ) below) has | dC | “
1. Thisassignment is completely analogous to the assignment of statistics to components in a su-perfield. To be completely clear, our conventions are such that also fermionic componentsand generators anticommute, so that if e.g. A “ A M E M and B “ B M E M are elementsat level 1 with | A | “ | B | “
0, then r A, B s “ r A M E M , B N E N s “ ´ A M B N r E M , E N s . Abosonic gauge parameter A M at level 1 sits in an element A with | A | “ r A, B s “ r A, B s , ( . ) r A, B s “ ´p´ q | B | p ad h q ´ rr h, A s , B s . ( . ) The notation r R p was used differently in ref. [3]. There, r R , r R , r R , . . . correspond to R , r R , rr R , . . . here, i.e. , the representations on the diagonal n “ p “ that themeanings of the notation coincide. ote that r A , B s has height 2 and lies in r R p A ` p B , if p A , p B are the levels of A, B . Thedecomposition r A, B s “ r A, B s ´ p´ q | B | p ad h q ´ rr h, A s , B s ( . )provides projections of R p p, q “ R p ‘ r R p on the two subspaces.We will initially consider fields (ghosts) in the positive levels of B p g r q , embeddedin B p g r ` q at zero height. They can thus be characterised as elements with positive(integer) eigenvalues of ad h and zero eigenvalue of the adjoint action of the element q ineq. ( . ). Unless explicitly stated otherwise, elements in B p g r ` q will be “bosonic”, in thesense that components multiplying generators that are fermions will also be fermionic,as in a superfield. This agrees with the statistics of ghosts. With such conventions, thesuperalgebra bracket r¨ , ¨s is graded antisymmetric, r C, C s “ | C | “ We will consider elements in certain subspaces of the algebra B p g r ` q which are alsofunctions of coordinates transforming in R “ R p´ λ q , the coordinates of an extendedspace. The functional dependence is such that a (strong) section constraint is satisfied.A derivative is in R “ R p λ q . Given the commutation relations between F M and r F M (which both provide bases of R ), the section constraint can be expressed as r F M , F N sB M b B N “ , r F M , r F N sB M b B N “ , r r F M , r F N sB M b B N “ . ( . )The first equation expresses the vanishing of R in the symmetric product of two deriva-tives (acting on the same or different fields), the last one the vanishing of r R in theantisymmetric product, and the second one contains both the symmetric and antisym-metric constraint. The first and third constraints come from the subalgebras B p g r q and g r ` , respectively, which gives a simple motivation for the introduction of the double xtension. By the Jacobi identity, they imply rr x, F M s , F N sB p M b B N q “ , rr x, r F M s , r F N sB r M b B N s “ . )for any element x P B p g r ` q . We refer to refs. [ , ] for details concerning e.g. the im-portance of eqs. ( . ) for the generalised Lie derivative, and the construction of solutionsto the section constraint.The generalised Lie derivative, acting on an element in R , has the form L U V M “ U N B N V M ` Z P QMN B N U P V Q , ( . )where the invariant tensor Z has the universal expression [ , ] σZ “ ´ η αβ t α b t β ` p λ, λ q ´ . )( σ is the permutation operator), i.e. , Z P QMN “ ´ η αβ p t α q P N p t β q QM ` pp λ, λ q ´ q δ NP δ MQ .With the help of the structure constants of B p g r ` q it can now be written [ ] L U V “ rr U, r F N s , B N V s ´ rrB N U , r F N s , V s , ( . )where U “ U M E M , V “ V M E M , with U M and V M bosonic. The two terms in thisexpression corresponds to the first and second terms in eq. ( . ), respectively, using thefourth and seventh equations in ( . ). It becomes clear that the superalgebra B p g r q does not provide the structure needed to construct a generalised Lie derivative, but that B p g r ` q does. In the following Section we will show that this construction not only ismade possible, but that the generalised Lie derivative arises naturally from consideringthe properties of a derivative.We introduce the following notation for the antisymmetrisation, which will be the2-bracket in the L algebra,2 rr U, V ss “ L U V ´ L V U “ rr U, r F N s , B N V s ´ rrB N U , r F N s , V s ´ p U Ø V q . ( . ) or the symmetric part we have2 L U, V M “ L U V ` L V U “ rr U, r F N s , B N V s ´ rrB N U , r F N s , V s` rr V, r F N s , B N U s ´ rrB N V , r F N s , U s“ rr U, B M V s , r F M s ´ rrB M U , V s , r F M s , ( . )where we have used the Jacobi identity. If r R “
0, then r r E M , E N s “ r r E N , E M s “ ´r E M , r E N s ( . )so that rB M r U , V s “ ´rB M U, r V s and 2 L U, V M “ B M rr U, r V s , r F M s .In the cases where L U L V ´ L U L V “ L rr U,V ss we get2 L rr U, V ss , W M “ L rr U,V ss W ` L W rr U, V ss“ L U L V W ` L W L U V “ L U L V W “ p L U L V W ´ L W L U V q“ p L rr U,V ss W ´ L W rr U, V ssq “ rrrr U, V ss , W ss ( . )antisymmetrised in U, V, W . These expressions, and their generalisations, will returnwith ghosts as arguments in Section . Note however that U and V have bosonic com-ponents. They will be replaced by fermionic ghosts, which together with fermionic basiselements build bosonic elements. The bracket will be graded symmetric. In this Section, we will start to examine operators on elements at height 0, which arefunctions of coordinates in R . Beginning with a derivative, and attempting to get asclose as possible to a derivation property, we are naturally led to the generalised Liederivative, extended to all positive levels. The generalised Lie derivative is automaticallyassociated with a graded symmetry, as opposed to the graded antisymmetry of thealgebra bracket. This will serve as a starting point for the L brackets. Other operatorsarise as obstructions to various desirable properties, and will represent contributions rom ancillary ghosts. Various identities fulfilled by the operators will be derived; theywill all be essential to the formulation of the L brackets and the proof of their identities. Define a derivative d : R p p, q Ñ R p p ´ , q ( p ą
0) by dA “ , A P R p , q , rB M A , r F M s , A P R p p, q , p ą . ( . )It fulfils d “ p ą dA p “ p rB M A, F M s . ( . )This follows from r A p , r F M s “ p rr A p , e s , r F M s “ p r A p , F M s ` p rr A p , r F M s , e s , ( . )where r A p , r F M s “ p ą p in eq. ( . ). The subsequent considerations will howeverdepend crucially on the coefficient. The derivative is not a derivation, but its failure to be one is of a useful form. It consistsof two parts, one being connected to the generalised Lie derivative, and the other tothe appearance of modules r R p . The almost-derivation property is derived using eq.( . ), which allows moving around raising operators at the cost of introducing height1 elements. Let p A , p B be the levels of A, B . One can then use the two alternative forms r A, B s “ r A, B s ` p´ q | B | p A p A ` p B r A , B s , p´ q | B | r A , B s ´ p´ q | B | p B p A ` p B r A , B s ( . ) o derive d r A, B s “ rr A, B M B s , r F M s ` rrB M A, B s , r F M s“ rr A, B M B s , r F M s ` p´ q | B | p A p A ` p B rr A , B M B s , r F M s` p´ q | B | rrB M A , B s , r F M s ´ p´ q | B | p B p A ` p B rrB M A , B s , r F M s“ rr A, r F M s , B M B s ` r A, rB M B , r F M ss` p´ q | B | rB M A , r B, r F M ss ` p´ q | B | rrB M A , r F M s , B s` p´ q | B | p A B p B q M ´ p B B p A q M p A ` p B rr A , B s , r F M s “ rr A, r F M s , B M B s ` r A, dB s ` δ p B , r A, rB M B , r F M ss` p´ q | B | rB M A , r B, r F M ss ` p´ q | B | r dA, B s ` δ p A , p´ q | B | rrB M A , r F M s , B s` p´ q | B | p A B p B q M ´ p B B p A q M p A ` p B rr A , B s , r F M s “ r A, dB s ` p´ q | B | r dA, B s` δ p A , ´ rr A, r F M s , B M B s ` p´ q | B | rrB M A , r F M s , B s ¯ ´ p´ q | A || B | δ p B , ´ rr B, r F M s , B M A s ` p´ q | A | rrB M B , r F M s , A s ¯ ` p´ q | B | p A B p B q M ´ p B B p A q M p A ` p B rr A , B s , r F M s . ( . )where superscript on derivatives indicate on which field they act. We recognise thegeneralised Lie derivative from eq. ( . ) in the second and third lines in the last step,and we define, for arbitrary A, B , L A B “ δ p A , ´ rr A, r F M s , B M B s ` p´ q | B | rrB M A , r F M s , B s ¯ . ( . )The extension is natural: a parameter A with p A ą . ) and ( . ) agree. The last termin eq. ( . ) is present only if r R p A ` p B is non-empty, since r A , B s is an element at height2 with r A , B s “
0. We will refer to such terms as ancillary terms, and denote them R p A, B q , i.e. , R p A, B q “ ´p´ q | B | p A B p B q M ´ p B B p A q M p A ` p B rr A , B s , r F M s . ( . )A generic ancillary element will be an element K P R p at height 0 (or raised to K atheight 1) obtained from an element B M P r R p ` at height 1 as K “ r B M , r F M s . Theextra index on B M is assumed to be “in section”. See Section for a more completediscussion.The derivative is thus “almost” a derivation, but the derivation property is brokenby two types of terms, the generalised Lie derivative and an ancillary term: d r A, B s ´ r
A, dB s ´ p´ q | B | r dA, B s “ L A B ´ p´ q | A || B | L B A ´ R p A, B q . ( . )The relative factor with which the derivative acts on different levels is fixed by theexistence of the almost derivation property.Eq. ( . ) states that the symmetry of L A B is graded symmetric, modulo termswith “derivatives”, which in the end will be associated with exact terms. This is good,since it means that we, roughly speaking, have gone from the graded antisymmetryof the superalgebra bracket to the desired symmetry of an L bracket. The gradedantisymmetric part of the generalised Lie derivative appearing in eq. ( . ) representswhat, for bosonic parameters U, V , would be the symmetrised part L U V ` L V U , and itcan be seen as responsible for the violation of the Jacobi identities (antisymmetry andthe Leibniz property imply the Jacobi identities [ ]). The generalised Lie derivative (atlevel 1) will be the starting point for the L and .We note that L dA B “ L r A,B s C “
0, and that L A fulfils a Leibniz rule, L A r B, C s “ p´ q | C | r L A B, C s ` p´ q | A || B | r B, L A C s . ( . )Consider the expression ( . ) for the generalised Lie derivative. It agrees with eq.( . ) when p A “ p B “ | A | “ | B | “
1. It is straightforward to see that the expres-sion contains a factor p´ q | B |` compared to the usual expression for the generalisedLie derivative when expressed in terms of components.In the present paper, we will assume that the generalised Lie derivative, when actingon an element in B ` p g r q , close. This is not encoded in the Borcherds superalgebra. We ill indicate in the Conclusions what we think will be the correct procedure if this isnot the case. We thus assume p L A L B ` p´ q | A || B | L B L A q C “ p´ q | C |` L p L A B `p´ q | A || B | L B A q C , ( . )where the sign comes from the consideration above. When all components are bosonicand level 1, this becomes the usual expression p L A L B ´ L B L A q C “ L p L A B ´ L B A q C .If we instead consider a ghost C with | C | “
0, then L C L C C “ ´ L L C C C . ( . ) The generalised Lie derivative anticommutes with the derivative, modulo ancillary terms.This can be viewed as covariance of the derivative, modulo ancillary terms. Namely,combining eq. ( . ) with entries A and dB with the derivative of eq. ( . ) gives therelation d L A B ` L A dB “ p´ q | B | pr dA, dB s ´ d r dA, B sq ` p´ q | A || B | d L B A ` dR p A, B q ` R p A, dB q . ( . )The left hand side can only give a non-vanishing contribution for p A “ p B ą X A B as d L A B ` L A dB “ ´ X A B . ( . )The explicit form of X A is X A B “ ´p d L A ` L A d q B “ ´ δ p A , rrrB M B N A , B s , r F M s , r F N s . ( . )The notation X A B means p X A B q . Thus, X A B is an element in R p B ´ at height 1.It will be natural to extend the action of the derivative and generalised Lie derivative o elements K at height 1 by dK “ ´p dK q , L C K “ ´p L C K q . ( . )Then, d d “ L C L C “ X dA B “ X r A,B s C “
0, directly inherited from the generalised Liederivative. In addition, we always have L X A B C “ . ( . )If r R “ r R is non-empty (as e.g. for g r “ E ), X A B repre-sents a parameter which gives a trivial transformation without being a total derivative,thanks to the section constraint. The operator X A obeys the important property dX A B ´ X A dB “ . ( . )It follows from the definition of X A and the nilpotency of d as dX A B ´ X A dB “ ´ d p d L A B ` L A dB q ` p d L A ` L A d q dB “ . ( . )It can also be verified by the direct calculation dX A B ´ X A dB “ ´ δ p A , rrrB P rB M B N A , B s , r F M s , r F N s , r F P s ` δ p A , rrrB M B N A , rB P B , r F P s s , r F M s , r F N s “ δ p A , p p B ´ qp p B ´ q ´ rrrB P rB M B N A , B s , r F M s , F N s , F P s`rrrB M B N A , rB P B , F P ss , r F M s , F N s ¯ , ( . )where the action of the raising operators have been expanded. In the first term, B P must hit B , the other term vanishes due to the section constraint. In the second term, rB M B N A , rB P B , F P ss “ rrB M B N A , B P B s , F P s , and the two terms cancel. Note that e are now dealing with identities that hold exactly, not only modulo ancillary terms(they are identities for ancillary terms).An equivalent relation raised to height 1 is p dX A ` X A d q B “ . ( . )A relation for the commutator of X with L is obtained directly from the definition( . ) of X , ´ L A X B ´ X A L B ` p´ q | A || B | p L B X A ´ X B L A q ¯ C “ p´ q | C | X p L A B `p´ q | A || B | L B A q C , ( . )or ´ L A X B ` X A L B ` p´ q | A || B | p L B X A ´ X B L A q ¯ C “ p´ q | C |` X p L A B `p´ q | A || B | L B A q C . ( . )For a ghost C the relation reads L C X C C ´ X C L C C “ X L C C C , ( . )or equivalently, p L C X C ` X C L C q C “ ´ X L C C C . ( . )Further useful relations expressing derivation-like properties, derived using the defi-nitions of X A B and R p A, B q , together with eq. ( . ), are: dR p A, B q ´ R p A, dB q ´ p´ q | B | R p dA, B q “ X A B ´ p´ q | A || B | X B A ( . )and L A R p B, C q ´ p´ q | A || B | R p B, L A C q ´ p´ q | C | R p L A B, C q“ ´ X A r B, C s ` p´ q | A || B | r B, X A C s ` p´ q | C | r X A B, C s . ( . )Although R p A, B q is non-vanishing for A and B at all levels (as long as r R p A ` p B is on-empty), we will sometimes use the notation R A B “ R p A, B q . Thanks to the Jacobiidentity for the Borcherds superalgebra and the Leibniz property of the generalised Liederivative, R p A, B q satisfies a cyclic identity,0 “ R p A, r B, C sq ´ R pr A, B s , C q ´ p´ q | A || B | R p B, r A, C sq` r
A, R p B, C qs ´ r R p A, B q , C s ´ p´ q | A || B | r B, R p A, C qs . ( . ) L algebras Let C P V be a full set of ghosts, including ghosts for ghosts etc. If the “algebra” ofgauge transformations does not contain any field dependence, the Batalin–Vilkovisky(BV) action [ ] can be truncated to ghosts and their antifields C ‹ . We denote thisghost action S p C, C ‹ q , and assume further that it is linear in C ‹ . The ghost action S can be (formally, if needed) expanded as a power series in C , S p C, C ‹ q “ ÿ n “ x C ‹ , rr C n ssy , ( . )where x¨ , ¨y is the natural scalar product on the vector space of the ghosts and its dual,and where rr C n ss “ rr C, C, . . . , C looooomooooon n ss ( . )is a graded symmetric map from b n V to V . This map is, roughly speaking, the L n -bracket. The 1-bracket is the BRST operator. The BV variation of C is p S, C q “ ÿ n “ rr C n ss . ( . )The BV master equation p S, S q “ p S, ¨q , the relation p S, p S, C qq “
0, which in the series expansion turns into a setof identities for the brackets [ , – ], n ´ ÿ i “ p i ` qrr C i , rr C n ´ i ssss “ . ( . ) ften, L algebras are presented with other conventions (see ref. [ ] for an overview).This includes a shifted notion of level, equalling ghost number minus 1. Then the n -bracket carries level n ´
2. In our conventions, all L brackets carry ghost number ´ L brackets are simply graded symmetric andthe statistics of the ghosts, inherited from the superalgebra, is taking care of all signsautomatically.Since the relation between the BV ghost variation and the L brackets seems to beestablished, but not common knowledge among mathematical physicists, we would liketo demonstrate the equivalence explicitly. (See also refs. [ , ].In order to go from the compact form ( . ) to a version with n arbitrary elements,let C “ ř k “ C k and take the part of the identity containing each of the terms in thesum once. We then get ÿ i,j ě i ` j “ n ` j ÿ σ rr C σ p i ` q , . . . , C σ p n q , rr C σ p q , . . . , C σ p i q ssss “ , ( . )where the inner sum is over all permutations σ of t , . . . , n u . The standard definition ofthe L identities does not involve the sum over all permutations, but over the subset of“unshuffles”, permutations which are ordered inside the two subsets: σ p q ă . . . ă σ p i q ,σ p i ` q ă . . . ă σ p n q . ( . )Reexpressing the sum in terms of the sum over unshuffles gives a factor i ! p n ´ i q !, whichcombined with the factor j in eq. ( . ) gives i ! j !, Rescaling the brackets according to n ! rr C , . . . , C n ss “ ¯ ℓ p C , . . . , C n q ( . )turns the identity into ÿ i,j ě i ` j “ n ` ÿ σ ¯ ℓ p C σ p q , . . . , C σ p j ´ q , ¯ ℓ p C σ p j q , . . . , C σ p n q qq “ , ( . )where the primed inner sum denotes summation over unshuffles. t remains to investigate the sign factors induced by the statistics of the elementsin the superalgebra. We therefore introduce a basis t c i u which consists of fermionic el-ements with odd ghost numbers and bosonic elements with even ghost numbers. Sincea ghost is always totally bosonic, this means that ghosts with odd ghost numbers havefermionic components in this basis and ghosts with even ghost numbers have bosoniccomponents. Furthermore, we include the x -dependence of the ghosts in the basis ele-ments c i (“DeWitt notation”) and thus treat the components as constants that we canmove out of the brackets. Then, our identities take the form ÿ i,j ě i ` j “ n ` ÿ σ ϕ j ´ p σ ; c q ¯ ℓ p c σ p q , . . . , c σ p j ´ q , ¯ ℓ p c σ p j q , . . . , c σ p n q qq “ , ( . )where ϕ j ´ p σ ; c q is the sign factor for the permutation σ in the graded symmetrisa-tion of the elements t c , . . . , c n , F u to t c σ p q , . . . , c σ p j ´ q , F, c σ p j q , . . . , c σ p n q u . Here, F isa fermionic element used to define the sign factor, which comes from the fact that thebrackets are fermionic.We now turn to the standard definition of L identities. The Koszul sign factor ε p σ ; x q for a permutation σ of n elements t x , . . . , x n u is defined inductively by anassociative and graded symmetric product x i ˝ x j “ p´ q | x i || x j | x j ˝ x i , ( . )where | x i | “ x i and 1 for “fermionic”. Then, x σ p q ˝ . . . ˝ x σ p n q “ ε p σ ; x q x ˝ . . . ˝ x n . ( . )Multiplying by a factor p´ q σ gives a graded antisymmetric product, which can be seenas a wedge product of super-forms, x σ p q ^ . . . ^ x σ p n q “ p´ q σ ε p σ ; x q x ^ . . . ^ x n . ( . )The standard form of the identities for an L bracket is ÿ i,j ě i ` j “ n ` p´ q i p j ´ q ÿ σ p´ q σ ε p σ ; x q ℓ p ℓ p x σ p q , . . . , x σ p i q q , x σ p i ` q , . . . , x σ p n q q “ . ( . ) he two equations ( . ) and ( . ) look almost identical. However, the assignment of“bosonic” and “fermionic” for the c ’s is opposite to the one for the x ’s. On the other hand,the brackets of x ’s are graded antisymmetric, while those of c ’s are graded symmetric.Seen as tensors, such products differ in sign when exchanging bosonic with fermionicindices. There is obviously a difference between a tensor being graded antisymmetric (the“ x picture”) and “graded symmetric with opposite statistics” (the “ c picture”). The twotypes of tensors are however equivalent as modules (super-plethysms) of a general linearsuperalgebra. As a simple example, a 2-index tensor which is graded antisymmetric canbe represented as a matrix ˜ a α ´ α t s ¸ , ( . )where a is antisymmetric and s symmetric, while a 2-index tensor which is gradedsymmetric in the opposite statistics is ˜ a α p α q t s ¸ . ( . )The tensor product V b V of a graded vector space V with itself can always be decom-posed as the sum of the two plethysms, graded symmetric and graded antisymmetric, i.e. , in the sum of the two super-plethysms. Equivalently, the same decomposition, asmodules of the general linear superalgebra gl p V q , is the sum of the graded antisymmetricand graded symmetric modules with the opposite assignment of statistics. The same istrue for higher tensor products b n V .This means that, as long as the brackets ℓ and ¯ ℓ are taken to be proportional up tosigns, the equations ( . ) and ( . ) contain the same number of equations in the same g -modules, but not that the signs for the different terms in the identities are equivalent.In order to show this, one needs to introduce an explicit invertible map, a so calledsuspension, from the “ x picture” to the “ c picture”, i.e. , between the two presentationsof the plethysms of the general linear superalgebra.Let us use a basis where all basis elements are labelled by an index A “ p a, α q , where a and and α correspond to fermionic and bosonic basis elements, respectively. We choosean ordering where the a indices are “lower” than the α ones. Any unshuffle then hasthe index structure t a . . . a k α . . . α k , a k ` . . . a ℓ α k ` . . . α ℓ u . If the brackets ℓ and ¯ ℓ re expressed in terms of structure constants, ℓ p x A , . . . , x A n q “ f A ...A n B x B , ¯ ℓ p c A , . . . , c A n q “ ¯ f A ...A n B c B , ( . )the respective identities contain terms of the forms p´ q i p j ´ q p´ q σ ε p a . . . a k α . . . α k a k ` . . . a ℓ α k ` . . . α ℓ qˆ f a ...a k α ...α k B f Ba k ` ...a ℓ α k ...α ℓ A , p´ q m ϕ j ´ p a m ` . . . a ℓ α m ` . . . α ℓ a . . . a m α . . . α m q ( . ) ˆ ¯ f a m ` ...a ℓ α m ...α ℓ B ¯ f a ...a m α ...α m BA , where k ` m “ ℓ , k ` m “ ℓ , k ` k “ i , m ` m “ j ´ i, j being the same variables asin the sums ( . ) and ( . )). Now, both expressions need to be arranged to the sameindex structure, which we choose as a . . . a ℓ α . . . α ℓ . This gives a factor p´ q k m forthe f term, and p´ q km for ¯ f . In order to compare the two brackets, we also need tomove the summation index B to the right on f when B “ β and to the left on ¯ f when B “ b . All non-vanishing brackets have a total odd number of “ a indices”, including theupper index, so B “ b when k is even, and B “ β when k is odd. This gives a factor p´ q m for the f expression when k is odd, and p´ q m for ¯ f when k is even.The task is now to find a relation¯ f a ...a k α ...α k B “ ̺ p k, k q f a ...a k α ...α k B ( . )for some sign ̺ p k, k q . The resulting relative sign between the two expressions in eq.( . ) must then be the same for all terms in an identity, i.e. , it should only depend on ℓ “ k ` m and ℓ “ k ` m . Taking the factors above into consideration, this conditionreads k even : p´ q p k ` k q m ̺ p k, k q ̺ p m ` , m q “ τ p k ` m, k ` m q ,k odd : p´ q p k ` k q m ̺ p k, k q ̺ p m, m ` q “ τ p k ` m, k ` m q . ( . )This is satisfied for ̺ p k, k q “ p´ q k p k ´ q , ( . ) ith τ p ℓ, ℓ q “ ̺ p ℓ, ℓ q . The last relation is natural, considering that the equations inturn belong to the two different presentations of the same super-plethysm. This givesthe explicit translation between the two pictures.All structure constants carry an odd number of a indices (including the upper one).This is a direct consequence of the fact that all brackets are fermionic in the c picture(since the BV antibracket is fermionic). The relation between the structure constants inthe two pictures implies, among other things, that¯ f aβ “ f aβ , ¯ f αb “ f αb , ¯ f a a b “ f a a b , ( . )¯ f aαβ “ f aαβ , ¯ f α α b “ ´ f α α b . The first two of these equations relate the 1-bracket (derivative) in the two pictures, andthe remaining three the 2-bracket. Using these relations we can give an explicit exampleof how identities in the two pictures are related to each other. Let us write | c a | “ | c α | “
0. We then have¯ ℓ p c A , c B q “ p´ q | c A || c B | ¯ ℓ p c B , c A q , ℓ p x A , x B q “ ´p´ q p| c A |` qp| c B |` q ℓ p x B , x A q . ( . )Furthermore, the relations ( . ) imply that under the inverse of the suspension,¯ ℓ p c A q ÞÑ ℓ p x A q , ¯ ℓ p c A , c B q ÞÑ p´ q | c A |` ℓ p x A , x B q . ( . )In the c picture, we have the identity¯ ℓ p ¯ ℓ p c A , c B qq ` p´ q | c A | ¯ ℓ p c A , ¯ ℓ p c B qq ` p´ q p| c A |` q| c B | ¯ ℓ p c B , ¯ ℓ p c A qq “ . ( . )Moving the inner 1-bracket to the left, the left hand side is equal to the expression¯ ℓ p ¯ ℓ p c A , c B qq ` p´ q | c A || c B | ¯ ℓ p ¯ ℓ p c B q , c A q ` ¯ ℓ p ¯ ℓ p c A q , c B q , ( . ) hich, according to ( . ), is mapped to p´ q | c A |` ℓ p ℓ p x A , x B qq ` p´ q p| c A |` q| c B | ℓ p ℓ p x B q , x A q ` p´ q | c A | ℓ p ℓ p x A q , x B q ( . ) “ p´ q | c A |` ´ ℓ p ℓ p x A , x B qq ` p´ q p| c A |` qp| c B |` q ℓ p ℓ p x B q , x A q ´ ℓ p ℓ p x A q , x B q ¯ “ p´ q | c A |` ˆ ℓ p ℓ p x A , x B qq ´ ´ ℓ p ℓ p x A q , x B q ´ p´ q p| c A |` qp| c B |` q ℓ p ℓ p x B q , x A q ¯˙ . Setting this to zero gives the identity in the x picture corresponding to the identity( . ) in the c picture.Note that the issue with the two pictures arises already when constructing a BRSToperator in a situation where one has a mixture of bosonic and fermionic constraints.In the rest of the paper, we stay within the c picture, i.e. , we work with ghosts withgraded symmetry. L structure, ignoring ancillary ghosts The following calculation will first be performed disregarding ancillary ghosts, i.e. , as ifall r R p “
0. The results will form an essential part of the full picture, but the structuredoes not provide an L subalgebra unless all r R p “ C which is totally bosonic, i.e. , | C | “
0, and which is a generalelement of B ` p g r q , i.e. , a height 0 element of B ` p g r ` q . This gives the correct statisticsof the components, namely the same as the basis elements in the superalgebra. All signsare taken care of automatically by the statistics of the ghosts. While the superalgebrabracket is graded antisymmetric, the L brackets (by which we mean the brackets in the c picture of the previous Section, before the rescaling of eq. ( . )) are graded symmetric.The a index of the previous Section labels ghosts with odd ghost number, and the α index those with even ghost number, and include also the coordinate dependence. The 1-bracket acting on a ghosts at height 0 is taken as rr C ss “ dC . ( . ) hen the 1-bracket identity rrrr C ssss “ c is rr c, c ss “ L c c , ( . )in order to reproduce the structure of the generalised diffeomorphisms. This alreadyassumes that there are no ancillary transformations, which also would appear on theright hand side of this equation, and have their corresponding ghosts (we will commenton this situation in the Conclusions). It is natural to extend this to arbitrary levels bywriting rr C, C ss “ L C C . ( . )Given the relations ( . ) between low brackets in the two pictures in the previousSection, this essentially identifies the - and -brackets between components with theones in the traditional L language (the x picture). Recall, however, that our ghosts C are elements in the superalgebra, formed as sums of components times basis elements,which lends a compactness to the notation, which becomes index-free.There are potentially two infinities to deal with, one being the level of the ghosts,the other the number of arguments in a bracket. In order to deal with the first one, weare trying to derive a full set of 2-brackets before going to higher brackets. Of course,the existence of higher level ghosts is motivated by the failure of higher identities, so itmay seem premature to postulate eq. ( . ) before we have seen this happen. However,it is essential for us to be able to deal with brackets for arbitrary elements, withoutsplitting them according to level. The identity for the 2-bracket is then satisfied, since rrrr C, C ssss ` rr C, rr C ssss “ d L C C ` ¨ L C dC “ . ( . )Notice that this implies that the 2-bracket between ghosts which are both at level 2 orhigher vanishes.There is of course a choice involved every time a new bracket is introduced, and thechoices differ by something exact. The choice will then have repercussions for the restof the structure. The first choice arises when the need for a level 2 ghost C becomesclear (from the 3-bracket identity as a modification of the Jacobi identity), and its 2- racket with the level 1 ghost is to be determined. Instead of choosing rr c, C ss “ L c C ,corresponding to eq. ( . ), we could have taken rr c, C ss “ ´ r c, dC s , since the derivativeof the two expressions are the same (modulo ancillary terms) according to eq. ( . ).The latter is the type of choice made in e.g. ref. [ ]. Any linear combination of thetwo choices with weight 1 is of course also a solution. However, it turns out that otherchoices than the one made here lead to expressions that do not lend themselves to unifiedexpressions containing C as a generic element in B ` p g r q . Thus, this initial choice andits continuation are of importance.We now turn to the 3-bracket. The identity is rrrr C, C, C ssss ` rr C, rr C, C ssss ` rr C, C, rr C ssss “ . ( . )The second term (the Jacobiator) equals L C rr C, C ss ` L rr C,C ss C . Here we must assumethe closure of the transformations, acting on something, i.e. , the absence of ancillarytransformations in the commutator of two level 1 transformations. Then, L C rr C, C ss “ L C L C C “ ´ L rr C,C ss C , ( . )and the second term in eq. ( . ) can be written expressed in terms of the (graded)antisymmetric part instead of the symmetric one, so that the derivation property maybe used: 2 rr C, rr C, C ssss “ ´ p L C L C C ´ L L C C C q“ ´ p d r C, L C C s ´ r C, d L C C s ` r dC, L C C sq“ ´ p d r C, L C C s ` r C, L C dC s ` r dC, L C C sq ( . )(modulo ancillary terms). If one takes rr C, C, C ss “ r C, L C C s , ( . )the identity is satisfied, since then rrrr C, C, C ssss “ d r C, L C s , ( . ) nd 3 rr C, C, rr C ssss “ ¨ p r C, L C dC s ` r dC, L C C sq . ( . )Starting from the 4-bracket identity rrrr C, C, C, C ssss ` rr C, rr C, C, C ssss ` rr C, C, rr C, C ssss ` rr C, C, C, rr C ssss “ , ( . )a calculation gives at hand that the second and third terms cancel (still modulo ancillaryterms). This would allow rr C, C, C, C ss “
0. The calculation goes as follows. We use thebrackets and identities above to show rr C, rr C, C, C ssss “ rr C, r C, L C C sss “ L C r C, L C C s“ ´ r L C C, L C C s ` r C, L C L C C s“ ´ r L C C, L C C s ´ r C, L L C C C s ( . )and rr C, C, rr C, C ssss “ pr L C C, L C C s ` r C, L L C C C s ` r C, L C L C C sq“ r L C C, L C C s ` r C, L L C C C s . ( . )This does not imply that all higher brackets vanish. Especially, the middle term3 rr C, C, rr C, C, C ssss in the 5-bracket identity is non-zero, which requires a 5-bracket.
In order to go further, we need to perform calculations at arbitrary order. There isessentially one possible form for the n -bracket, namely rr C n ss “ k n p ad C q n ´ L C C . ( . )It turns out that the constants k n are given by Bernoulli numbers, k n ` “ n B ` n n ! , ( . ) here B ` n “ p´ q n B n (which only changes the sign for n “
1, since higher odd Bernoullinumbers are 0).We will first show that it is consistent to set all rr C n ss “ n ě
2. Then the2 p n ` q -identity reduces to0 “ rr C, rr C n ` ssss ` p n ` qrr C n , rr C, C ssss . ( . )Evaluating the two terms gives rr C, rr C n ` ssss “ rr C, k n ` p ad C q n ´ L C C ss“ k n ` L C p ad C q n ´ L C C ( . ) “ k n ` ´ p ad C q n ´ L C L C C ´ n ´ ÿ i “ p ad C q i ad L C C p ad C q n ´ ´ i L C C ¯ “ k n ` ´ ´ p ad C q n ´ L L C C C ´ n ´ ÿ i “ p ad C q i ad L C C p ad C q n ´ ´ i L C C ¯ , rr C n , rr C, C ssss “ k n ` n ` ´ p ad C q n ´ L C L C C ` p ad C q n ´ L L C C C ` n ´ ÿ i “ p ad C q i ad L C C p ad C q n ´ ´ i L C C ¯ ( . ) “ k n ` n ` ´ p ad C q n ´ L L C C C ` n ´ ÿ i “ p ad C q i ad L C C p ad C q n ´ ´ i L C C ¯ , which shows that eq. ( . ) is fulfilled.We then turn to the general n -identities, n ě n ). They are0 “ rrrr C n ssss ` n ´ ÿ i “ p i ` qrr C i , rr C n ´ i ssss ` n rr C n ´ , rr C ssss . ( . )The first term equals k n d p ad C q n ´ L C C . Repeated use of eq. ( . ) (without the ancillary erm) gives d p ad C q n ´ L C C “ ´ n ´ ÿ i “ p ad C q i ad dC p ad C q n ´ i ´ L C C ´ n p ad C q n ´ L L C C C ´ n ´ ÿ i “ p i ` qp ad C q i ad L C C p ad C q n ´ i ´ L C C . ( . )The first sum cancels the last term in eq. ( . ). We now evaluate the middle termsunder the summation sign in eq. ( . ). rr C i , rr C n ´ i ssss “ k i ` k n ´ i i ` ´ ´ p ad C q n ´ L L C C C ` i ´ ÿ j “ p ad C q j ad pp ad C q n ´ i ´ L C C qp ad C q i ´ j ´ L C C ´ n ´ i ´ ÿ j “ p ad C q i ` j ´ ad L C C p ad C q n ´ i ´ j ´ L C C ¯ . ( . )Here we have ignored the insertion of the 2-bracket in the argument of the generalisedLie derivative in the p n ´ q -bracket (which changes the sign of the term with L L C C C ),since this already has been taken care of in eqs. ( . ) and ( . ). It does not appear inthe identity for odd n .Let n “ m ` i “ j . There is a single term containing L L C C C , namely ´ k j ` k p m ´ j q` p j ` q p ad C q m ´ L L C C C . ( . )The total coefficient of this term in eq. ( . ) demands that k n ` “ ´ n ` n ´ ÿ j “ k j ` k p n ´ j q` . ( . )It is straightforward to show that the Bernoulli numbers satisfy the identity m ´ ÿ j “ B j B p m ´ j q p j q ! p p m ´ j qq ! “ ´p m ` q B m p m q ! . ( . ) t follows from the differential equation ddt r t p f ´ t qs ` f “
0, satisfied by f p t q “ te t ´ ` t ´ , where te t ´ “ ÿ n “ B n n ! t n . ( . )The p m ` q -identity ( . ) then is satisfied with the coefficients given by eq. ( . ). Theinitial value k “ fixes the coefficients to the values in eq. ( . ). Bernoulli numbersas coefficients of L brackets have been encountered earlier [ , ].In order to show that the identities are satisfied at all levels, we use the methoddevised by Getzler [ ] (although our expressions seem to be quite different from theones in that paper). All expressions remaining after using the derivation property andidentifying the coefficients using the L L C C C terms are of the form Z n,j,k “ p ad C q n ´ ´ j ´ k rp ad C q j L C C, p ad C q k L C C s . ( . )There are however many dependencies among these expressions. First one observes that,since L C C is fermionic, Z n,j,k “ Z n,k,j . Furthermore, the Jacobi identity immediatelygives Z n,j,k “ Z n,j ` ,k ` Z n,j,k ` ( . )for j ` k ă n ´
4. If one associates the term Z n,j,k with the monomial s j t k , the Jacobiidentity implies s j t k « s j ` t k ` s j t k ` , i.e. , p s ` t ´ q s j t k «
0. We can then replace s by 1 ´ t , so that s j t k becomes p ´ t q j t k . The symmetry property is taken care of bysymmetrisation, so that the final expression corresponding to Z n,j,k is pp ´ t q j t k ` t j p ´ t q k q . ( . )All expressions are reduced to polynomials of degree up to n ´ t Ø ´ t . An independent basis consists of even powers of t ´ . In addition tothe equations with L L C C C that we have already checked, there are m ´ p L C C q in the p m ` q -identity, involving k m ` andproducts of lower odd k ’s.We will now show that all identities are satisfied by translating them into polynomialswith Getzler’s method, using the generating function for the Bernoulli numbers.Take the last sum in eq. ( . ). It represents the contribution from the first and last erms in the identity. It translates into the polynomial ´ k n n ´ ÿ i “ p i ` q t n ´ i ´ “ ´ k n n ´ ´ p n ´ q t ` t n ´ p ´ t q . ( . )The terms from the middle terms in the identity (eq. ( . )) translate into n ´ ÿ i “ k i ` k n ´ i ´ i ´ ÿ j “ s n ´ i ´ t i ´ j ´ ´ n ´ i ´ ÿ j “ t n ´ i ´ j ´ ¯ “ n ´ ÿ i “ k i ` k n ´ i ´ s n ´ i ´ ´ t i ´ ´ t ´ ´ t n ´ i ´ ´ t ¯ . ( . )Let f p x q be the generating function for the coefficients k n , i.e. , f p x q “ ÿ n “ k n x n “ ÿ n “ n B ` n n ! x n ` “ x ´ e ´ x ´ x “ x ` x ´ x ` x ´
14 725 x `
293 555 x ´ x ` ¨ ¨ ¨ ( . )We now multiply the contributions from eqs. ( . ) and ( . ), symmetrised in s and t , by x n and sum over n , identifying the function f when possibility is given. This gives12 p ´ t q ´ ´p ´ t q xf p x q ` p ´ t q f p x q ´ f p tx q t ¯ ` p ´ t q x ´ ´ f p x q ` f p x q f p sx q s ` f p x q f p tx q t ´ f p sx q f p tx q s t ¯ ` p s Ø t q . ( . )When the specific function f is used, this becomes, after some manipulation, φ p s, t, x q “ p s ` t ´ q x st ´ p s ` t ´ qp ´ s q x t p ´ s q sinh pp ´ s q x q sinh x sinh p sx q´ p s ` t ´ qp ´ t q x s p ´ t q sinh pp ´ t q x q sinh x sinh p tx q` p s ` t ´ q x p ´ s qp ´ t q x ´ ´ sinh pp ´ s q x q sinh pp ´ t q x q sinh p sx q sinh p tx q ¯ . ( . )This expression clearly vanishes when s ` t ´ “
0, which proves that the identities forthe brackets hold to all orders.The function φ p s, t, x q “ ř n “ φ n p s, t q , with the coefficient functions φ n p s, t q given y the sum of eqs. ( . ) and ( . ), symmetrised in s and t , will appear again in manyof the calculations for the full identities in Section .The complete variation p S, C q “ ř n “ rr C n ss can formally be written as p S, C q “ dC ` g p ad C q L C C , ( . )where g is the function g p x q “ x f p x q “ ´ e ´ x ´ x . ( . )Then p S, p S, C qq “
0. This concludes the analysis in the absence of ancillary terms.
We have already encountered “ancillary terms”, whose appearance in various identitiesfor the operators, such as the deviation of d from being a derivation and the deviationof d from being covariant, rely on the existence of modules r R p . Note that the Borcherdssuperalgebra always has r R “ H , i.e. , R p , q “ R ; this is what prevents us from treatingsituations where already the gauge “algebra” of generalised Lie derivatives contains an-cillary transformations. The ancillary terms at level p appear as r B M , r F M s “ r B M , r F M s ‚where B M is an element in r R p ` at height 1 ( i.e. , B M “ B M carries an extra R index, which is “in section”, meaning that the relations ( . ) are fulfilled also when oneor two B M ’s are replaced by a B M .The appearance of ancillary terms necessitates the introduction of ancillary ghosts.We will take them as elements K p P R p at height 1 constructed as above. The idea isthen to extend the 1-bracket to include the operator , which makes it possible to cancelancillary terms in identities (ignored in the previous Section) by a “derivative” of otherterms at height 1.The derivative d and the generalised Lie derivative L C are extended to level 1 as inSection . . This implies that anticommutes with d and with L C . Since d “ d d “ R p at height 0 and 1, it can be used in the construction of a1-bracket, including the ancillary ghosts. The generic structure is shown in Table .Ancillary elements form an ideal A of B ` p g r q . Let K “ r B M , r F M s as above, and et A P B ` p g r q . Then, r A, K s “ rr A, B M s , r F M s ` p´ q | A || B | r B M , r A, r F M ss . ( . )The first term is ancillary, since the height 1 element r A, B M s is an element in r R p A ` p B ,thanks to r A, B M s “
0, and the section property of the M index remains. The secondterm has r A, r F M s ‰ p A “
1‚ but vanishes thanks to r B M , f s “
0. This showsthat r B ` p g r q , A s Ă A . An explicit example of this ideal, for the E exceptional fieldtheory in the M-theory section, is given in Section , Table .Let us consider the action of d on ancillary ghosts K at height 1. Let B M P r R p ` with height 1, and let K “ r B M , r F M s P R p at height 0. We will for the moment assumethat r B M , F M s “ . ( . )This is a purely algebraic condition stating that R p but not r R p is present in the tensorproduct r R p ` b R in B M . Then, K “ r B M , r F M s . Acting with the derivative gives dK “ p ´ rrB N B M , r F M s , F N s “ p ´ rrB N B M , F N s , r F M s “ r B M , r F M s , ( . )where B M “ p ´ rB N B M , F N s . The derivative preserves the structure, thanks to thesection constraint. Also, the condition ( . ) for B , r B M , F M s “
0, is automaticallysatisfied.The appearance of modules r R p can be interpreted in several ways. One is as aviolation of covariance of the exterior derivative, as above. Another is as a signal thatPoincaré’s lemma does not hold. In this sense, ancillary modules encode the presenceof “local cohomology”, i.e. , cohomology present in an open set. It will be necessary tointroduce ghosts removing this cohomology.Let the lowest level p for which r R p ` is non-empty be p . Then it follows that anancillary element K p at level p will be closed, dK p “
0, and consequently dK p “ K p does not need to be a total derivative, since B M does not need to equal B M Λ. Indeed, our ancillary terms are generically not total derivatives. An ancillaryelement at level p represents a local cohomology, a violation of Poincaré’s lemma.The algebraic condition ( . ) was used to show that the ancillary property is pre-served under the derivative. Consider the expression X A B from eq. ( . ). Raised to p (cid:15) (cid:15) o o d K p ` (cid:15) (cid:15) o o d K p ` (cid:15) (cid:15) o o d ¨ ¨ ¨ o o d C o o d ¨ ¨ ¨ o o d C p ´ o o d C p o o d C p ` o o d C p ` o o d ¨ ¨ ¨ Table : The typical structure of the action of the -bracket between the ghost modules,with ancillary ghosts appearing from level p ě . height 1 it gives an expression K “ rr β MN , F N s , r F M s “ r B M , r F M s ( . )with B M “ r β MN , F N s , where β MN is symmetric and where both its indices are insection. Then, r B M , F M s “
0, and the condition is satisfied. The same statement cannot be made directly for any term R p A, B q , since it contains only one derivative. One canhowever rely the identities ( . ) and ( . ), which immediately show (in the latter casealso using the property that ancillary expressions form an ideal) that the derivatives andgeneralised Lie derivatives of an ancillary expression (expressed as R p A, B q ) is ancillary.This is what is needed to consistently construct the brackets in the following Section.The section property of B M implies that L K A “ K is an ancillary expres-sion (see eq. ( . )). This identity is also used in the calculations for the identities ofthe brackets. L structure We will now display the full L structure, including ancillary ghosts. The calculationsfor the L brackets performed in Section will be revised in order to include ancillaryterms. .1 Some low brackets The 1-bracket, which now acts on the ghosts C at height 0, and also on ancillary ghosts K at height 1, is d ` 5 : rr C ` K ss “ dC ` K ` dK . ( . )Since d “ 5 “ d d “
0, the identity rrrr C ` K ssss “ d .The 2-bracket identity was based on “ d L C C ` L C dC “ rr C, C ss “ L C C ` X C C , rr C, K ss “ L C K , rr K, K ss “ . ( . )Then, rrrr C, C ssss ` rr C, rr C ssss“ rr L C C ` X C C ss ` rr C, dC ss“ d L C C ` X C C ` dX C C ` L C dC ` X C dC “ . ( . )thanks to eqs. ( . ) and ( . ), and rrrr C, K ssss ` ¨ rr C, rr K ssss ` ¨ rr K, rr C ssss“ ´ d L C K ` p L C K q ` L C dK ` L C K ` X C K ¯ “ . ( . )The terms at height 1 cancel using X C K “ X C K , where the sign follows from passingboth a d and an L C . Here, we have of course used L K “
0. Note that the height0 identity involving one K is trivial, while the identity at height 1 identity with one K is equivalent to the height 0 identity with no K ’s. These are both general features,recurring in all bracket identities. In addition rr K, K ss “ L K K “
0, implying thatthe bracket with two K ’s consistently can be set to 0.Consider the middle term in the 3-bracket identity. Including ancillary terms, we ave 2 rr C, rr C, C ssss “ rr C, L C C ` X C C ss“ L C L C C ` X C L C C ` L L C C C ` X L C C C ` L C X C C “ p L L C C C ` X L C C C q . ( . )We know that rr C, C, C ss contains the non-ancillary term r C, L C C s . Calculating thecontribution from this term to rrrr C, C, C ssss ` rr C, C, rr C ssss gives d r C, L C C s ` r C, L C dC s ` r dC, L C C s“ ´ ´ L L C C ´ r C, X C C s ´ R p C, L C C q ¯ ( . )There is still no sign of something cancelling the second term in eq. ( . ), but thepresence of lowered ancillary terms implies that it is necessary to include the ancillaryterms pr C, X C C s ` R p C, L C C qq in the 3-bracket. The term in rrrr C, C, C ssss from the part of the 1-bracket will then cancel these. We still need to check the terms at height1. The height 1 contribution to rrrr C, C, C ssss ` rr C, C, rr C ssss from r C, X C C s is ` d r C, X C C s ` r C, X C dC s ` r dC, X C C s ˘ “ ` L C X C C ` R p C, X C C q ˘ , ( . )and from R p C, L C C q , using eq. ( . ): ` dR p C, L C C q ` R p C, L C dC q ` R p dC, L C C q ˘ “ ` X C L C C ´ X L C C C ´ R p C, X C C q ˘ . ( . )The complete height 1 terms in the 3-bracket identity become p ´ q X L C C C ` p L C X C ` X C L C q C “ . ( . )Checking the 3-bracket identity with two C ’s and one K becomes equivalent to theheight 0 identity for the bracket with three C ’s when rr C, C, K ss “ pr C, L C K s ` r K, L C C sq . ( . )There is also a height 0 part of the CCK identity, which is trivial since generates no ncillary terms. Again, there is no need for a bracket with CKK , since rr C, K, K ss “ pr K , L C K s ` r K, L C K sq “ . ( . )These properties will be reflected at all orders, and we do not necessarily mention themevery time.The 4-bracket identity with four C ’s reads rrrr C, C, C, C ssss ` rr C, rr C, C, C ssss ` rr C, C, rr C, C ssss ` rr C, C, C, rr C ssss “ . ( . )We will now show that the vanishing of the 4-bracket persists when ancillary terms aretaken into account. The height 1 terms in 2 rr C, rr C, C, C ssss are ` X C r C, L C s ` L C r C, X C C s ` L C R p C, L C C q ˘ , ( . )and those in 3 rr C, C, rr C, C ssss become ` r C, X C L C C s ` r C, L C X C C s ` r C, X L C C C s ` r L C C, X C C s` R p C, L L C C C q ` R p L C C, L C C q ˘ ( . )The terms cancel, using eqs. ( . ) and ( . ). The structure encountered so far can be extended to arbitrarily high brackets. Knowingthe height 0 part of rr C n ss “ k n p ad C q n ´ L C C enables us to deduce the ancillary part.Namely, keeping ancillary terms when applying eq. ( . ) sequentially, calculating thefirst and last terms in the n -bracket identity gives, apart from the second row of eq.( . ), d p ad C q n ´ L C C “ . . . ´ ´ p ad C q n ´ X c C ` n ´ ÿ i “ p ad C q i R C p ad C q n ´ i ´ L C C ¯ . ( . )This forces the n -bracket to take the form rr C n ss “ k n ´ p ad C q n ´ p L C C ` X C C q ` n ´ ÿ i “ p ad C q i R C p ad C q n ´ i ´ L C C ¯ . ( . ) t is then reasonable to assume that rr C n ´ , K ss is obtained from the symmetrisation ofthe height 0 part of rr C n ss , i.e. , rr C n ´ , K ss “ k n n ´ p ad C q n ´ L C K ` n ´ ÿ i “ p ad C q i ad K p ad C q n ´ i ´ L C C ¯ , ( . )and that brackets with more than one K vanish.We will show that the set of non-vanishing brackets above is correct and complete.The height 0 identity with only C ’s is already satisfied, thanks to the contribution from in rrrr C n ssss . The height 1 identity with one K contains the same calculation. The height 0identity with one K is trivial, and just follows from moving ’s in and out of commutatorsand through derivatives and generalised Lie derivatives. The vanishing of the bracketswith more than one K is consistent with the vanishing of rr C n ´ , K , K ss . Lowering thisbracket gives rr C n ´ , K , K ss which vanishes by statistics, since K is fermionic.The only remaining non-trivial check is the height 1 part of the identity with only C ’s. This is a lengthy calculation that relies on all identities exposed in Section . Wewill go through the details by collecting the different types of terms generated, one byone.A first result of the calculation is that all terms containing more than one ancillaryexpression X or R cancel. This important consistency condition relies on the precisecombination of terms in the n -bracket, but not on the relation between the coefficients k n . It could have been used as an alternative means to obtain possible brackets.We then focus on the terms containing X . In addition to its appearance in thebrackets, X arises when a derivative or a generalised Lie derivative is taken through an R , according to eqs. ( . ) and ( . ). It turns out that all terms where X C appears in an“inner” position in terms of the type p ad C q i X C p ad C q n ´ i ´ L C C , with n ´ i ą
3, cancel.This again does not depend on the coefficients k n . Collecting terms p ad C q n ´ L C X C C and p ad C q n ´ X C L C C , the part rrrr C n ssss ` n rr C n ´ , rr C ssss gives a contribution k n p n ´ qp ad C q n ´ L C X C C ( . )from the X term in the bracket, and k n ` p n ´ qp ad C q n ´ X C L C C ´ p ad C q n ´ X L C C C ˘ ( . ) rom the R term, together giving ´ n k n p ad C q n ´ X L C C C . ( . )A middle term in the identity, rr C i , rr C n ´ i ssss contains ´ k i ` k n ´ i p ad C q n ´ X L C C C . ( . )The total contribution cancels, thanks to the relation ( . ) between the coefficients.The remaining terms with X are of the types p ad C q j ad L C C p ad C q n ´ ´ j X C C and p ad C q j ad X C C p ad C q n ´ ´ j L C C and similar. The first and last term in the identitygives a contribution ´ k n n ´ ÿ j “ p j ` qp ad C q j ad L C C p ad C q n ´ ´ j X C C ( . )from the X term in the n -bracket, and ´ k n n ´ ÿ j “ p j ` qp ad C q j ad X C C p ad C q n ´ ´ j L C C ( . )from the R term. A middle term rr C i , rr C n ´ i ssss gives k i ` k n ´ i ´ ´ n ´ i ´ ÿ j “ p ad C q i ` j ´ ad L C C p ad C q n ´ i ´ j ´ X C C ´ n ´ i ´ ÿ j “ p ad C q i ` j ´ ad X C C p ad C q n ´ i ´ j ´ L C C ` i ´ ÿ j “ p ad C q j ad pp ad C q n ´ i ´ L C C qp ad C q i ´ j ´ X C C ` i ´ ÿ j “ p ad C q j ad pp ad C q n ´ i ´ X C C qp ad C q i ´ j ´ L C C ¯ . ( . )Note the symmetry between X C and L C in all contributions. We can now representa term p ad C q n ´ ´ j ´ k rp ad C q j L C C, p ad C q k X C C s by a monomial s j t k , exactly as inSection . . Since we have the symmetry under s Ø t , the same rules apply as in thatcalculation. Indeed, precisely the same polynomials are generated as in eqs. ( . ) and . ). The terms cancel.Finally, there are terms of various structure with one R and two L ’s. One suchstructure is p ad C q j R C p ad C q n ´ j ´ L L C C C . For each value of j , the total coefficient ofthe term cancels thanks to nk n ` ř i k i ` k n ´ i “
0. Of the remaining terms, many have C as one of the two arguments of R , but some do not. In order to deal with the latter,one needs the cyclic identity ( . ). Let F j “ p ad C q j L C C and S n,j,k “ p ad C q n ´ j ´ k ´ R p F j , F k q . ( . )Taking the arguments in the cyclic identity as C , F j and F k turns it into S n,j,k ´ S n,j ` ,k ´ S n,j,k ` “ ´p ad C q n ´ j ´ k ´ ` R C r F j , F k s ´ r F p j , R C F k q s ˘ . ( . )We need to verify that terms containing S n,j,k , i.e. , not having C as one of the argumentsof R , combine into the first three terms of this equations, and thus can be turned intoexpressions with R C . Note that this relation is analogous to eq. ( . ) for Z n,j,k inSection . , but with a remainder term. We now collect such terms. They are ´ k n n ´ ÿ j “ p n ´ ´ j q S n,j, ` n ´ ÿ i “ k i ` k n ´ i ´ i ´ ÿ j “ S n,j,n ´ i ´ ´ n ´ i ´ ÿ j “ S n,j, ¯ . ( . )This is the combination encountered earlier (eqs. ( . ) and ( . )), which means thatthese terms can be converted to terms with R C . However, since the “ s ` t ´ «
0” relationin the form ( . ) now holds only modulo R C terms, we need to add the corresponding R C terms to the ones already present.Let us now proceed to the last remaining terms. They are of two types: U n,r,j,k “ p ad C q r R C p ad C q n ´ r ´ j ´ k ´ rp ad C q j L C C, p ad C q k L C C s ,V n,r,j,k “ p ad C q n ´ r ´ j ´ k ´ rp ad C q j L C C, p ad C q k R C p ad C q r L C C s . ( . )If the j and k indices in both expressions are translated into monomials s j t k as before,both expressions should be calculated modulo s ` t ´ « U , symmetryunder s Ø t can be used, but not in V . Both types of terms need to cancel for all valuesof r , since there is no identity that allows us to take ad C past R C . he terms of type U n,r,j,k obtained directly from rrrr C n ssss ` n rr C n ´ , rr C ssss are ´ k n n ´ ÿ r “ n ´ r ´ ÿ k “ p n ´ k ´ q U n,r, ,k , ( . )and those from rr C i , rr C n ´ i ssss are k i ` k n ´ i ´ i ´ ÿ r “ i ´ r ´ ÿ k “ U n,r,n ´ i ´ ,k ´ i ´ ÿ r “ n ´ i ´ ÿ k “ U n,r, ,k ´ n ´ ÿ r “ i ´ n ´ r ´ ÿ k “ U n,r, ,k ¯ . ( . )To these contributions must be added the remainder term corresponding to the first termon the right hand side of eq. ( . ), with the appropriate coefficients from eq. ( . ).Let U n,r,j,k correspond to the monomial s j t k u r . According to eq. ( . ), the remainderterms then become u n ´ φ n p su , tu q su ` tu ´ « u n ´ ´ u φ n ` su , tu ˘ , ( . )where φ p s, t, x q “ ř n “ φ n p s, t q x n , and where s ` t ´ « n -bracket identity then is ” k n ´ ´ n ´ ÿ r “ n ´ r ´ ÿ k “ p n ´ k ´ q u r t k ´ u n ´ ´ u n ´ ÿ k “ p n ´ k ´ q ` tu ˘ k ¯ ` n ´ ÿ i “ k i ` k n ´ i ´ i ´ ÿ r “ i ´ r ´ ÿ k “ s n ´ i ´ t k u r ´ i ´ ÿ r “ n ´ i ´ ÿ k “ t k u r ´ n ´ ÿ r “ i ´ n ´ r ´ ÿ k “ t k u r ` u n ´ ´ u i ´ ÿ k “ ` su ˘ n ´ i ´ ` tu ˘ k ´ u n ´ ´ u n ´ i ´ ÿ k “ ` tu ˘ k ¯ı ` p s Ø t q“ ” ´ k n n ´ ´ p n ´ q t ` t n ´ p ´ t q p ´ u q` n ´ ÿ i “ k i ` k n ´ i p ´ t i ´ q s n ´ i ´ ´ p ´ t n ´ i ´ qp ´ t qp ´ u q ı ` p s Ø t q“ φ n p s, t q ´ u . ( . )The U n,r,s,t terms thus cancel for all values of r . he terms of type V n,r,s,t obtained directly from rrrr C n ssss ` n rr C n ´ , rr C ssss are ´ k n n ´ ÿ r “ n ´ r ´ ÿ k “ p n ´ r ´ k ´ q V n,r, ,k ( . )and the ones from rr C i , rr C n ´ i ssss are k i ` k n ´ i ´ ´ n ´ i ´ ÿ r “ n ´ i ´ r ´ ÿ k “ V n,r, ,k ` i ´ ÿ j “ n ´ i ´ ÿ k “ V n,n ´ i ´ k ´ ,j,k ` i ´ ÿ r “ i ´ r ´ ÿ k “ V n,r,n ´ i ´ ,k ¯ . ( . )In addition, there is a remainder term from the second term on the right hand side ofeq. ( . ). If V n,r,j,k is represented by s j t k u r , the remainder term becomes ´ φ n p s, u q s ` u ´ . ( . )The total contribution of terms of type V to the n -bracket is then represented by thefunction v n p s, t, u q : v n p s, t, u q “ ´ k n n ´ ÿ r “ n ´ r ´ ÿ k “ p n ´ r ´ k ´ q t k u r ` n ´ ÿ i “ k i ` k n ´ i ´ ´ n ´ i ´ ÿ r “ n ´ i ´ r ´ ÿ k “ t k u r ` i ´ ÿ j “ n ´ i ´ ÿ k “ s j t k u n ´ i ´ k ´ ` i ´ ÿ r “ i ´ r ´ ÿ k “ s n ´ i ´ t k u r ¯ ( . ) ` s ` u ´ ” k n n ´ ÿ ℓ “ p n ´ ℓ ´ qp s ℓ ` u ℓ q` n ´ ÿ i “ k i ` k n ´ i ´ n ´ i ´ ÿ ℓ “ p s ℓ ` u ℓ q ´ i ´ ÿ ℓ “ p s n ´ i ´ u ℓ ` s ℓ u n ´ i ´ q ¯ı . Performing the sums, except the ones over i , and replacing s by 1 ´ t , this function turnsinto v n p ´ t, t, u q “ φ n p ´ t, t q t ´ u . ( . ) herefore, these terms cancel. Note that the symmetrisation s Ø t in φ n is automatic,and not imposed by hand. This concludes the proof that all the identities are satisfied.The series ř n “ k n p ad C q n ´ appearing in the variation of the ghosts, the sum of allbrackets, can be written in the concise form g p ad C q , where g p x q “ ´ e ´ x ´ x . Likewise,the sum ř n “ k n n p ad C q n ´ becomes h p ad C q , where h p x q “ x x ż dy yg p y q “ ´ x ` log p ´ e ´ x q ´ x ` Li p e ´ x q ´ π ˘ . ( . )The terms in the brackets containing sums of type ř n ´ i “ p ad C q i O p ad C q n ´ i ´ can beformally rewritten, e.g. , ÿ n “ k n n ´ ÿ i “ p ad C q i O p ad C q n ´ i ´ “ ÿ n “ k n p ad C q n ´ L ´ p ad C q n ´ R p ad C q L ´ p ad C q R O “ g pp ad C q L q ´ g pp ad C q R qp ad C q L ´ p ad C q R O , ( . )where subscripts L, R stands for action to the left or to the right of the succeedingoperator ( O ). Then, the full ghost variation takes the functional form p S, C ` K q “ p d ` 5qp C ` K q ` g p ad C qp L C ` X C q C ` h p ad C q L C K ( . ) ` ” g pp ad C q L q ´ g pp ad C q R qp ad C q L ´ p ad C q R R C ı L C C ` ” h pp ad C q L q ´ h pp ad C q R qp ad C q L ´ p ad C q R ad K ı L C C .
The criterion that no ancillary transformations appear in the commutator of two gen-eralised diffeomorphisms is quite restrictive. It was shown in ref. [ ] that this happensif and only if g r is finite-dimensional and the derivative module is R p λ q where λ is afundamental weight dual to a simple root with Coxeter label 1. The complete list is( i ) g r “ A r , λ “ Λ p , p “ , . . . , r ( p -form representations);( ii ) g r “ B r , λ “ Λ (the vector representation);( iii ) g r “ C r , λ “ Λ r (the symplectic-traceless r -form representation);( iv ) g r “ D r , λ “ Λ , Λ r ´ , Λ r (the vector and spinor representations); “ ´ p “ p “ q “ q “ v 1 ‘ adj ‘ q “ ´ v 1 Table : The decomposition of A p r ` | q « sl p r ` | q in A p r q « sl p r ` q modules. ( v ) g r “ E , λ “ Λ , Λ (the fundamental representations);( vi ) g r “ E , λ “ Λ (the fundamental representation).If g r ` has a 5-grading or higher with respect to the subalgebra g r (in particular, if itis infinite-dimensional), r R will be non-empty (see Table ), and there will be ancillaryghosts starting from level 1 (ghost number 2).Ordinary diffeomorphisms provide a simple and quite degenerate example, where g r “ A r and λ “ Λ . In this case, both R and r R are empty, so both g r ` and B p g r q are -gradings. Still, the example provides the core of all other examples. The algebraof vector fields in r ` B p A r ` q « A p r ` | q « sl p r ` | q . ( . )There is of course neither any reducibility nor any ancillary ghosts, and the only ghostsare the ones in the vector representation v in R p , q . The double grading of the superal-gebra is given in Table .The double diffeomorphisms, obtained from g r “ D r , have a singlet reducibility,and no ancillary transformations. The L structure (truncating to an L algebra) wasexamined in ref. [ ]. The Borcherds superalgebra is finite-dimensional, B p D r ` q « D p r ` | q « osp p r ` , r ` | q . ( . )The double grading of this superalgebra is given in Table . The only ghosts are the “ ´ p “ ´ p “ p “ p “ q “ q “ ‘ adj ‘ q “ ´ Table : The decomposition of D p r ` | q « osp p r ` , r ` | q in D p r q « so p r, r q modules. (double) vector in R p , q and the singlet in R p , q .The extended geometry based on g r “ B r follows an analogous pattern, and is alsodescribed by Table , but with the doubly extended algebra B p r ` , q « osp p r ` , r ` | q being decomposed into modules of B p r q « so p r, r ` q .Together with the ordinary diffeomorphisms, these are the only cases with finitereducibility and without ancillary transformations at ghost number 1. In order for thereducibility to be finite, it is necessary that B p g r q is finite-dimensional. The remainingfinite-dimensional superalgebras in the classification by Kac [ ] are not represented byDynkin diagrams where the grey node connects to a node with Coxeter label 1. Therefore,even if there are other examples with finite-dimensional B p g r q , they all have ancillarytransformations appearing in the commutator of two generalised Lie derivatives. Suchexamples may be interesting to investigate in the context of the tensor hierarchy algebra(see the discussion in Section ).We now consider the cases g r “ E r for r ď
7. The level decompositions of theBorcherds superalgebras are described in ref. [ ]. There are always ancillary ghosts,starting at level 8 ´ r (ghost number 9 ´ r ). In Table , we give the double gradingin the example g r “ E p q « so p , q . Modules r R p are present for p ě
4, signalling aninfinite tower of ancillary ghost from ghost number 4. Table gives the correspondingdecomposition for g r “ E p q . This is as far as the construction of the present paperapplies. Note that for g r “ E p q already r R “ , which leads to ancillary ghosts in the at p p, q q “ p , q . In Table , we have divided the modules R p for the E p q example “ ´ p “ p “ p “ p “ p “ p “ q “ q “ ‘ ‘ q “
16 1 ‘ ‘ q “ ´
16 1
Table : Part of the decomposition of B p E p q q in E p q « so p , q modules. Note theappearance of modules r R p for p ě . p “ p “ p “ p “ p “ q “ q “ ‘ ‘ ¨ q “ ‘ ‘
56 8645 ‘ ¨ ‘ ‘ q “ ‘ ‘ ‘ q “ ´ Table : Part of the decomposition of B p E p q q in E p q modules. f Table into A modules with respect to a choice of section. Below the solid dividingline are the usual sequences of ghosts for diffeomorphisms and 2-form and 5-form gaugetransformations. Above the line are sequences that contain tensor products of formswith some other modules, i.e. , mixed tensors. All modules above the line are effectivelycancelled by the ancillary ghosts. They are however needed to build modules of g r . Inthe example, there is nothing below the line for p ě
7, which means that the operationfrom ancillary to non-ancillary ghosts at these levels becomes bijective.Reducibility is of course not an absolute concept; it can depend on the amount ofcovariance maintained. If a section is chosen, the reducibility can be made finite bythrowing away all ghosts above the dividing line. One then arrives at the situation inref. [ ]. If full covariance is maintained, reducibility is infinite. Since the modules abovethe line come in tensor products of some modules with full sets of forms of alternatingstatistics, they do not contribute to the counting of the degrees of freedom. This showswhy the counting of refs. [ , ], using only the non-ancillary ghosts, gives the correctcounting of the number of independent gauge parameters.This picture of the reduction of the modules R p in a grading with respect to thechoice of section also makes the characterisation of ancillary ghosts clear. They areelements in R p above a certain degree (for which the degree of the derivative is 0). Thedotted line in the table indicates degree 0. If we let A be the subalgebra of ancillaryelements above the solid line, it is clear that A forms an ideal in B ` p g r q (which wasalso shown on general grounds in Section ). The grading coincides with the gradingused in ref. [ ] to show that the commutator of two ancillary transformations again isancillary. “ p “ p “ p “ p “ p “ v “ p ‘ q b Λ v “ b Λ b Λ ‘ p ‘ q b Λ v “ b Λ b Λ ‘ b Λ b Λ ‘ b Λ ‘ b Λ v “ b Λ b Λ b Λ b Λ v “ Λ Λ Λ Λ Λ v “ Λ Λ v “ Table : Part of the decomposition of R p for the E p q exceptional geometry with respect to a section sl p q . The derivativeacts horizontally to the left and Λ k denote the k -form modules of sl p q , such that Λ , Λ , . . . “ , , . . . and Λ “ Λ “ .The degree v is such that the relative weights in the extension to gl p q are given by v ` p . The Λ in the lower leftcorner is the vector module corresponding to the ordinary coordinates with this choice of section. s an aside, the regularised dimension, twisted with fermion number, of B ` p g r ` q can readily be calculated using the property that all modules at p ‰ e and f , without need of any further regularisation( e.g. through analytic continuation). Using the cancellation of these doublets, inspectionof Table gives at hand that the “super-dimension” (where fermionic generators countwith a minus sign) ´ sdim p B ` p g r ` qq “ ` dim p R q ` dim p r R q ` dim p rr R q ` . . . “ ` dim p g r ` , ` q , ( . )where g r ` , ` is the positive level part of the grading of g r ` with respect to g r . Thisimmediately reproduces the counting of the effective number of gauge transformations inref. [ ]. In the example B p E q above, we get 1 ` “
17, which is the correct countingof gauge parameters for diffeomorphisms, 2- and 5-form gauge transformations in 6dimensions.
10 Conclusions
We have provided a complete set of bracket giving an L algebra for generalised dif-feomorphisms in extended geometry, including double geometry and exceptional geom-etry as special cases. The construction depends crucially on the use of the underlyingBorcherds superalgebra B p g r ` q , which is a double extension of the structure algebra g r . This superalgebra is needed in order to form the generalised diffeomorphisms, andhas a natural interpretation in terms of the section constraint. It also provides a clearcriterion for the appearance of ancillary ghosts.The full list of non-vanishing brackets is: rr C ss “ dC , rr K ss “ dK ` K , rr C n ss “ k n ´ p ad C q n ´ p L C C ` X C C q ` n ´ ÿ i “ p ad C q i R C p ad C q n ´ i ´ L C C ¯ ( . ) rr C n ´ , K ss “ k n n ´ p ad C q n ´ L C K ` n ´ ÿ i “ p ad C q i ad K p ad C q n ´ i ´ L C C ¯ , here the coefficients have the universal model-independent expression in terms ofBernoulli numbers k n ` “ n B ` n n ! , n ě . ( . )All non-vanishing brackets except the 1-bracket contain at least one level 1 ghost c . Nobrackets contain more than one ancillary ghost.The violation of covariance of the derivative, that modifies already the 2-bracket,has a universal form, encoded in X C in eq. ( . ). It is not unlikely that this makes itpossible to covariantise the whole structure, as in ref. [ ]. However, we think that it isappropriate to let the algebraic structures guide us concerning such issues.The characterisation of ancillary ghosts is an interesting issue, that may deservefurther attention. Even if the construction in Section makes the appearance of ancillaryghosts clear (from the existence of modules r R p ) it is indirect and does not containan independent characterisation of the ancillary ghosts, in terms of a constraint. Thisproperty is shared with the construction of ancillary transformations in ref. [ ]. Thecharacterisation in Section in terms of the grading induced by a choice of sectionis a direct one, in this sense, but has the drawback that it lacks full covariance. Inaddition, there may be more than one possible choice of section. This issue may becomemore important when considering situations with ancillary ghosts at ghost number 1(see below). Then, with the exception of some simpler cases with finite-dimensional g r ,ancillary transformations are not expected to commute.We have explicitly excluded from our analysis cases where ancillary transformationsappear already at ghost number 1 [ – , ]. The canonical example is exceptional ge-ometry with structure group E p q . If we should trust and extrapolate the results of thepresent paper, this would correspond to the presence of a module r R . However, there isnever such a module in the Borcherds superalgebra. If we instead turn to the tensor hier-archy algebra [ – ] we find that a module r R indeed appears in cases when ancillarytransformations are present in the commutator of two generalised diffeomorphisms.As an example, Table contains a part of the double grading of the tensor hierarchyalgebra W p E q (following the notation of ref. [ ]), which we believe should be used inthe construction of an L algebra for E generalised diffeomorphisms. The E modulesthat are not present in the B p E q superalgebra are marked in blue colour. The singlet at p p, q q “ p , q is the extra element appearing at level 0 in W p E q that can be identified “ ´ p “ p “ p “ q “ q “ ‘ ‘
248 1 ‘
248 248 ‘ ‘ ‘ q “ ‘ ‘ ‘
248 1 ‘ ‘ ‘ q “ ´
248 1
Table : Part of the decomposition of the tensor hierarchy algebra W p E q into E mod-ules. The modules not present in B p E q are marked blue. Note the presence of r R “ . with the Virasoro generator L (as can be seen in the decomposition under gl p q [ ]).The elements at q ´ p “ ´ W p E q (theembedding tensor or big torsion module). For an affine g r ` this is a shifted fundamentalhighest weight module, with its highest weight at p p, q q “ p , q , appearing in W p g r ` q in addition to the unshifted one with highest weight at p p, q q “ p , q appearing alsoin the Borcherds superalgebra B p g r ` q . In the E example, it contains the at p p, q q “ p , q which will accommodate parameters of the ancillary transformations. Insituations when ancillary transformations are absent at ghost number 1 (the subject ofthe present paper), using W p g r ` q is equivalent to using B p g r ` q , so all results derivedhere will remain unchanged.We take this as a very strong sign that the tensor hierarchy algebra is the correct un-derlying algebra, and hope that a generalisation of the present approach to the use of anunderlying tensor hierarchy algebra will shed new light on the properties of generaliseddiffeomorphisms in situations where ancillary transformations are present. Acknowledgements
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