# \(L_{\infty }\) Algebras for Extended Geometry from Borcherds Superalgebras

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## Abstract

We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin–Vilkovisky framework, or equivalently, an \(L_\infty \) algebra. The \(L_\infty \) brackets are given as derived brackets constructed using an underlying Borcherds superalgebra \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\), which is a double extension of the structure algebra \({{\mathfrak {g}}}_r\). The construction includes a set of “ancillary” ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra.

## 1 Introduction

The ghosts in exceptional field theory [1], and generally in extended field theory with an extended structure algebra \({{\mathfrak {g}}}_r\) [2], are known to fall into \({\mathscr {B}}_+({{\mathfrak {g}}}_r)\), the positive levels of a Borcherds superalgebra \({\mathscr {B}}({{\mathfrak {g}}}_r)\) [3, 4]. We use the concept of ghosts, including ghosts for ghosts etc., as a convenient tool to encode the structure of the gauge symmetry (structure constants, 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 formulation in terms of the structure constants of the Borcherds superalgebra \({\mathscr {B}}(E_{r+1})\). This generalises to extended geometry in general [2]. The more precise rôle of the Borcherds superalgebra has not been spelt out, and one of the purposes of the present paper is to fill this gap. The gauge structure of extended geometry will be described as an \(L_\infty \) algebra, governed by an underlying Borcherds superalgebra \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\). The superalgebra \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\) generalises \({\mathscr {B}}(E_{r+1})\) in Ref. [3], and is obtained from the structure algebra \({{\mathfrak {g}}}_r\) 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, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], the structure simplifies to an \(L_{n<\infty }\) algebra [20, 21, 22], and the reducibility becomes finite.

It is likely that a consistent treatment of quantum extended geometry will require a full Batalin–Vilkovisky treatment of the ghost sector, which is part of the motivation behind our work. Another, equally strong motivation is the belief that the underlying superalgebras carry much information about the models—also concerning physical fields and their dynamics—and that this can assist us in the future when investigating extended geometries bases on infinite-dimensional structure algebras.

The first \(8-r\) levels in \({\mathscr {B}}(E_r)\) consist of \(E_r\)-modules for form fields in exceptional field theory [1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 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 \(R_{p-1}\) at level \(p-1\) [31]. Above the window, the modules, when decomposed as \(\mathfrak {gl}(r)\) modules with respect to a local choice of section, start to contain mixed tensors, and covariance is lost. For \(E_8\), the window closes, not even the generalised diffeomorphisms are covariant [39] and there are additional restricted local \(E_8\) transformations [38]. Such transformations were named “ancillary” in ref. [2]. In the present paper, we will not treat the situation where ancillary transformations arise in the commutator of two generalised diffeomorphisms, but we will extend the concept of ancillary ghosts to higher ghost number. It will become clear from the structure of the doubly extended Borcherds superalgebra \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\) why and when such extra restricted ghosts appear, and what their precise 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 structure has not yet been formulated, we will perform all our calculations in the general setting with arbitrary structure group (which for simplicity will be taken to be simply laced, although non-simply laced groups present no principal problem). The general formulation of ref. [2] introduces no additional difficulty compared to any special case, and in fact provides the best unifying formalism also for the different exceptional groups. We note that the gauge symmetries of exceptional generalised geometry have been dealt with in the \(L_\infty \) algebra framework earlier [42]. However, this was done in terms of a formalism where ghosts are not collected into modules of \(E_r\), but consist of the diffeomorphism parameter together with forms for the ghosts of the tensor gauge transformations (*i.e.*, in generalised geometry, not in extended geometry).

*i.e.*, the part where ghosts and brackets belong to the \({\mathscr {B}}_+({{\mathfrak {g}}}_{r})\) subalgebra, is derived in Section 6. The complete non-ancillary variation \((S,C)=\sum _{n=1}^\infty [\![C^n]\!]\) can formally be written as

*g*is the function

## 2 The Borcherds Superalgebra

For simplicity we assume the structure algebra \({{\mathfrak {g}}}_r\) to be simply laced, and we normalise the inner product in the real root space by \((\alpha _i,\alpha _i)=2\). We let the coordinate module, which we denote \(R_1=R(-\lambda )\), be a lowest weight module^{1} with lowest weight \(-\lambda \). Then the derivative module is a highest weight module \(R(\lambda )\) with highest weight \(\lambda \), and \(R(-\lambda )=\overline{R(\lambda )}\).

*i.e.*, odd) root, which means that the associated Chevalley generators \(e_0\) and \(f_0\) belong to the fermionic subspace of the resulting Lie superalgebra \(\mathscr {B}({{\mathfrak {g}}}_r)\). In both cases, the inner product of the additional simple root with those of \({{\mathfrak {g}}}_r\) is given by the Dynkin labels of \(\lambda \), with a minus sign,

^{2}We will then get two different Dynkin diagrams (two different sets of simple roots) corresponding to the same Lie superalgebra \(\mathscr {B}({{\mathfrak {g}}}_{r+1})\). These are shown in Figure 1 in the case when \({{\mathfrak {g}}}=E_r\) and \(\lambda \) is the highest weight of the derivative module in exceptional geometry. The line between the two grey nodes in the second diagram indicate that the inner product of the two corresponding simple roots is \((\beta _{-1},\beta _0)=1\), not \(-1\) as when one or both of the nodes are white.

*e*,

*f*and

*h*. They satisfy the (anti-)commutation relations

*h*on \(e_0\) and \(f_0\) we have

*e.g.*\(\{\cdot ,\cdot \}\), common in the physics literature) for brackets between a pair of fermionic elements.

*k*be an element in the Cartan subalgebra of \(\mathscr {B}({{\mathfrak {g}}}_r)\) that commutes with \({{\mathfrak {g}}}_r\) and satisfies \([k,e]=e\) and \([k,f]=-f\) when we extend \(\mathscr {B}({{\mathfrak {g}}}_r)\) to \(\mathscr {B}({{\mathfrak {g}}}_{r+1})\). In the Cartan subalgebra of \(\mathscr {B}({{\mathfrak {g}}}_{r+1})\), set \({\widetilde{k}}= k+h\), so that \([e,f]=h={\widetilde{k}}-k\). We then have

*p*and

*q*for \(\beta _0\) and \(\beta _{-1}\), 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 action of \(h={\widetilde{k}}- k\) and the Cartan element

*m*and

*n*, corresponding to \(\gamma _0\) and \(\gamma _{-1}\), respectively, are related to the level and height by \(m=p\) and \(n=p-q\). The \(L_\infty \) structure on \(\mathscr {B}({{\mathfrak {g}}}_{r+1})\) that we are going to introduce is based on yet another \(\mathbb {Z}\)-grading,

The general structure of the superalgebra \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\)

*i.e.*, where also the right hand side belongs to level \(-1\), 0 or 1) that follow from the Chevalley–Serre relations are

*e*and

*f*on \([[E_{M}, {\widetilde{F}}^{N}],E_{P}]=0\) and \([[{\widetilde{E}}_{M}, F^{N}],{\widetilde{E}}_{P}]=0\), respectively.

*e.g.*on the connection to minimal orbits and to a denominator formula for the Borcherds superalgebra, we refer to refs. [2, 3, 4]. The (anti-)commutation relations with generators at level \(\pm 1\) acting on those in (2.18) at level \({\mp } 2\) follow from Eqs. (2.14) by the Jacobi identity.

*e*,

*f*and

*h*. This follows from Eqs. (2.3). An element at positive level and height 0 is annihilated by \(\text {ad}\,f\). It can be “raised” to height 1 by \(\text {ad}\,e\) and lowered back by \(\text {ad}\,f\). We define, for any element at a non-zero level

*p*,

As explained above \(\mathscr {B}({{\mathfrak {g}}}_{r+1})\) decomposes into \({{\mathfrak {g}}}_r\) modules, where we denote the one at level *p* and height *q* by \(R_{(p,q)}\). Every \({{\mathfrak {g}}}_r\)-module \(R_p=R_{(p,0)}\) at level \(p>0\) and height 0 exists also at height 1. In addition there may be another module. We write \(R_{(p,1)}=R_p\oplus {\widetilde{R}}_p\). Sometimes, \({\widetilde{R}}_p\) may vanish. The occurrence of non-zero modules \({\widetilde{R}}_p\) is responsible for the appearance of “ancillary ghosts”.^{3}

Let *A* and *B* be elements at positive level and height 0 (or more generally, annihilated by \(\text {ad}\,f\)), 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* always has \(|C|=0\), while its derivative (to be defined in Eq. (4.1) below) has \(|dC|=1\). This assignment is completely analogous to the assignment of statistics to components in a superfield. To be completely clear, our conventions are such that also fermionic components and generators anticommute, so that if *e.g.*\(A=A^ME_M\) and \(B=B^ME_M\) are elements at level 1 with \(|A|=|B|=0\), then \([A,B]=[A^ME_M,B^NE_N]=-A^MB^N[E_M,E_N]\). A bosonic gauge parameter \(A^M\) at level 1 sits in an element *A* with \(|A|=1\).

*A*,

*B*. The decomposition

We will initially consider fields (ghosts) in the positive levels of \({\mathscr {B}}({{\mathfrak {g}}}_r)\), embedded in \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\) at zero height. They can thus be characterised as elements with positive (integer) eigenvalues of \(\text {ad}\,h\) and zero eigenvalue of the adjoint action of the element *q* in Eq. (2.8). Unless explicitly stated otherwise, elements in \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\) will be “bosonic”, in the sense 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, the superalgebra bracket \([\cdot ,\cdot ]\) is graded antisymmetric, \([C,C]=0\) when \(|C|=0\).

## 3 Section Constraint and Generalised Lie Derivatives

*e.g.*the importance of Eqs. (3.1) for the generalised Lie derivative, and the construction of solutions to the section constraint.

*Z*has the universal expression [2, 40]

*i.e.*, \(Z_{PQ}{}^{MN}=-\eta _{\alpha \beta }(t^\alpha )_P{}^N(t^\beta )_Q{}^M +((\lambda ,\lambda )-1)\delta _P^N\delta _Q^M\). With the help of the structure constants of \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\) it can now be written [3]

*U*,

*V*,

*W*. These expressions, and their generalisations, will return with ghosts as arguments in Section 6. Note however that

*U*and

*V*have bosonic components. They will be replaced by fermionic ghosts, which together with fermionic basis elements build bosonic elements. The bracket will be graded symmetric.

## 4 Derivatives, Generalised Lie Derivatives and Other Operators

In this Section, we will start to examine operators on elements at height 0, which are functions of coordinates in \(R_1\). Beginning with a derivative, and attempting to get as close as possible to a derivation property, we are naturally led to the generalised Lie derivative, extended to all positive levels. The generalised Lie derivative is automatically associated with a graded symmetry, as opposed to the graded antisymmetry of the algebra bracket. This will serve as a starting point for the \(L_\infty \) brackets. Other operators arise as obstructions to various desirable properties, and will represent contributions from ancillary ghosts. Various identities fulfilled by the operators will be derived; they will all be essential to the formulation of the \(L_\infty \) brackets and the proof of their identities.

### 4.1 The derivative

*d*: \(R_{(p,0)}\rightarrow R_{(p-1,0)}\) (\(p>0\)) by

Only insisting on having a nilpotent derivative does not determine the relative coefficients depending on the level *p* in Eq. (4.2). The subsequent considerations will however depend crucially on the coefficient.

### 4.2 Generalised Lie derivative from “almost derivation”

*A*,

*B*. One can then use the two alternative forms

*A*,

*B*,

*A*with \(p_A>1\) generates a vanishing transformation, while the action on arbitrary elements is the one which follows from demanding a Leibniz rule for the generalised Lie derivative. Note that bosonic components at level 1 implies fermionic elements, hence the signs in Eqs. (3.5) and (4.6) agree. The last term in Eq. (4.5) is present only if \({\widetilde{R}}_{p_A+p_B}\) is non-empty, since \([A^\sharp ,B^\sharp ]\) is an element at height 2 with \([A^\sharp ,B^\sharp ]^\sharp =0\). We will refer to such terms as ancillary terms, and denote them \(-R^\flat (A,B)\),

*i.e.*,

*K*at height 1) obtained from an element \(B_M\in {\widetilde{R}}_{p+1}\) at height 1 as \(K^\flat =[B_M,{\widetilde{F}}^M]\). The extra index on \(B_M\) is assumed to be “in section”. See Section 7 for a more complete discussion.

Equation (4.8) states that the symmetry of \({\mathscr {L}}_AB\) is graded symmetric, modulo terms with “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 antisymmetry of the superalgebra bracket to the desired symmetry of an \(L_\infty \) bracket. The graded antisymmetric part of the generalised Lie derivative appearing in Eq. (4.8) represents what, for bosonic parameters *U*, *V*, would be the symmetrised part \({\mathscr {L}}_UV+{\mathscr {L}}_VU\), and it can be seen as responsible for the violation of the Jacobi identities (antisymmetry and the Leibniz property imply the Jacobi identities [8]). The generalised Lie derivative (at level 1) will be the starting point for the \(L_\infty \) 2-bracket in Sections 6 and 8.

*not*encoded in the Borcherds superalgebra. We will indicate in the Conclusions what we think will be the correct procedure if this is not the case. We thus assume

*C*with \(|C|=0\), then

### 4.3 “Almost covariance” and related operators

*A*and

*dB*with the derivative of Eq. (4.8) gives the relation

*K*at height 1 by

*e.g.*for \({{\mathfrak {g}}}_r=E_7\)), \(X^\flat _AB\) represents a parameter which gives a trivial transformation without being a total derivative, thanks to the section constraint.

### 4.4 More operator identities

*d*as

*B*, the other term vanishes due to the section constraint. In the second term, \([\partial _M\partial _NA^\sharp ,[\partial _PB^\sharp ,F^P]]=[[\partial _M\partial _NA^\sharp ,\partial _PB^\sharp ],F^P]\), and the two terms cancel. Note that we are now dealing with identities that hold exactly, not only modulo ancillary terms (they are identities

*for*ancillary terms).

*X*,

*C*the relation reads

*R*(

*A*,

*B*), together with Eq. (4.10), are:

*R*(

*A*,

*B*) is non-vanishing for

*A*and

*B*at all levels (as long as \({\widetilde{R}}_{p_A+p_B}\) is non-empty), we will sometimes use the notation \(R_AB=R(A,B)\). Thanks to the Jacobi identity for the Borcherds superalgebra and the Leibniz property of the generalised Lie derivative,

*R*(

*A*,

*B*) satisfies a cyclic identity,

## 5 Batalin–Vilkovisky Ghost Actions and \(L_\infty \) Algebras

*S*can be (formally, if needed) expanded as a power series in

*C*,

*n*-bracket. The 1-bracket is the BRST operator. The BV variation of

*C*is

*n*-bracket carries level \(n-2\). In our conventions, all \(L_\infty \) brackets carry ghost number \(-1\), and the superalgebra bracket preserves ghost number. Also, the properties of the brackets under permutation of elements are sometimes presented as governed by “Koszul sign factors”. In our conventions, the \(L_\infty \) brackets are simply graded symmetric and the statistics of the ghosts, inherited from the superalgebra, is taking care of all signs automatically.

Since the relation between the BV ghost variation and the \(L_\infty \) brackets seems to be established, but not common knowledge among mathematical physicists, we would like to demonstrate the equivalence explicitly. (See also refs. [21, 48].

*n*arbitrary elements, let \(C=\sum _{k=1}^\infty C_k\) and take the part of the identity containing each of the terms in the sum once. We then get

*j*in Eq. (5.5) gives

*i*!

*j*!, Rescaling the brackets according to

*x*-dependence of the ghosts in the basis elements \(c_i\) (“DeWitt notation”) and thus treat the components as constants that we can move out of the brackets. Then, our identities take the form

*F*is a fermionic element used to define the sign factor, which comes from the fact that the brackets are fermionic.

*n*elements \(\{x_1,\ldots ,x_n\}\) is defined inductively by an associative and graded symmetric product

*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 fermionic indices. There is obviously a difference between a tensor being graded antisymmetric (the “

*x*picture”) and “graded symmetric with opposite statistics” (the “

*c*picture”). The two types of tensors are however equivalent as modules (super-plethysms) of a general linear superalgebra. As a simple example, a 2-index tensor which is graded antisymmetric can be represented as a matrix

*a*is antisymmetric and

*s*symmetric, while a 2-index tensor which is graded symmetric in the opposite statistics is

*V*with itself can always be decomposed 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, as modules of the general linear superalgebra \(\mathfrak {g}\mathfrak {l}(V)\), is the sum of the graded antisymmetric and graded symmetric modules with the opposite assignment of statistics. The same is true for higher tensor products \(\otimes ^nV\).

This means that, as long as the brackets \(\ell \) and \({\bar{\ell }}\) are taken to be proportional up to signs, the equations (5.9) and (5.13) contain the same number of equations in the same \({{\mathfrak {g}}}\)-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 called suspension, from the “*x* picture” to the “*c* picture”, *i.e.*, between the two presentations of the plethysms of the general linear superalgebra.

*a*and and \(\alpha \) correspond to fermionic and bosonic basis elements, respectively. We choose an ordering where the

*a*indices are “lower” than the \(\alpha \) ones. Any unshuffle then has the index structure \(\{a_1\ldots a_k\alpha _1\ldots \alpha _{k'}, a_{k+1}\ldots a_\ell \alpha _{k'+1}\ldots \alpha _{\ell '}\}\). If the brackets \(\ell \) and \({\bar{\ell }}\) are expressed in terms of structure constants,

*i*,

*j*being the same variables as in the sums (5.9) and (5.13)). Now, both expressions need to be arranged to the same index structure, which we choose as \(a_1\ldots a_\ell \alpha _1\ldots \alpha _{\ell '}\). This gives a factor \((-1)^{k'm}\) for the \(f^2\) term, and \((-1)^{km}\) for \({\bar{f}}^2\). In order to compare the two brackets, we also need to move the summation index

*B*to the right on

*f*when \(B=\beta \) and to the left on \({\bar{f}}\) when \(B=b\). All non-vanishing brackets have a total odd number of “

*a*indices”, including the upper index, so \(B=b\) when

*k*is even, and \(B=\beta \) when

*k*is odd. This gives a factor \((-1)^m\) for the \(f^2\) expression when

*k*is odd, and \((-1)^m\) for \({\bar{f}}^2\) when

*k*is even.

*i.e.*, it should only depend on \(\ell =k+m\) and \(\ell '=k'+m'\). Taking the factors above into consideration, this condition reads

*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 in the two pictures implies, among other things, that

*c*picture, we have the identity

*x*picture corresponding to the identity (5.24) in the

*c*picture.

Note that the issue with the two pictures arises already when constructing a BRST operator 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 with graded symmetry.

## 6 The \(L_\infty \) Structure, Ignoring Ancillary Ghosts

The following calculation will first be performed disregarding ancillary ghosts, *i.e.*, as if all \({\widetilde{R}}_p=0\). The results will form an essential part of the full picture, but the structure does not provide an \(L_\infty \) subalgebra unless all \({\widetilde{R}}_p=0\).

We use a ghost *C* which is totally bosonic, *i.e.*, \(|C|=0\), and which is a general element of \({\mathscr {B}}_+({{\mathfrak {g}}}_r)\), *i.e.*, a height 0 element of \({\mathscr {B}}_+({{\mathfrak {g}}}_{r+1})\). This gives the correct statistics of the components, namely the same as the basis elements in the superalgebra. All signs are taken care of automatically by the statistics of the ghosts. While the superalgebra bracket is graded antisymmetric, the \(L_\infty \) brackets (by which we mean the brackets in the *c* picture of the previous Section, before the rescaling of Eq. (5.7)) are graded symmetric. The *a* index of the previous Section labels ghosts with odd ghost number, and the \(\alpha \) index those with even ghost number, and include also the coordinate dependence.

### 6.1 Some low brackets

*c*is

*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 is of course a choice involved every time a new bracket is introduced, and the choices differ by something exact. The choice will then have repercussions for the rest of the structure. The first choice arises when the need for a level 2 ghost \(C_2\) becomes clear (from the 3-bracket identity as a modification of the Jacobi identity), and its 2-bracket with the level 1 ghost is to be determined. Instead of choosing \([\![c,C_2]\!]=\tfrac{1}{2}{\mathscr {L}}_cC_2\), corresponding to Eq. (6.4), we could have taken \([\![c,C_2]\!]=-\tfrac{1}{2}[c,dC_2]\), since the derivative of the two expressions are the same (modulo ancillary terms) according to Eq. (4.8). The latter is the type of choice made in *e.g.* Ref. [21]. Any linear combination of the two choices with weight 1 is of course also a solution. However, it turns out that other choices than the one made here lead to expressions that do not lend themselves to unified expressions containing *C* as a generic element in \({\mathscr {B}}_+({{\mathfrak {g}}}_r)\). Thus, this initial choice and its continuation are of importance.

*i.e.*, the absence of ancillary transformations in the commutator of two level 1 transformations. Then,

### 6.2 Higher brackets

*n*-bracket, namely

*n*-identities, \(n\ge 2\) (the remaining ones are those with odd

*n*). They are

*n*.

*i.e.*, \((s+t-1)s^jt^k\approx 0\). We can then replace

*s*by \(1-t\), so that \(s^jt^k\) becomes \((1-t)^jt^k\). The symmetry property is taken care of by symmetrisation, so that the final expression corresponding to \(Z_{n,j,k}\) is

*k*’s.

We will now show that all identities are satisfied by translating them into polynomials with Getzler’s method, using the generating function for the Bernoulli numbers.

*f*(

*x*) be the generating function for the coefficients \(k_n\),

*i.e.*,

*s*and

*t*, by \(x^n\) and sum over

*n*, identifying the function

*f*when possibility is given. This gives

*f*is used, this becomes, after some manipulation,

The function \(\phi (s,t,x)=\sum _{n=2}^\infty \phi _n(s,t)\), with the coefficient functions \(\phi _n(s,t)\) given by the sum of Eqs. (6.29) and (6.30), symmetrised in *s* and *t*, will appear again in many of the calculations for the full identities in Section 8.

*g*is the function

## 7 Ancillary Ghosts

We have already encountered “ancillary terms”, whose appearance in various identities for the operators, such as the deviation of *d* from being a derivation and the deviation of *d* from being covariant, rely on the existence of modules \({\widetilde{R}}_p\). Note that the Borcherds superalgebra always has \({\widetilde{R}}_1=\emptyset \), *i.e.*, \(R_{(1,1)}=R_1\); this is what prevents us from treating situations where already the gauge “algebra” of generalised Lie derivatives contains ancillary transformations. The ancillary terms at level *p* appear as \([B_M^\sharp ,{\widetilde{F}}^M]^\flat =[B_M,{\widetilde{F}}^M]\), where \(B_M\) is an element in \({\widetilde{R}}_{p+1}\) at height 1 (*i.e.*, \(B_M^\flat =0\)). \(B_M\) carries an extra \({\overline{R}}_1\) index, which is “in section”, meaning that the relations (3.1) are fulfilled also when one or two \(\partial _M\)’s are replaced by a \(B_M\).

The typical structure of the action of the 1-bracket between the ghost modules, with ancillary ghosts appearing from level \(p_0\ge 1\)

The derivative *d* and the generalised Lie derivative \({\mathscr {L}}_C\) are extended to level 1 as in Section 4.3. This implies that \(\flat \) anticommutes with *d* and with \({\mathscr {L}}_C\). Since \(d^2=0\) and \(d\flat +\flat d=0\) on elements in \(R_p\) at height 0 and 1, it can be used in the construction of a 1-bracket, including the ancillary ghosts. The generic structure is shown in Table 2.

*M*index remains. The second term has \([A,{\widetilde{F}}^M]\ne 0\) only for \(p_A=1\), but vanishes thanks to \([B_M,f]=0\). This shows that \([{\mathscr {B}}_+({{\mathfrak {g}}}_r),{\mathscr {A}}]\subset {\mathscr {A}}\). An explicit example of this ideal, for the \(E_5\) exceptional field theory in the M-theory section, is given in Section 9, Table 7.

*d*on ancillary ghosts

*K*at height 1. Let \(B_M\in {\widetilde{R}}_{p+1}\) with height 1, and let \(K^\flat =[B^\sharp _M,{\widetilde{F}}^M]^\flat \in R_p\) at height 0. We will for the moment assume that

The appearance of modules \({\widetilde{R}}_p\) can be interpreted in several ways. One is as a violation of covariance of the exterior derivative, as above. Another is as a signal that Poincaré’s lemma does not hold. In this sense, ancillary modules encode the presence of “local cohomology”, *i.e.*, cohomology present in an open set. It will be necessary to introduce ghosts removing this cohomology.

Let the lowest level *p* for which \({\widetilde{R}}_{p+1}\) is non-empty be \(p_0\). Then it follows that an ancillary element \(K_{p_0}\) at level \(p_0\) will be closed, \(dK_{p_0}=0\), and consequently \(dK^\flat _{p_0}=0\). However, \(K_{p_0}\) does not need to be a total derivative, since \(B_M\) does not need to equal \(\partial _M\Lambda \). Indeed, our ancillary terms are generically not total derivatives. An ancillary element at level \(p_0\) represents a local cohomology, a violation of Poincaré’s lemma.

*R*(

*A*,

*B*), since it contains only one derivative. One can however rely the identities (4.25) and (4.26), which immediately show (in the latter case also using the property that ancillary expressions form an ideal) that the derivatives and generalised Lie derivatives of an ancillary expression (expressed as \(R^\flat (A,B)\)) is ancillary. This is what is needed to consistently construct the brackets in the following Section.

The section property of \(B_M\) implies that \({\mathscr {L}}_{K^\flat }A=0\) when \(K^\flat \) is an ancillary expression (see Eq. (4.16)). This identity is also used in the calculations for the identities of the brackets.

## 8 The Full \(L_\infty \) Structure

We will now display the full \(L_\infty \) structure, including ancillary ghosts. The calculations for the \(L_\infty \) brackets performed in Section 6 will be revised in order to include ancillary terms.

### 8.1 Some low brackets

*C*at height 0, and also on ancillary ghosts

*K*at height 1, is \(d+\flat \):

*d*.

*d*and an \({\mathscr {L}}_C\). Here, we have of course used \({\mathscr {L}}_{K^\flat }=0\). Note that the height 0 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 \([\![K,K^\flat ]\!]=\tfrac{1}{2}{\mathscr {L}}_{K^\flat }K=0\), implying that the bracket with two

*K*’s consistently can be set to 0.

*C*’s and one

*K*becomes equivalent to the height 0 identity for the bracket with three

*C*’s when

*CCK*identity, which is trivial since \(\flat \) generates no ancillary terms. Again, there is no need for a bracket with

*CKK*, since

*C*’s reads

### 8.2 Higher brackets

*n*-bracket identity gives, apart from the second row of Eq. (6.20),

*n*-bracket to take the form

*i.e.*,

*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 \(\flat \) in \([\![[\![C^n]\!]]\!]\). The height 1 identity with one *K* contains the same calculation. The height 0 identity with one *K* is trivial, and just follows from moving \(\flat \)’s in and out of commutators and through derivatives and generalised Lie derivatives. The vanishing of the brackets with more than one *K* is consistent with the vanishing of \([\![C^{n-2},K^\flat ,K]\!]\). Lowering this bracket gives \([\![C^{n-2},K^\flat ,K^\flat ]\!]\) which vanishes by statistics, since \(K^\flat \) 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 4. We will go through the details by collecting the different types of terms generated, one by one.

A first result of the calculation is that all terms containing more than one ancillary expression *X* or *R* cancel. This important consistency condition relies on the precise combination 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.

*X*. In addition to its appearance in the brackets,

*X*arises when a derivative or a generalised Lie derivative is taken through an

*R*, according to Eqs. (4.25) and (4.26). It turns out that all terms where \(X_C\) appears in an “inner” position in terms of the type \((\text {ad}\,C)^iX_C(\text {ad}\,C)^{n-i-3}{\mathscr {L}}_CC\), with \(n-i>3\), cancel. This again does not depend on the coefficients \(k_n\). Collecting terms \((\text {ad}\,C)^{n-3}{\mathscr {L}}_CX_CC\) and \((\text {ad}\,C)^{n-3}X_C{\mathscr {L}}_CC\), the part \([\![[\![C^n]\!]]\!]+n[\![C^{n-1},[\![C]\!]]\!]\) gives a contribution

*X*term in the bracket, and

*R*term, together giving

*X*are of the types \((\text {ad}\,C)^j\text {ad}\,{\mathscr {L}}_CC(\text {ad}\,C)^{n-4-j}X_CC\) and \((\text {ad}\,C)^j\text {ad}\,X_CC(\text {ad}\,C)^{n-4-j}{\mathscr {L}}_CC\) and similar. The first and last term in the identity gives a contribution

*X*term in the

*n*-bracket, and

*R*term. A middle term \([\![C^i,[\![C^{n-i}]\!]]\!]\) gives

*R*and two \({\mathscr {L}}\)’s. One such structure is \((\text {ad}\,C)^jR_C(\text {ad}\,C)^{n-j-4}{\mathscr {L}}_{{\mathscr {L}}_CC}C\). For each value of

*j*, the total coefficient of the term cancels thanks to \(nk_n+\sum _ik_{i+1}k_{n-i}=0\). 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 (4.27). Let

*C*, \(F_j\) and \(F_k\) turns it into

*i.e.*, not having

*C*as one of the arguments of

*R*, combine into the first three terms of this equations, and thus can be turned into expressions with \(R_C\). Note that this relation is analogous to Eq. (4.27) for \(Z_{n,j,k}\) in Section 6.2, but with a remainder term. We now collect such terms. They are

*j*and

*k*indices in both expressions are translated into monomials \(s^jt^k\) as before, both expressions should be calculated modulo \(s+t-1\approx 0\) as before. In

*U*, symmetry under \(s\leftrightarrow t\) can be used, but not in

*V*. Both types of terms need to cancel for all values of

*r*, since there is no identity that allows us to take \(\text {ad}\,C\) past \(R_C\).

*n*-bracket identity then is

*r*.

*V*to the

*n*-bracket is then represented by the function \(v_n(s,t,u)\):

*i*, and replacing

*s*by \(1-t\), this function turns into

*e.g.*,

*L*,

*R*stands for action to the left or to the right of the succeeding operator (\({\mathscr {O}}\)). Then, the full ghost variation takes the functional form

## 9 Examples

- (
*i*) -
\({{\mathfrak {g}}}_r=A_r\), \(\lambda =\Lambda _p\), \(p=1,\ldots ,r\) (

*p*-form representations); - (
*ii*) -
\({{\mathfrak {g}}}_r=B_r\), \(\lambda =\Lambda _1\) (the vector representation);

- (
*iii*) -
\({{\mathfrak {g}}}_r=C_r\), \(\lambda =\Lambda _r\) (the symplectic-traceless

*r*-form representation); - (
*iv*) -
\({{\mathfrak {g}}}_r=D_r\), \(\lambda =\Lambda _1,\Lambda _{r-1},\Lambda _r\) (the vector and spinor representations);

- (
*v*) -
\({{\mathfrak {g}}}_r=E_6\), \(\lambda =\Lambda _1,\Lambda _5\) (the fundamental representations);

- (
*vi*) -
\({{\mathfrak {g}}}_r=E_7\), \(\lambda =\Lambda _1\) (the fundamental representation).

The decomposition of \(A(r+1|0)\approx \mathfrak {sl}(r+2|1)\) in \(A(r)\approx \mathfrak {sl}(r+1)\) modules

\({p}=-1\) | \({p}=0\) | \({p}=1\) | |
---|---|---|---|

\(q=1\) | | | |

\(q=0\) | \({\overline{\mathbf{v}}}\) | \(\mathbf{1}\oplus \mathbf{adj}\oplus \mathbf{1}\) | |

\(q=-1\) | \({\overline{\mathbf{v}}}\) | |

The extended geometry based on \({{\mathfrak {g}}}_r=B_r\) follows an analogous pattern, and is also described by Table 4, but with the doubly extended algebra \(B(r+1,0)\approx \mathfrak {osp}(r+1,r+2|2)\) being decomposed into modules of \(B(r)\approx \mathfrak {so}(r,r+1)\).

The decomposition of \(D(r+1|0)\approx {\mathfrak {osp}}(r+1,r+1|2)\) in \(D(r)\approx \mathfrak {so}(r,r)\) modules

\({p}=-2\) | \({p}=-1\) | \({p}=0\) | \({p}=1\) | \({p}=2\) | |
---|---|---|---|---|---|

\({q}=1\) | | | | ||

\({q}=0\) | | | \(\mathbf{1}\oplus \mathbf{adj}\oplus \mathbf{1}\) | | |

\({q}=-1\) | | | |

Part of the decomposition of \({\mathscr {B}}(E_{6(6)})\) in \(E_{5(5)}\approx \mathfrak {so}(5,5)\) modules. Note the appearance of modules \({\widetilde{R}}_p\) for \(p\ge 4\)

\({p}=-1\) | \({p}=0\) | \({p}=1\) | \({p}=2\) | \({p}=3\) | \({p}=4\) | \({p}=5\) | |
---|---|---|---|---|---|---|---|

\({q}=2\) | | | |||||

\({q}=1\) | | | | \(\overline{\mathbf{16}}\) | \(\mathbf{45}\oplus \mathbf{1}\) | \(\overline{\mathbf{144}}\oplus \mathbf{16}\) | |

\({q}=0\) | \({\overline{\mathbf{16}}}\) | \(\mathbf{1}\oplus \mathbf{45}\oplus \mathbf{1}\) | | | \(\overline{\mathbf{16}}\) | | \(\overline{\mathbf{144}}\) |

\({q}=-1\) | \(\overline{\mathbf{16}}\) | |

Part of the decomposition of \({\mathscr {B}}(E_{8(8)})\) in \(E_{7(7)}\) modules

\({p}=0\) | \({p}=1\) | \({p}=2\) | \({p}=3\) | \({p}=4\) | |
---|---|---|---|---|---|

\({q}=3\) | | ||||

\({q}=2\) | | | \(\mathbf{1539}\oplus \mathbf{133}\oplus 2\cdot \mathbf{1}\) | ||

\({q}=1\) | | | \(\mathbf{133}\oplus \mathbf{1}\) | \(\mathbf{912}\oplus \mathbf{56}\) | \(\mathbf{8645}\oplus 2\cdot \mathbf{133}\oplus \mathbf{1539}\oplus \mathbf{1}\) |

\({q}=0\) | \(\mathbf{1}\oplus \mathbf{133}\oplus \mathbf{1}\) | | | | \(\mathbf{8645}\oplus \mathbf{133}\) |

\({q}=-1\) | |

Part of the decomposition of \(R_p\) for the \(E_{5(5)}\) exceptional geometry with respect to a section \(\mathfrak {sl}(5)\)

\({p}=1\) | \({p}=2\) | \({p}=3\) | \({p}=4\) | \({p}=5\) | \({p}=6\) | |
---|---|---|---|---|---|---|

\({v}=6\) | \((\overline{\mathbf{15}}\oplus \mathbf{40})\otimes \Lambda _5\) | |||||

\({v}=5\) | \(\mathbf{24}\otimes \Lambda _5\) | \(\mathbf{24}\otimes \Lambda _4 \oplus (\mathbf{5}\oplus \overline{\mathbf{45}})\otimes \Lambda _5\) | ||||

\({v}=4\) | \(\mathbf{10}\otimes \Lambda _5\) | \(\mathbf{10}\otimes \Lambda _4\oplus \overline{\mathbf{15}}\otimes \Lambda _5\) | \(\mathbf{10}\otimes \Lambda _3 \oplus \overline{\mathbf{15}}\otimes \Lambda _4 \oplus \overline{\mathbf{5}}\otimes \Lambda _5\) | |||

\({v}=3\) | \(\overline{\mathbf{5}}\otimes \Lambda _5\) | \(\overline{\mathbf{5}}\otimes \Lambda _4\) | \(\overline{\mathbf{5}}\otimes \Lambda _3\) | \(\overline{\mathbf{5}}\otimes \Lambda _2\) | ||

\({v}=2\) | \(\Lambda _5\) | \(\Lambda _4\) | \(\Lambda _3\) | \(\Lambda _2\) | \(\Lambda _1\) | \(\Lambda _0\) |

\({v}=1\) | \(\Lambda _2\) | \(\Lambda _1\) | \(\Lambda _0\) | |||

\({v}=0\) | \(\Lambda _4\) |

In Table 7, we have divided the modules \(R_p\) for the \(E_{5(5)}\) example of Table 5 into \(A_4\) modules with respect to a choice of section. Below the solid dividing line are the usual sequences of ghosts for diffeomorphisms and 2-form and 5-form gauge transformations. Above the line are sequences that contain tensor products of forms with some other modules, *i.e.*, mixed tensors. All modules above the line are effectively cancelled by the ancillary ghosts. They are however needed to build modules of \({{\mathfrak {g}}}_r\). In the example, there is nothing below the line for \(p\ge 7\), which means that the \(\flat \) operation from ancillary to non-ancillary ghosts at these levels becomes bijective.

Reducibility is of course not an absolute concept; it can depend on the amount of covariance maintained. If a section is chosen, the reducibility can be made finite by throwing away all ghosts above the dividing line. One then arrives at the situation in ref. [42]. If full covariance is maintained, reducibility is infinite. Since the modules above the line come in tensor products of some modules with full sets of forms of alternating statistics, they do not contribute to the counting of the degrees of freedom. This shows why the counting of refs. [1, 4], using only the non-ancillary ghosts, gives the correct counting of the number of independent gauge parameters.

This picture of the reduction of the modules \(R_p\) in a grading with respect to the choice of section also makes the characterisation of ancillary ghosts clear. They are elements in \(R_p\) above a certain degree (for which the degree of the derivative is 0). The dotted line in the table indicates degree 0. If we let \({\mathscr {A}}\) be the subalgebra of ancillary elements above the solid line, it is clear that \({\mathscr {A}}\) forms an ideal in \({\mathscr {B}}_+({{\mathfrak {g}}}_r)\) (which was also shown on general grounds in Section 7). The grading coincides with the grading used in ref. [2] to show that the commutator of two ancillary transformations again is ancillary.

*e*and

*f*, without need of any further regularisation (

*e.g.*through analytic continuation). Using the cancellation of these doublets, inspection of Table 1 gives at hand that the “super-dimension” (where fermionic generators count with a minus sign)

## 10 Conclusions

We have provided a complete set of bracket giving an \(L_\infty \) algebra for generalised diffeomorphisms in extended geometry, including double geometry and exceptional geometry as special cases. The construction depends crucially on the use of the underlying Borcherds superalgebra \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\), which is a double extension of the structure algebra \({{\mathfrak {g}}}_r\). This superalgebra is needed in order to form the generalised diffeomorphisms, and has a natural interpretation in terms of the section constraint. It also provides a clear criterion for the appearance of ancillary ghosts.

*c*. No brackets 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. (4.14). It is not unlikely that this makes it possible to covariantise the whole structure, as in ref. [39]. However, we think that it is appropriate to let the algebraic structures guide us concerning such issues.

The characterisation of ancillary ghosts is an interesting issue, that may deserve further attention. Even if the construction in Section 7 makes the appearance of ancillary ghosts clear (from the existence of modules \({\widetilde{R}}_p\)) it is indirect and does not contain an independent characterisation of the ancillary ghosts, in terms of a constraint. This property is shared with the construction of ancillary transformations in ref. [40]. The characterisation in Section 9 in terms of the grading induced by a choice of section is a direct one, in this sense, but has the drawback that it lacks full covariance. In addition, there may be more than one possible choice of section. This issue may become more important when considering situations with ancillary ghosts at ghost number 1 (see below). Then, with the exception of some simpler cases with finite-dimensional \({{\mathfrak {g}}}_r\), ancillary transformations are not expected to commute.

We have explicitly excluded from our analysis cases where ancillary transformations appear already at ghost number 1 [38, 39, 40, 52]. The canonical example is exceptional geometry with structure group \(E_{8(8)}\). If we should trust and extrapolate the results of the present paper, this would correspond to the presence of a module \({\widetilde{R}}_1\). However, there is never such a module in the Borcherds superalgebra. If we instead turn to the tensor hierarchy algebra [53, 54, 55] we find that a module \({\widetilde{R}}_1\) indeed appears in cases when ancillary transformations are present in the commutator of two generalised diffeomorphisms.

As an example, Table 8 contains a part of the double grading of the tensor hierarchy algebra \(W(E_9)\) (following the notation of ref. [54]), which we believe should be used in the construction of an \(L_\infty \) algebra for \(E_8\) generalised diffeomorphisms. The \(E_8\) modules that are not present in the \({\mathscr {B}}(E_9)\) superalgebra are marked in blue colour. The singlet at \((p,q)=(1,1)\) is the extra element appearing at level 0 in \(W(E_9)\) that can be identified with the Virasoro generator \(L_1\) (as can be seen in the decomposition under \(\mathfrak {gl}(9)\) [55]). The elements at \(q-p=1\) come from the “big” module at level \(-1\) in \(W(E_9)\) (the embedding tensor or big torsion module). For an affine \({{\mathfrak {g}}}_{r+1}\) this is a shifted fundamental highest weight module, with its highest weight at \((p,q)=(1,2)\), appearing in \(W({{\mathfrak {g}}}_{r+1})\) in addition to the unshifted one with highest weight at \((p,q)=(0,1)\) appearing also in the Borcherds superalgebra \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\). In the \(E_8\) example, it contains the \(\mathbf{248}\) at \((p,q)=(0,1)\) which will accommodate parameters of the ancillary transformations. In situations when ancillary transformations are absent at ghost number 1 (the subject of the present paper), using \(W({{\mathfrak {g}}}_{r+1})\) is equivalent to using \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\), so all results derived here will remain unchanged.

Part of the decomposition of the tensor hierarchy algebra \(W(E_9)\) into \(E_8\) modules. The modules not present in \({\mathscr {B}}(E_9)\) are marked italic

\({p}=-1\) | \({p}=0\) | \({p}=1\) | \({p}=2\) | |
---|---|---|---|---|

\({q}=2\) | | 248 | ||

\({q}=1\) | \({ 1}\oplus { 3875}\oplus { 248}\) | \({ 1}\oplus { 248}\) | \({248}\oplus { 1}\) | \({1}\oplus {3875}\oplus {248}\) |

\({q}=0\) | \({248}\oplus { 1}\oplus { 3875}\oplus { 248}\) | \({ 1}\oplus {248}\oplus { 1}\) | 248 | \({1}\oplus {3875}\) |

\({q}=-1\) | 248 | 1 |

## Footnotes

- 1.
In refs. [2, 40], the coordinate module was taken to be a highest weight module. We prefer to reverse these conventions (in agreement with ref. [3]). With the standard basis of simple roots in the superalgebra, its positive levels consists of

*lowest*weight \({{\mathfrak {g}}}_r\)-modules. In the present paper the distinction is not essential, since the cases treated all concern finite-dimensional \({{\mathfrak {g}}}_r\) and finite-dimensional \({{\mathfrak {g}}}_r\)-modules. - 2.
In ref. [2], the algebras \({{\mathfrak {g}}}_{r+1}\), \(\mathscr {B}({{\mathfrak {g}}}_r)\) and \(\mathscr {B}({{\mathfrak {g}}}_{r+1})\) were called \(\mathscr {A}\), \(\mathscr {B}\) and \(\mathscr {C}\), respectively.

- 3.
The notation \({\widetilde{R}}_p\) was used differently in ref. [3]. There, \({\widetilde{R}}_1, {\widetilde{R}}_2, {\widetilde{R}}_3, \ldots \) correspond to \(R_1, {\widetilde{R}}_2, \widetilde{{\widetilde{R}}}_3, \ldots \) here,

*i.e.*, the representations on the diagonal \(n=0\) in Table 1. Thus it is only for \(p=2\) that the meanings of the notation coincide.

## Notes

### Acknowledgements

We would like to thank David Berman and Charles Strickland-Constable for collaboration in an early stage of this project. We also acknowledge discussions with Alex Arvanitakis, Klaus Bering, Olaf Hohm, and Barton Zwiebach. This research is supported by the Swedish Research Council, project no. 2015-04268.

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