Superalgebras, constraints and partition functions
aa r X i v : . [ h e p - t h ] A ug MI-TH-1508Gothenburg preprintarXiv:1503.06215
Superalgebras, constraintsand partition functions
Martin Cederwall
Department of Fundamental PhysicsChalmers University of TechnologySE-412 96 G¨oteborg, Sweden [email protected]
Jakob Palmkvist
Mitchell Institute for Fundamental Physics and AstronomyTexas A&M UniversityCollege Station, TX 77843, USA [email protected]
Abstract
We consider Borcherds superalgebras obtained from semisimple finite-dimensionalLie algebras by adding an odd null root to the simple roots. The additional Serrerelations can be expressed in a covariant way. The spectrum of generators at positivelevels are associated to partition functions for a certain set of constrained bosonicvariables, the constraints on which are complementary to the Serre relations in thesymmetric product. We give some examples, focusing on superalgebras related topure spinors, exceptional geometry and tensor hierarchies, of how construction ofthe content of the algebra at arbitrary levels is simplified. ontents D = 8 pure spinors and null vectors . . . . . . . . . . . . . . . . . . . 207.3 D = 10 pure spinors and supergravity forms . . . . . . . . . . . . . . 217.4 Superalgebras and Lie algebras . . . . . . . . . . . . . . . . . . . . . 23 It is often useful in physics to describe a spectrum of states that appear at variousinteger levels by means of an associated partition function, especially if the spectrumis infinite. If the states at each level transform in a representation of a Lie algebra,the spectrum of representations may also be obtained from an extended (possiblyinfinite-dimensional) algebra by a level decomposition. In the present paper we willrelate these two approaches to each other, and also to a third important tool inphysics: the BRST treatment of reducible constraints.Our main example is the spectrum of dynamical forms in D -dimensional maximalsupergravity, which transform in representations of the U-duality group E n , where n = 11 − D . Remarkably, these representations form a Lie superalgebra, which canbe extended to an infinite-dimensional Borcherds superalgebra [1–3]. Decomposingit with respect to the E n subalgebra gives back the spectrum of dynamical forms atthe positive levels, and also precisely the additional non-dynamical forms allowedby supersymmetry, first determined for D = 10 in refs. [4–6]. The consistency withsupersymmetry was shown in refs. [7–10] using a superspace formulation, general-ising bosonic forms to superforms with arbitrary high degrees. However, already in Throughout the paper, we use the notation E n for the split real form, and also for the corre-sponding Lie algebra. All arguments are however equally valid for the complex Lie algebras. p the representations can be also obtained from a level decompositionof the Kac–Moody algebra E n + p [3, 10–12]. This generalises results for E , which asa special case contains the form spectrum up to p = D [13–15]. However, E is notenough to accommodate forms with higher degrees, and a rendition of all the rep-resentations coming from the Borcherds superalgebra would require a considerationof the infinite-rank algebra E ∞ .With a few exceptions, the level decomposition of the Borcherds superalgebrafurthermore agrees with the tensor hierarchy of form potentials, field strengths andgauge parameters that arises in the embedding tensor approach to gauged super-gravity [16–19]. The tensor hierarchy can be continued to infinity, but misses someof the representations coming from the Borcherds superalgebra. Perfect agreementis instead given by a tensor hierarchy algebra, where the embedding tensor is in-terpreted as an element at level minus one [20]. Using this algebra all the Bianchiidentities and gauge transformations for the gauged theory can be derived in a sim-ple way [10, 21]. This demonstrates the efficiency of organising representations intoa level decomposition of a Lie (super)algebra.Yet another context where the same infinite sequence of representations appears,and where it cannot be truncated, is exceptional geometry. The exceptional (gen-eralised) diffeomorphisms have infinite reducibility, and the sequence arises as thetower of ghosts for ghosts, describing this reducibility [22]. The connection to parti-tion functions of constrained objects, of which pure spinors [23] is one example, wasconjectured already in ref. [22], and used there to correctly regularise the infinitesums arising when counting the degrees of freedom. The same representations occurfor tensor fields in exceptional geometry [24], and in the tensor hierarchies consid-ered in [25–28]. The somewhat heuristic approach of ref. [22] provided one of themotivations for the present investigation, which puts the correspondence betweenthe algebra and the constrained objects on a firmer footing.For 3 ≤ D ≤ E n is extended to the infinite-dimensionalBorcherds superalgebra B by adding an odd null root β to the simple roots of E n .This is the special case that we focus on in this paper, with E n generalised to anysemisimple finite-dimensional Lie algebra g . The inner products of β with the simpleroots of g are assumed to be such that the Serre relations of the Borcherds super-algebra are at most quadratic in the odd Chevalley generators e , f correspondingto β . Denoting the representation at level p by R p , the Serre relations quadraticin e (say) thus belong to a representation of g contained in the symmetric tensorproduct of R with itself, with R as its complement . It generates an ideal of the With the complement of a representation R in another R ′ we mean the quotient R ′ − R . R , and at each level p ≥
2, the representation R p is the complement that is left when this ideal is factored out. A recursive studyof the ideal thus gives all information about the representations R p at any level p . Inthis paper, we will show that the representations R p alternatively, and often moredirectly, can be determined from the partition function for a bosonic object λ in R , subject to the constraint λ | R = 0. As our main result, we will show that thispartition function is the inverse of the partition function for the universal envelopingalgebra of B + , the subalgebra of B at positive levels.The paper is organised as follows. In Section 2 we describe in more detail the Liesuperalgebras that we consider, and how they are constructed from the Chevalleygenerators and the Serre relations. In Section 3 we introduce the partition functionsthat we use in Section 4 to state our results and give them an interpretation interms of a BRST operator. The argument of Section 4 corresponds roughly to theheuristic argument of ref. [22]. We then prove the result in Section 5 using thedenominator formula for Borcherds superalgebras. Section 6 addresses the questionwhy the method is not applicable to Lie algebras (extensions by an additional evenreal root). In Section 7 we present and discuss some examples. Let g be a semisimple finite-dimensional Lie algebra of rank r with simple roots α i ( i = 1 , . . . , r ). We recall that they form a basis of a euclidean space, and from theirmutual inner products we get the Cartan matrix a ij of g by a ij = ( α j , α i ∨ ) = 2 ( α j , α i )( α i , α i ) , (2.1)where α i ∨ = 2 α i / ( α i , α i ) is the coroot of α i .The construction of a Lie algebra from a basis of simple roots can be generalisedto inner product spaces which are not necessarily euclidean, and even from Liealgebras to Lie superalgebras. Semisimple finite-dimensional Lie algebras are thengeneralised to Kac-Moody (super)algebras, which in turn are generalised further toBorcherds (super)algebras. Thus Borcherds superalgebras is a very general concept,but in this paper we only consider the special cases described below, motivatedby their simplicity and by their appearance in the examples that we will study inSection 7. We refer the reader to refs. [29–32] for more general definitions and otherdetails about Borcherds and Kac-Moody (super)algebras.The Borcherds superalgebras that we consider are infinite-dimensional superex-tensions of semisimple finite-dimensional Lie algebras, obtained by adding an oddnull root to the simple roots. Let B be such an extension of g , with simple roots4 I ( I = 0 , , . . . , r ). Thus β is odd and null, ( β , β ) = 0, whereas β i = α i are evenand real, ( β i , β i ) >
0. The Cartan matrix B IJ of B is obtained from a ij by addingan extra column B i = ( β , β i ∨ ) = 2 ( β , β i )( β i , β i ) (2.2)and an extra row B I = ( β I , β ), including the diagonal entry B = ( β , β ) = 0.The additional off-diagonal entries B i are required to be non-positive integers, like B ij = a ij for i = j . We assume furthermore that B IJ is non-degenerate, and for each i = 1 , . . . , r , either B i = 0 or B i = − g corresponds toadding an extra node to the Dynkin diagram of g , connected with | B i | lines tonode i . Following ref. [29] we indicate that β is both null and odd by painting thecorresponding node “grey” (which means that it looks like ⊗ ), and let the othernodes, representing real even roots, be white. For example, the Dynkin diagram corresponds to the Cartan matrix B IJ = − − − − − −
10 0 − − . (2.3)We will come back to this algebra, among other examples, in Section 7.To each simple root β I of B we associate Chevalley generators e I , f I and h I , and B is then defined as the Lie superalgebra generated by these elements (of which e and f are odd and the others even) modulo the Chevalley relations[ h I , e J ] = B IJ e J , [ h I , f J ] = − B IJ f J , [ e I , f J } = δ IJ h J , (2.4)and the Serre relations(ad e I ) − B IJ ( e J ) = (ad f I ) − B IJ ( f J ) = 0 (2.5)5or I = J . For I = 0 the Serre relations (2.5) can equivalently be replaced by { e , e } = { f , f } = 0 , (2.6)since, by the Jacobi identity, [ { e , e } , e J ] = { e , [ e , e J ] } , [ { f , f } , f J ] = { f , [ f , f J ] } , (2.7)which gives[ { e , [ e , e J ] } , f J ] = { e , e } , [ { f , [ f , f J ] } , e J ] = { f , f } (2.8)if B J = −
1. Thus in this case the ideal generated by (2.5) is contained in the idealgenerated by (2.6), and conversely. If B J = 0, there is already a redundance in (2.5)because of the antisymmetry of the bracket, so replacing (2.5) by (2.6) in this casesimply amounts to removing one of two equivalent relations in (2.5).For any integer p , let B p be the subspace of B spanned by all root vectorscorresponding to roots β = pβ + α , where α is a linear combination of the realsimple roots β i = α i , and, if p = 0, by the Cartan elements h I . Since B is thedirect sum of all these subspaces, and [ B p , B q } ⊆ B p + q , this decomposition is a Z -grading of B , leading to a level decomposition of its adjoint representation underthe subalgebra g ⊂ B , with B p consisting of a (maybe reducible) representation of g at level p . We will throughout the paper denote this representation R p .Let B + and B − be the subalgebras of B spanned by elements at positive andnegative levels, respectively, and let ˜ B ± be the free Lie superalgebra generated by B ± . The Serre relations (2.6) generate an ideal of ˜ B which is the direct sum of twosubalgebras D ± , where D ± ⊂ ˜ B ± (and is the maximal such ideal). The Borcherdssuperalgebra B is then obtained by factoring out this ideal from ˜ B , and in particular B + is obtained by factoring out D + from ˜ B + , the free Lie superalgebra generatedby B . The ideal D + of ˜ B is generated by the element { e , e } at level two, which isset to zero in one of the Serre relations (2.6). However, considered as an ideal of ˜ B + only, it is generated by all elements at level two in D + , which are not only { e , e } but also those obtained from { e , e } by successively acting with g . These elementsform a representation R ⊥ , which is the complement of R in ∨ R (the symmetrisedtensor product of R with itself) with a lowest weight vector { e , e } . It then followsfrom the Chevalley relations that the Dynkin labels of the lowest weight of R ⊥ aregiven by λ i = 2 B i .Using a basis E M of B (so that the index M corresponds to the representa-tion R ), we can summarize the above construction of B + by saying that it is the Following the physics terminology, we use the term “representation” also for the module of therepresentation, i.e. , the vector space it acts on. E M modulo the “covariant Serrerelations” { E M , E N }| R ⊥ = 0 . (2.9)Recursive use of these relations (and of course of the Jacobi identity) gives completeinformation about the representation R p at arbitrarily high levels p . In Section 4 wewill describe how this information can be efficiently encoded into partition functions,which will be discussed next. The purpose of this section is to introduce and define notation for the partitionfunctions we use to state our results.The partition functions we will consider count the number of bosonic and fermi-onic objects occurring with some Z -weight, or level, and some additional quantumnumbers. In an “unrefined” partition function, only counting the number of statesper level, the presence of some set of N linearly independent objects with weight p corresponds to a term σ p N t p , where σ = 1 for bosons and σ = − Z F ( t ) = 1 − t , (3.1)and that of a bosonic one, Z B ( t ) = (1 − t ) − , (3.2)are each other’s inverses.More refined partition functions may be defined if additional quantum numbersare available. In a typical case, a variable or operator will transform in some rep-resentation R of a Lie algebra g . A refined partition function encodes completelythe representations of all states, and is a formal power series in a variable t (cor-responding to the grading) with coefficients in the unit ring of g -representations(under tensor product).The basic examples are the refined partition functions for fermionic and bosonic7reation operators in R : Z FR ( t ) = | R | M p =0 ( − t ) p ∧ p R , Z BR ( t ) = ∞ M p =0 t p ∨ p R . (3.3)Here, we use ∧ and ∨ for antisymmetric and and symmetric products, respectively,and | R | denotes the dimension of a representation R . These two functions are alsothe inverses of each other, when multiplication is taken as the tensor product withthe trivial representation as the identity. This can be seen explicitly at any order in t by observing that the tensor product ( ∧ p R ) ⊗ ( ∨ q R ) generically contains exactly theplethysms described by the two different “hook” Young tableaux of sl ( | R | ) obtainedby gluing together the column and the row describing the two factors. One thus has Z FR ( t ) ⊗ Z BR ( t ) = 1 , (3.4)It is then reasonable to use the formal notation Z FR ( t ) = (1 − t ) R , Z BR ( t ) = (1 − t ) − R . (3.5)A fermion in R can be seen as a boson in − R and vice versa. It is important tounderstand the notation of eq. (3.5) as the shorthand it is, with eq. (3.3) being itsdefining expression.All considerations of the refined partition functions may also be performed usingcharacters, since they provide a ring homomorphism. Writing the character of therepresentation R as χ ( R ) = P µ ∈ Λ R e µ , where Λ R is the set of weights for R , countingweights with multiplicities m > m “distinct” weights, we have χ ( ∧ k R ) = X { µ ,...,µ k } e µ + ··· + µ k , (3.6)where the sum is over sets of distinct (in the sense above) weights in Λ R . Thus, χ FR ( t ) ≡ χ ( Z FR ( t )) = | R | X k =1 ( − t ) k X { µ ,...,µ k } e µ + ··· + µ k = Y µ ∈ Λ R (1 − te µ ) , (3.7)which of course is just the product of the characters for the individual fermionsmaking up the representation R . It then follows that χ BR ( t ) ≡ χ ( Z BR ( t )) = ( χ FR ( t )) − = Y µ ∈ Λ R (1 − te µ ) − . (3.8)8he character picture will be used for a proof of our result in Section 5.The examples above used for setting the notation are valid only for unconstrainedvariables (creation operators). We will use such refined partition functions to encodethe spectrum of generators in the Borcherds superalgebras described in Section 2.Before going into the construction of partition functions for algebras and forconstrained objects, we will consider two other situations, which will be of use later.The first is when a fermionic or bosonic variable is “maximally constrained”, so thatany bilinear vanishes. Then the partition function just contains a linear term: Z ( t ) = 1 + σRt (3.9)(where again σ = ± R occur (but with odd levels still labeled as bosonic orfermionic by a sign σ ). Then the partition function is Z ( t ) = ∞ M p =0 ( σt ) p ⊗ p R = (1 − σRt ) − . (3.10)The observation that the partition functions (3.9) and (3.10) are each other’s inversesfor opposite choices of σ is one, somewhat trivial, example of our main result whichwill be demonstrated in the following sections. In this case the algebra is freelygenerated by the representation at level one. Consider the subalgebra B + of elements at positive levels of a Z -graded Borcherdssuperalgebra B , as defined in Section 2. In the generic case, the algebra will beinfinite-dimensional, and contain elements at arbitrarily high levels. However, as wesaw in Section 2, all this information is contained in the covariant Serre relations { E M , E N }| R ⊥ = 0 , (4.1)where R ⊥ is the complement to R in ∨ R . At level two, we thus have generators E MN = { E M , E N } in R .As announced in Section 1, we will argue that all information about the repre-sentations occurring at each level can be obtained in an alternative way, which oftenprovides a more direct answer, namely by considering a bosonic object λ M in R ,subject to the constraint λ | R = 0 . (4.2)9otice that the object λ M has opposite statistics (bosonic) to E M (thinking of oddelements in a superalgebra as fermionic), and that its constraint is in the symmetricrepresentation complementary to that of the Serre relations. The precise relation wewill establish, and which is the main result of this paper, is: The partition function of the universal enveloping algebra U ( B + ) is the in-verse of the partition function for the constrained object λ , i.e., Z U ( B + ) ( t ) ⊗ Z λ ( t ) = 1 . (4.3)Since the partition functions used are completely refined, in the sense of Section 3,this provides complete information of the generators at each level of the Borcherdssuperalgebra B . The refined partition function for λ , if λ is seen as a complex object,can be seen as encoding holomorphic functions of λ .The way we will argue for this equality in the present section is by identifyingthe action of the BRST operator for the (conjugated) constraint with the operation“ d ” of the coalgebra B ∗ + . This will not constitute a full proof (which would requirea consideration of cohomology of B ∗ + ), but provides a clear picture of the correspon-dence. The proof, based on the denominator formula for Borcherds superalgebras,is given in Section 5.Let us first consider the coalgebra, repeat some well known facts and set thenotation. For simplicity, we do this for the case of an ordinary Lie algebra; thegeneralisation to graded brackets and Lie superalgebras is trivial. The coalgebra ofa Lie algebra a is defined on the vector space a ∗ dual to a . It is equipped with amap d : a ∗ → a ∗ ∧ a ∗ , which is dual to the Lie bracket [ · , · ] in the sense that for any A, B ∈ a and X ∈ a ∗ , h dX | A ∧ B i = h X | [ A, B ] i , (4.4)where h·|·i is the canonical scalar product, naturally extended to tensor products. If E a and E ∗ a are dual bases for a and a ∗ , and [ E a , E b ] = f abc E c , eq. (4.4) reads dE ∗ a = f bca E ∗ b ∧ E ∗ c . (4.5)The action of d is naturally extended to tensor products of elements by defining itto act as a derivation. The Jacobi identity is equivalent to the nilpotency, d = 0, of d . The above can be generalised to a Lie superalgebra with the appropriate gradedinterpretation of wedge products, brackets and derivations.We now specialise on the Borcherds superalgebras at hand. The first two levels10f the coalgebra B ∗ + read dE ∗ M = 0 ,dE ∗ MN = E ∗ M ∨ E ∗ N | ¯ R . (4.6)The Serre relations manifest themselves as the absence of generators in ¯ R ⊥ at leveltwo. What is the procedure for the continued construction? Of course, knowledgeof the algebra directly provides the full information of the coalgebra. But it is alsopossible to use eq. (4.6) as a starting point for recursively deriving the contentat each level as well as the coproduct. One must then allow for the most generalrepresentation for E ∗ (3) and the most general form of dE ∗ (3) ∼ E ∗ (2) ∧ E ∗ (1) consistentwith d = 0. A general Ansatz consists of letting E ∗ MNP belong to a representation R ⊂ R ⊗ R and writing dE ∗ MNP = E ∗ M ∧ E ∗ NP | ¯ R . (4.7)The nilpotency of d then determines the allowed representation R . For example,a totally symmetric representation is always excluded from R , since it will van-ish due to the Jacobi identity. This procedure can then be continued to all levels,where dE ∗ ( p ) will contain sums of terms E ∗ ( q ) ∧ E ∗ ( p − q ) (wedge here denoting gradedantisymmetrisation).The unique result of the procedure can be understood by the following argu-ment, which also provides a conceptual idea behind the result stated in eq. (4.3).Everything starts from, and is generated from, the basic set of generators E M in the g -representation B . Since they are odd elements of a superalgebra, they are nor-mally thought of as fermionic. It is however useful to think of E M as not carrying adefinite statistics. Indeed, considering the Serre relations (4.1), the only constrainton a bilinear in E M (including both symmetric and anti-symmetric parts) is thata certain representation R ⊥ of the symmetric bilinear vanishes. The identificationof the symmetric part in the complement R with “new” generators E MN is inthis sense optional. Without this identification, and corresponding identificationsat higher levels, the universal algebra U ( B + ) can be constructed from the tensoralgebra of B by factoring out the ideal generated by E M ⊗ E N | R ⊥ . This provides away of constructing an arbitrary element, not in the algebra B + , but in its universalenveloping algebra U ( B + ), in terms of powers of E M only. The partition functionof the universal enveloping algebra will be that of an object E M in R of indefinitestatistics (although the elements at odd levels are labelled as fermionic in the par-tition function, see Section 3), modulo the ideal generated by the Serre relations.Seen this way, our main result can be phrased in the following way: The partition function for a bosonic object ( λ ) in R subject to a bilinear con-straint in R is the inverse of the partition function for an object (the set of level-one enerators in B ) with indefinite statistics, where odd powers are labeled as fermionic,subject to a bilinear constraint in R ⊥ . This statement provides an interpolating generalisation for partitions of con-strained objects of the ones made for unconstrained and maximally constrainedones in Section 3. However, unlike in those limiting cases, the statistics here maynot be switched, which we will comment on in Section 6.Now, consider an object ¯ λ in ¯ R , with the constraint ¯ λ | ¯ R = 0. The constraintcan be treated using a BRST formalism. For convenience, we change our notation anduse c M instead of ¯ λ M . The first term in the BRST operator Q is Q (2) = b MN c M c N ,where b MN in R is the ghost for the constraint . However, if the constraint happensto be reducible, there will be higher order ghosts compensating for the reducibility.Such reducibility will be captured by the introduction of a new bc pair, and a term Q (3) = b MNP c MN c P in Q . The representation of b MNP is everything that is allowed by Q = 0. This should be continued, as long as the reducibility continues, i.e. , as longas further such terms can be added. A generic term will be of the form b ( p + q ) c ( p ) c ( q ) ,where the ghosts are alternatingly fermionic and bosonic. From this trilinear form ofthe BRST operator it is immediately clear that its action on the c ghosts defines thecoalgebra of a Lie superalgebra. An infinite reducibility corresponds to an infinite-dimensional algebra.We now recognise the exact parallel between on one hand the construction of thecoalgebra, given the Serre relations (and nothing more), and on the other hand theconstruction of the BRST operator. The difference is only a matter of notation. Thecogenerators E ∗ ( p ) correspond to the ghosts c ( p ) , and the graded wedge products areautomatically implied by the “wrong” statistics of the ghosts. The operator d is theadjoint action (graded commutator) of Q , so that dE ∗ ( p ) ↔ [ Q, c ( p ) } .This means that if we calculate the partition function of λ as a constrainedobject, which is obtained as the conjugate of the tensor product of the partitionfunctions of all the ghosts, Z λ ( t ) = ∞ O p =1 (1 − t p ) ( − p R p , (4.8)it will coincide with the inverse partition function of the universal enveloping algebra It would maybe be more conventional to use a notation where c is the ghost multiplying theconstraint, and b its conjugate. Here, however, it turns out that all terms will be of the form bcc ,which corresponds to the standard form of “algebra” ghost terms in a BRST operator. The concept of reducibility is not absolute, but may depend on the degree of covariance. Here,we always consider reducibility as expressed in terms of representations of the finite-dimensional Liealgebra g (but should of course not be confused with the possible reducibility of the representationsthemselves). ( B + ), which by definition is Z U ( B + ) ( t ) = ∞ O p =1 (1 − t p ) ( − p +1 R p , (4.9)using the shorthand notation of (3.5). The inverse simply appears since the corre-spondence E ∗ ( p ) ↔ c ( p ) changes statistics.The above argument does not provide a strict proof of eq. (4.3). The missingstep is the proof that the BRST operator Q ∼ bcc correctly encodes the degrees offreedom of the constrained object, or, equivalently, that no other unwanted coho-mology arises. We refrain from doing this, but we will present a different proof inSection 5.Neither of the two above methods of finding the spectrum of generators has anadvantage over the other, since we just demonstrated that they contain exactly thesame calculational steps. However, knowing that the partition function is that of aconstrained object λ can often provide an alternative, more direct, and simpler wayof obtaining the answer. Provided that we know from the constraint which repre-sentation S p appear at any power λ p , the partition function is directly constructedas Z λ ( t ) = ∞ M p =0 S p t p . (4.10)Expanding this partition function in a product form gives information about all theghost representations, and thus about the generators of the algebra. This calculationbecomes especially simple in cases where S = R is an irreducible representationof some Lie algebra with highest weight λ , and S = R ⊥ = ∨ R ⊖ R is therepresentation with highest weight 2 λ . Then S p will have highest weight pλ . Indeed,the class of Borcherds superalgebras we consider all have this property, as will beshown in Section 5. We will give some examples of such situations in Section 7,among which are pure spinors and their associated superalgebras. Although therepresentations R p are complicated, they can be calculated from the more readilyavailable representations S p by inserting eqs. (4.9)–(4.10) into (4.3), which gives ∞ O p =1 (1 − t p ) ( − p R p = ∞ M q =0 S q t q . (4.11)The explicit solution of this relation for the spectrum of the Borcherds superalgebra, i.e. , the representations R p in terms of the known S p , can be obtained by recursion, This is the method used by Berkovits and Nekrasov in ref. [23] to obtain detailed informationon the partition functions of pure spinors.
13r by a M¨obius inversion. By comparing the left and right hand sides for the firstfew powers of t we get R = S ,R = ∨ R ⊖ S ,R = ( R ⊗ R ) ⊖ ∨ R ⊕ S ,R = (cid:0) ( R ⊗ R ) ⊕ ∧ R (cid:1) ⊖ ( ∨ R ⊗ R ) ⊕ ∨ R ⊖ S . (4.12)We will display some explicit examples of varying complexity in Section 7. This section will provide a proof of our main result (4.3), using the denominatorformula for Borcherds superalgebras [31–34]. It is known for general Borcherds su-peralgebras but here we only need a simplified version given below, valid for thespecial cases of Borcherds superalgebras under consideration.Let Φ be the root system of B , and for any integer p , let Φ p be the subset of Φconsisting of all roots β = pβ + α , where α is a linear combination of the real simpleroots β i = α i . Thus Φ is the root system of the subalgebra g , and B p is the directsum of all root spaces B β such that β ∈ Φ p , and, if p = 0, the Cartan subalgebra.We will show that the eq. (4.11), with the lowest weights of the representations S q given by the Dynkin labels λ i = q ( β , β i ∨ ) = q · β , β i )( β i , β i ) = qB i , (5.1)is equivalent to the denominator formula for B [31–34], which reads Q β ∈ Φ +(0) (1 − e − β ) mult β Q β ∈ Φ +(1) (1 + e − β ) mult β = X w ∈ W ∞ X q =0 ( − | w | ( − q e w ( ρ − qβ ) − ρ . (5.2)Here Φ +(0) and Φ +(1) consist of all even and odd positive roots, respectively, ρ is theWeyl vector of B , defined by ( ρ, β I ) = ( β I , β I )2 , (5.3)and the Weyl group W of B is generated by all fundamental Weyl reflections r i : β β − ( α i ∨ , β ) = β − α i , β )( α i , α i ) . (5.4)14he length | w | of an element w in W is the minimal number of fundamental Weylreflections (not necessarily distinct), which, applied after each other, give w .Note that the representations S p are given by the Dynkin labels λ i of their lowest weights, since we consider positive levels of B . However, we are going torelate the denominator formula (5.2) for B to the character formula for g , which isusually expressed in terms of the highest weight of a representation. Therefore it isconvenient to replace eq. (4.11) by the equivalent equation obtained by conjugatingall representations, ∞ O p =1 (1 − t p ) ( − p ¯ R p = ∞ M q =0 ¯ S q t q , (5.5)where now the highest weight of ¯ S q is − λ i = − qB i . What we will actually show isthat this eq. (5.5) is equivalent to the denominator formula (5.2). Let ˜Λ be the element in the weight space of B such that ( ˜Λ , α i ) = 0 for all i = 1 , . . . , r , and the componenent of ˜Λ corresponding to β in the basis of simpleroots is equal to one (this element exists uniquely since both B IJ and B ij = a ij arenon-degenerate). Thus ˜Λ − β is an element in the weight space of g (considered asa subspace of the weight space of B ). More generally, a root β ∈ Φ p can be written β = p ˜Λ + µ , where µ = β − p ˜Λ is an element in the weight space of g . We then get Y β ∈ Φ p (1 − e − β ) mult β = Y µ ∈ R p (1 − e − p ˜Λ e − µ ) mult µ = | R p | X k =0 ( − k X e − ( µ + ··· + µ k ) ( e − ˜Λ ) kp , (5.6)where the second sum goes over all sets of k distinct weights µ , . . . , µ k among theweights of R p , counting (as in Section 3) a weight with multiplicity m as m “distinct”weights. This sum can be obtained from the character for ∧ k R p by inverting eachterm, which corresponds to conjugating the representation R p . Thus Y β ∈ Φ p (1 − e − β ) mult β = | ¯ R p | X k =0 ( − k χ ( ∧ k ¯ R p ) s kp (5.7)where we have set s = e − ˜Λ . In the same way, Y β ∈ Φ p (1 + e − β ) mult β = | ¯ R p | X k =0 χ ( ∧ k ¯ R p ) s kp (5.8) Instead of considering positive levels in B and conjugating the representations we could ofcourse also have considered negative levels and only highest weights from the beginning. Y β ∈ Φ p (1 + e − β ) − mult β = ∞ X k =0 ( − k χ ( ∨ k ¯ R p ) s kp . (5.9)Here the character of ∨ k ¯ R p is given by the sum of all terms e − ( µ + ··· + µ k ) , where µ , . . . , µ k are weights of R p , this time not necessarily distinct. Following the notationin Section 3, we write this as Y β ∈ Φ p (1 ± e − β ) ∓ mult β = χ (cid:0) (1 ± s p ) ∓ ¯ R p (cid:1) , (5.10)and the left hand side of eq. (5.2) becomes Q β ∈ Φ +(0) (1 − e − β ) mult β Q β ∈ Φ +(1) (1 + e − β ) mult β = Y α ∈ Φ +0 (1 − e − α ) mult α ∞ Y p =1 χ (cid:16)(cid:0) − ( − p s p (cid:1) ( − p ¯ R p (cid:17) , (5.11)where Φ +0 consists of the positive roots of g .We now turn to the right hand side of the denominator formula, X w ∈ W ∞ X q =0 ( − | w | ( − q e w ( ρ − qβ ) − ρ . (5.12)Here W is the Weyl group of B , but since it is generated by the fundamental Weylreflections corresponding to the real roots only, it coincides with the Weyl group of g . In order to use the character formula for g we also need to replace the Weyl vectorof B with the one of g , but this requires some more consideration. The Weyl vector ρ = ρ B of B is defined as the element in the weight space of B satisfying( ρ B , β I ) = ( β I , β I )2 , (5.13)whereas the Weyl vector ρ g of g only has to satisfy( ρ g , α i ) = ( α i , α i )2 , (5.14)but on the other hand it must have a zero component corresponding to β in thebasis of simple roots. Thus the Weyl vectors of B and g are different (in general),but since their difference ρ B − ρ g is orthogonal to the real roots, ( ρ B − ρ g , α i ) = 0,it is invariant under the Weyl group, w ( ρ B − ρ g ) = ρ B − ρ g . We then get w ( ρ B − qβ ) − ρ B = w (cid:0) ρ g + ( ρ B − ρ g ) − qβ (cid:1) − ρ g − ( ρ B − ρ g )= w ( ρ g − qβ ) − ρ g + w ( ρ B − ρ g ) − ( ρ B − ρ g )= w ( ρ g − qβ ) − ρ g (5.15)16nd we can indeed replace ρ = ρ B by ρ g in eq. (5.12). To simplify the notation, wewill henceforth write ρ = ρ g . Furthermore, since also ˜Λ is orthogonal to the realroots, we have w ( ρ − qβ ) − ρ = w (cid:0) ρ − q ˜Λ + q ( ˜Λ − β ) (cid:1) − ρ = w (cid:0) ρ + q ( ˜Λ − β ) (cid:1) − ρ − q ˜Λ (5.16)and then X w ∈ W ∞ X q =0 ( − | w | ( − q e w ( ρ − qβ ) − ρ = X w ∈ W ∞ X q =0 ( − | w | ( − q e w ( ρ + q (˜Λ − β )) − ρ − q ˜Λ = X w ∈ W ∞ X q =0 ( − | w | ( − q e w ( ρ + q (˜Λ − β )) − ρ s q . (5.17)Equating eqs. (5.11) and (5.17) we get ∞ Y p =1 χ (cid:16)(cid:0) − ( − p s p (cid:1) ( − p ¯ R p (cid:17) = ∞ X q =0 P w ∈ W ( − | w | e w ( ρ + q (˜Λ − β )) − ρ Q β ∈ Φ +0 (1 − e − β ) mult β ( − q s q , (5.18)where we recognise P w ∈ W ( − | w | e w ( ρ + q (˜Λ − β )) − ρ Q β ∈ Φ +0 (1 − e − β ) mult β (5.19)as the character of the representation of g with highest weight q ( ˜Λ − β ) given bythe Dynkin labels (cid:0) q ( ˜Λ − β ) , β i ∨ (cid:1) = − q ( β , β i ∨ ) = − q · β , β i )( β i , β i ) = − qB i = − λ i , (5.20)and thus ∞ Y p =1 χ (cid:16)(cid:0) − ( − p s p (cid:1) ( − p ¯ R p (cid:17) = ∞ X q =0 χ ( ¯ S q )( − q s q . (5.21)Finally, substituting s by − t we arrive at the equation ∞ Y p =1 χ (cid:0) (1 − t p ) ( − p ¯ R p (cid:1) = ∞ X q =0 χ ( ¯ S q ) t q , (5.22)which is the character version of (and thus equivalent to) eq. (5.5).17 Why is the method not applicable to Lie alge-bras?
Let us replace β with an even simple root α , which is real, ( α , α ) >
0, but oth-erwise satisfies the same inner product relations as β , thus ( α , α i ) = ( β , α i ). TheChevalley-Serre relations (2.4)–(2.5), with all superbrackets being ordinary antisym-metric brackets, and with the Cartan matrix B IJ replaced by A IJ = ( α J , α I ∨ ) = 2 ( α J , α I )( α I , α I ) , (6.1)defines a Kac-Moody algebra A . This corresponds to adding an ordinary (white)node to the Dynkin diagram of g instead of a grey one, and in analogy with B theadjoint representation of A dcomposes into g -representations R p . The representation R is the same as in the case of B , but R is now a subrepresentation of ∧ R , the anti-symmetric tensor product of R with itself. Its complement is the direct sum ofrepresentations with lowest weights given by the Dynkin labels λ i = A ij + 2 A i forall j such that A j = 0.One might imagine that the statement (4.3) would apply both for the Lie su-peralgebra B and the ordinary Lie algebra A . This would potentially have made itpossible to extract precise information about the generators to all levels for classes ofinfinite-dimensional ( e.g. hyperbolic) Kac–Moody algebras. It turns out, however,that statistics can not just be changed. This is because the constrained object λ then would be fermionic. Having bilinear (bosonic) constraints on fermionic vari-ables is generically a strange situation, and leads to complicated structures, as wewill explain.Consider a fermionic λ in a representation R of g with a bilinear constraint in R of A , thus complementary to some Serre relations in the anti-symmetric product ∧ R . An algebraic “solution” to the constraints (in the sense that one considers apower series in λ modulo the constraint) will result in a polynomial partition func-tion, where the highest term is of order lower than or equal to | R | . Its factorisationin ghost contributions is however infinite. This is because the ghosts, like the originalvariables, are fermionic, and so are the ghosts for reducibility. Instead of removingfermionic degrees of freedom, the ghosts add more fermions, corresponding to theremoval of bosonic degrees of freedom (the constraint).This somewhat pathological behaviour is in itself not an obstruction for theexistence of a relation like eq. (4.3) — one might well imagine that a properlyregularised sum with strictly positive terms yields a negative value (although it is a valid argument against an analogous construction when the Lie algebra is finite-dimensional). What makes things go wrong is the fact that a bilinear constraint18n fermions inherently has some reducibility coming from the fermionic propertyof the variables. Whatever the bosonic constraint is, it is e.g. obvious that raisingit to a sufficient power will give zero due to saturation of fermions. This has nocounterpart in the bosonic situation, and will introduce ghosts in a BRST treatmentwhich do not enter Q in the “ bcc ” form. Therefore, a correct BRST treatment cannot be given an interpretation in terms of a Lie algebra. We have observed in anumber of examples that a naive treatment of the Serre relations as complementaryto a constraint gives agreement in the spectrum to a number of low levels, beforethe “saturation of fermions” becomes relevant. Whether there is a systematic wayof consistently defining partitions for fermions that circumvents this and correctlyencodes the Serre relations (and thereby the spectrum of the Lie algebra) is an openquestion. In any case it seems reasonable that the occurrence, in the superalgebracase, of the highest weights which are simply multiples of the defining one, hasparticularly simple structure without counterpart in the Lie algebra situation.Turning to the actual proof of the main result for the Lie superalgebra B inSection 5, it is easy to identify the step where the argument fails for Lie algebras.The Weyl group of the extended algebra A is not identical to that of g , and theproof in its present form fails, although the denominator formula is known. We will give a number of examples that illustrate the connection between the con-strained bosonic variable and the spectrum of generators in the superalgebra.We use the notation B + for the subalgebra of generators at positive levels, al-though in some examples (the freely generated algebras) it is not a subalgebra of aBorcherds superalgebra of the precise type described in Section 2. Consider first a freely generated superalgebra. Then the Serre relations are empty,and elements of the universal enveloping algebra U ( B + ) are given by arbitrarytensor products of R . Thus Z U ( B + ) is given by eq. (3.10) with σ = −
1, which is theinverse of the formal partition for a “maximally constrained” boson in R . This isone extreme case of the correspondence (in which the statistics can be interchanged).It appears if the additional simple root is not a null root, but has negative lengthsquared.The other extremal case is when the Serre relations fill the whole symmetricproduct ∨ R , so that { E M , E N } = 0. The superalgebra is then finite-dimensionalwith E M forming a basis for B + = B . The partition function Z U ( B + ) is the par-19ition function of fermions in R , which is the inverse of the partition function foran unconstrained boson. Also in this other special case the statistics can be inter-changed.Intermediary cases only work as a correspondence of the form (4.3) betweensuperalgebras and constrained bosons, and these provide less trivial illustrations ofour result. D = 8 pure spinors and null vectors Pure spinors provide well known and extensively studied examples of constrainedbosons. They lead to minimal spinor orbits under Spin groups, due to the fact thatonly a single irreducible representation appears in a spinor bilinear (the one whosehighest weight is twice the one of a spinor), and by induction there is only onerepresentation for each positive power of the spinor. We will give two examples ofpure spinors, in this subsection and the next.Let us first consider a pure spinor in D = 8, where the constraint is particularlysimple, λ α λ α = 0. This is via triality equivalent to a null vector. The Dynkin diagramof the corresponding superalgebra is given below.The analysis can equally well be performed for null vectors in general dimension D .The refined partition function for λ reads Z λ ( t ) = ∞ M p =0 ( p . . . t p , (7.1)where Dynkin labels of highest weights have been used for the representations, and(10 . . .
0) denotes the vector representation. The representation ( p . . .
0) consists ofsymmetric and traceless multi-vectors. Its dimension is readily calculated to be (cid:18) p + D − p (cid:19) − (cid:18) p + D − p − (cid:19) = (2 p + D − p + D − D − p ! , (7.2)so the unrefined partition function (just counting dimensions) is Z λ ( t ) = ∞ X p =0 (2 p + D − p + D − D − p ! t p = 1 + t (1 − t ) D − = 1 − t (1 − t ) D . (7.3)20he same result is obtained by constructing the partition function from the ghosts.An (unconstrained) variable λ contributes to Z with (1 − t ) − (10 ... , and the fermionicghost for the constraint with (1 − t ) (00 ... . The constraint is irreducible, so thereare no higher ghosts. The correspondence (4.3) tells us that the spectrum of theuniversal enveloping algebra U ( B + ) is given by Z U ( B + ) ( t ) = ( Z λ ( t )) − = (1 − t ) (10 ... ⊗ (1 − t ) − (00 ... , (7.4)corresponding to a fermionic generator in (10 . . .
0) at level one and a bosonic onein (00 . . .
0) at level two. The superalgebra B , which here comes with a 5-grading,is finite-dimensional, B ≈ osp ( D | λ is finite. D = 10 pure spinors and supergravity forms Let us turn to the more interesting cases of infinite-dimensional superalgebras, whichare related to the spectrum of forms in supergravity, and thereby also to the tensorhierarchies in gauged supergravity (see the discussion in Section 1).Pure spinors in D = 10 are relevant for the off-shell superfield formulation of D = 10 super-Yang–Mills theory (see e.g. refs. [35–38]). The partition function isdescribed in some detail in ref. [23], and is given by Z λ ( t ) = ∞ M p =0 (0000 k ) t p = (cid:2) (00000) ⊖ (10000) t ⊕ (00001) t ⊖ (00010) t ⊕ (10000) t ⊖ (00000) t (cid:3) ⊗ (1 − t ) − (00001) , (7.5)or, just counting dimensions, Z λ ( t ) = 1 − t + 16 t − t + 10 t − t (1 − t ) = (1 + t )(1 + 4 t + t )(1 − t ) . (7.6)The power 11 of the pole at t = 1 signals 11 degrees of freedom in a pure spinor.The Dynkin diagram of the corresponding superalgebra is given below.21his algebra is infinite-dimensional. Still, we know that the spectrum is determinedby Z U ( B + ) = Z λ − . The generators at each level in B + are obtained by rewritingthe partition function (7.5) on product form, which reflects the ghost structurecorresponding to the infinite reducibility:( Z λ ( t )) − = ∞ O p =1 (1 − t p ) ( − p +1 R p . (7.7)This can be done recursively as in eq. (4.12), with the following result for the firstfew representations: R = (00001) = , R = (10000) = , R = (00010) = ,R = (01000) = , R = (10010) = ,R = (11000) ⊕ (00020) ⊕ (10000) = ⊕ ⊕ , . . . (7.8)For the dimensionalities | R p | , an explicit M¨obius inversion formula can be found [23].The case of D = 10 pure spinors is relevant to exceptional field theory withU-duality group E ≈ Spin (5 , E n ( n ≤
8) is identical to the infinite spectrumof superforms in D -dimensional maximal supergravity ( D = 11 − n ), as was shownfor low levels in ref. [22]. Our results here, combined with those in refs. [8–10],establish this correspondence for all levels. Here we have shown that the ghosts for aconstrained object give rise to a Borcherds superalgebra by the action of the BRSToperator, and in refs. [8–10] it was shown that the forms in the supergravity theorysimilarly give rise to a Borcherds superalgebra by their Bianchi identities. In theextended field theory the constraint is directly associated with the section condition,and leads to the same Borcherds superalgebras as the supersymmetry constraint onthe supergravity side. Since the Borcherds superalgebras are the same, the sequencesof representations are the same as well.The (unrefined) partition functions corresponding to the constraint in the ex-ceptional field theories were give in ref. [22]. As an example, the E case gives aBorcherds superalgebra defined by a λ belonging to a cˆone over the Cayley plane [39].The partition function is Z λ ( t ) = ∞ M k =0 ( k t k = (cid:2) (000000) ⊖ (000001) t ⊕ (010000) t ⊖ (001000) t ⊕ (100001) t ⊖ (000002) t ⊖ (200000) t ⊕ (100001) t ⊖ (000010) t ⊕ (010000) t ⊖ (100000) t ⊕ (000000) t (cid:3) ⊗ (1 − t ) − (100000) , (7.9)22nd the spectrum of the Borcherds algebra is obtained recursively by rewriting Z λ on product form.With the same interpretation as gauge transformations and reducibility for gen-eralised diffeomorphisms, our earlier example with null vectors, corresponding to afinite-dimensional superalgebra, is relevant for double field theory with T-dualitygroup O ( d, d ). A curious observation, somewhat besides the main focus of this paper, is that thelast Dynkin diagram of the previous subsection has the same form as the one for E , had the extra node been white instead of grey. Similarly, had the extra node inthe superalgebra of Subsection 7.2 been white, we would have had the Lie algebrafor SO (10), or in the general case SO ( D + 2).Polynomials of a pure spinor, the constrained object encoding the spectrum ofthe Borcherds algebra in question, indeed form an infinite-dimensional “singleton”representation of E , which can be constructed as follows. Consider generators of E ⊃ so (10) ⊕ u (1). The adjoint splits as → − ⊕ ( ⊕ ) ⊕ . Call thespinorial generators λ α and µ α . With the conventions[ J ab , λ α ] = 14 ( γ ab λ ) α , [ Q, λ α ] = λ α , (7.10)the only non-manifestly covariant non-vanishing commutator is[ µ α , λ β ] = ( γ ab ) αβ J ab + 32 δ βα Q . (7.11)The relative coefficient is fixed by demanding the Jacobi identity on the form[[ µ α , λ [ β ] , λ γ ] ] = 0 . (7.12)Now, we start from an so (10)-scalar ”ground state” | i annihilated by µ α , anduse λ α as ”creation operators”, giving a Verma module of polynomials in λ . Let theground state have charge q , Q | i = q | i . We want to adjust the value of q so that( λγ a λ ) generates an E -invariant ideal. This happens if µ α ( λγ a λ ) | i = 0. A shortcalculation leads to µ α ( λγ a λ ) | i = 32 (2 q + 1)( γ a λ ) α | i + 14 ( γ ij γ a γ ij λ ) α | i = (cid:18)
32 (2 q + 1) − (cid:19) ( γ a λ ) α | i . (7.13)23f q = 4, this vanishes, and the ideal may be factored out without breaking E .This shows how the space of (holomorphic) polynomials in a pure spinor forms aninfinite-dimensional lowest-weight representation of E . It may be called a singletonrepresentation, since it only consists of a “leading trajectory” of Spin (10) repre-sentations with highest weights (0000 k ) at each U (1) charge 4 + k . The loweringoperator µ α can be identified with the gauge invariant (in the sense that it respectsthe ideal) derivative with respect to a pure spinor constructed in ref. [40].In the same way a singleton representation of SO ( D + 2) [41] is obtained bystarting from a ground state of a certain charge. We use the conventions[ J ab , λ c ] = 2 λ [ a η b ] c , [ Q, λ a ] = λ a , [ µ a , λ b ] = J ab − η ab Q . (7.14)The ideal generated by λ can be factored out for vacuum charge q = D − , leadingto a singleton representation.At least in these particular cases, the constrained object, i.e. , the pure spinor ornull vector, has a direct relation both to a Borcherds superalgebra and to a (finite-dimensional) Lie algebra. Both algebras are obtained by adding a node, grey andwhite, respectively, in the same position to the Dynkin diagram of a semi-simpleLie algebra, but the rˆole of the pure spinor is quite different in the two cases. Thephenomenon is certainly more general, but is in its present form limited to situationswhere the λ ’s commute, and thus are the only generators at positive level in the Liealgebra. It holds e.g. for the constrained objects λ occurring in relation to tensorhierarchies of E n × R + with n ≤
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