The conformal characters

Journal of High Energy Physics, Apr 2018

Antoine Bourget, Jan Troost

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The conformal characters

HJE The conformal characters Antoine Bourget 0 1 3 Jan Troost 0 1 2 Paris 0 1 France 0 1 0 CNRS, PSL Research University , Sorbonne Universites 1 Calle Federico Garc a Lorca , 8, 33007 Oviedo , Spain 2 Departement de Physique, Ecole Normale Superieure 3 Department of Physics, Universidad de Oviedo We revisit the study of the multiplets of the conformal algebra in any dimension. The theory of highest weight representations is reviewed in the context of the Bernstein-Gelfand-Gelfand category of modules. The Kazhdan-Lusztig polynomials code the relation between the Verma modules and the irreducible modules in the category and are the key to the characters of the conformal multiplets (whether nite dimensional, in - nite dimensional, unitary or non-unitary). We discuss the representation theory and review in full generality which representations are unitarizable. The mathematical theory that allows for both the general treatment of characters and the full analysis of unitarity is made accessible. A good understanding of the mathematics of conformal multiplets renders the treatment of all highest weight representations in any dimension uniform, and provides an overarching comprehension of case-by-case results. Unitary highest weight representations and their characters are classi ed and computed in terms of data associated to cosets of the Weyl group of the conformal algebra. An executive summary is provided, as well as look-up tables up to and including rank four. Conformal and W Symmetry; Conformal Field Theory; Supersymmetry and Duality 2.1 2.2 2.3 3.1 3.2 3.3 4.1 4.2 1 Introduction 2 Warm-up: the so(2; 3) algebra The representations of the so(3) algebra The representations of the so(2; 3) algebra The unitary representations 2.4 In physics conventions 3 The characters of irreducible representations The Kazhdan-Lusztig theory The Kazhdan-Lusztig polynomials The nite-dimensional representations 3.4 Examples 4 The unitary conformal multiplets Useful concepts 4.3 The algebras Dk = so(2; 2k 6 The unitary conformal characters 6.1 In odd space-time dimension 6.2 In even space-time dimension 7 Summary and comparison with the physics literature 7.1 The executive summary 7.2 A brief comparison to the physics literature 8 Apologia A The Lie algebra conventions A.1 The Lie algebra Bk A.2 The Lie algebra Dk B The structure of real simple Lie algebras B.1 The structure theory B.2 The classi cation of real simple Lie algebras B.3 The classi cation of Hermitian symmetric pairs C The character tables for integral unitary weights { i { Introduction Quantum eld theory is one of the most successful tools of theoretical physics. It is ubiquitous in our understanding of physical phenomena from the smallest to the largest scales. Conformal quantum eld theories can be viewed as simpli ed quantum eld theories that arise at very low or very high energy, or at critical points. Their symmetry algebra is enlarged. Relativistic conformal eld theories allow for a symmetry algebra which includes the conformal algebra so(2; n) in space-time dimension (n It is useful to gather the spectrum of a physical theory in terms of multiplets of the symmetry algebra. Hence it is crucial to study the representation theory of the conformal algebra so(2; n). In physical theories, often only highest weight representations will arise. Moreover, in unitary theories, these representations are required to be unitarizable. Thus, the study of the unitary highest weight representations of so(2; n) has been an integral part of the physics literature of the last fty years. Importantly, before physicists classi ed conformal multiplets in all cases of their interest, the mathematics literature yielded an overarching insight into the generic case, providing a complete classi cation of conformal multiplets, with proof. In particular, the representation theory will reduce to a theory of Weyl groups, and numbers associated to pairs of Weyl group elements. That provides an e cient coding of otherwise lengthy manipulations of conformal algebra generators. In most cases, the mathematics literature precedes the physics literature, which is indicative of the fact that physicists have found the mathematics literature hard to read. We intend to bridge this unfortunate gap in the treatment of this central problem in quantum eld theory by providing a physicist's guide to the mathematics literature. Our treatment will be practical yet generic, referring to the relevant mathematics books and papers for the complete proofs while still providing the conformal eld theorist with all the necessary tools to reconstruct a particular result using general principles only. Bridges between the mathematics literature and the physics literature have been constructed previously. We refer e.g. to [1] for the exploitation of the generic classi cation of unitary multiplets, and to [2] for a review of the salient properties of parabolic KazhdanLusztig polynomials relevant to conformal multiplets nitely represented on the compact subalgebra of the conformal algebra. We will provide a considerably more complete treatment, and hopefully a more accessible bridge. From the e ort we invested in identifying and marrying the mathematics and physics literature, we concluded that an introduction for physicists to the intricate mathematics of conformal multiplets remained overdue. It is possible to extend the scope of this work to include representations of superalgebras, but in that situation additional subtleties arise | in particular, the Weyl group geometry alone does not determine completely the representation theory | we plan to discuss this in future work. The structure of the paper is as follows. In section 2, we treat a warm-up example, namely the so(2; 3) conformal algebra in three space-time dimensions. We compute the characters of all irreducible highest weight multiplets, and gently introduce some of the mathematics necessary to understand the structure of the representation theory. We also { 1 { identify all multiplets that are unitarizable, and write out their characters in both a mathematical and physical language. After the warm-up section, we introduce the advanced mathematics to treat the generic case. In section 3, we summarize how to compute characters of all highest weight representations of the algebras so(2; n) with n arbitrary. This discussion will include nite dimensional, in nite dimensional, unitary and non-unitary representations. We review how the multiplicities of the irreducible modules in the Verma modules are given by the evaluation of Kazhdan-Lusztig polynomials at argument equal to one, and how the inversion of the decomposition xes the irreducible characters. It will be su cient to do calculations in the Weyl group (and Hecke algebra) of the conformal algebra to understand the full structure of the representation theory. In order to apply the formulas, we gather data on the Weyl groups of the Bk and Dk algebras, and the corresponding Kazhdan-Lusztig polynomials. In section 4, we review how to identify the unitarizable representations among all those studied in section 3. We will implicitly make use of necessary and su cient inequalities on the quantum numbers which are elegantly derived in the mathematics literature. In section 5 we exploit the speci c features of unitary representations to simplify the generic Kazhdan-Lusztig theory, and factorize a compact subalgebra Weyl group, which leads to the study of parabolic Kazhdan-Lusztig theory. That allows us to compute all unitary highest weight characters in section 6. Hasty readers can jump directly to section 7 where they will nd an executive summary, with references to an appendix containing low rank unitary character tables. 2 Warm-up: the so(2; 3) algebra In this section, before facing the representation theory of the so(2; n) algebras in all its complexity, we focus on the conformal algebra so(2; 3). We review the conformal multiplets which have a highest weight. We determine the structure of the irreducible representations, and also which irreducible highest weight representations are unitarizable. Our analysis is phrased in the mathematical language of the category of highest weight modules, and introduces a number of useful mathematical concepts. These serve as a warm-up for the introduction of more advanced concepts in section 3. References for proofs of the statements in this section are mostly postponed to section 3 as well. 2.1 The representations of the so(3) algebra We draw inspiration from the highest weight representation theory of the simplest Lie algebra so(3) = su(2), generated by three generators, so(3) = hJ1; J2; J3i. Its representation theory is obtained by rst choosing a Cartan subalgebra h = hJ3i, as well as raising and lowering operators J = J1 iJ2. Then, we pick a highest weight eigenvector of the Cartan generator J3 with eigenvalue , called the highest weight. The highest weight vector is by de nition annihilated by the raising operator. We then act on it with the lowering operator, generating new vectors, which generate a representation of so(3). As is well-known, if 2= Z 0 (in a given normalization), the process never stops and the representation is in nite dimensional. On the other hand, when 2 Z 0, we can consistently de ne a + 1 { 2 { dimensional (irreducible) representation, with a lowest weight vector which is annihilated by the lowering operator. In this subsection, we formalize these well-known facts in the language of modules, which is used in the rest of the paper. We de ne a Verma module M with weight as the representation of g where no constraint is imposed beyond the relations of the Lie algebra. This means that the character [M ] of the Verma module M is given by w so the character of the general irreducible module can be rewritten where the lowering operator has eigenvalue 2. In particular, a Verma module is always in nite-dimensional. It may happen that a Verma module contains other Verma modules. Here, this happens only when 2 Z 0, where the Verma module M 2 is included in the Verma module M . In that case, we can construct the quotient module M =M 2, which is nite-dimensional and irreducible. We call this irreducible module L , and its character is M is irreducible, and therefore we set L = M . is not a positive integer, the Verma module As we will see later, it is natural to introduce the Weyl vector = 1 of the so(3) Lie algebra, and the Weyl group W = f1; 1g. We further introduce the dot action of the Weyl group on the weight space through the formula (2.1) HJEP04(218)5 (2.3) (2.4) (2.5) 2For the exploration of other objects in the category of highest weight modules of so(3) in a physical context, see [3]. { 3 { We perform a similar analysis for the highest weight representations of the so(2; 3) = B2 algebra. The choice of real form of the algebra does not matter at this stage, but we must come back to this point when we consider the question of unitarity. We choose a Cartan subalgebra in the compact subalgebra so(2) so(3) of so(2; 3), corresponding to the dilatation operator and a spin component. The Verma module M with highest-weight generically has character generalizing (2.1), where the product over negative roots makes sure that we take into account the free action of the lowering operators on the highest weight state. Depending on the weight , the module M may be reducible, and the character of the irreducible module L will di er from the Verma character (2.6). This can happen only if another Verma module M is a strict submodule of M for some weight . Integral regular weights. Firstly, we consider for simplicity an integral weight : for each root the product h ; _i satis es h ; _i 2 Z. The weight space is two-dimensional, and the position of with respect to the negative Weyl vector is characterized by the sign of the integers h i + ; _ . This de nes eight (shifted) Weyl chambers, as shown in gure 1. One can label the chambers with elements of the Weyl group W , associating the identity element to the chamber that contains the weight 2 , as in gure 1. On the Weyl group, one can de ne a partial order, the Bruhat order [4]. This order can be summarized in a Bruhat graph represented in gure 2, as well as on gure 1. In a minimal representation of a Weyl group element in terms of simple re ections, the length of the element is equal to the number of simple re ections. Any integral weight in the interior of one of the (shifted) Weyl chambers can be written in a unique way as = w , where w 2 W and is antidominant, meaning that h + ; _i 2= Z>0 for each positive root . The partial Bruhat order is instrumental in our understanding of the structure of Verma modules [5]. Indeed, for an integral weight lying in the interior of the antidominant Weyl chamber, and any Weyl group element w, we have that the irreducible module (and character) can be understood in terms of the Verma modules (and characters) associated to the same antidominant weight, and Weyl group elements smaller than w in the Bruhat order: X w0 w [Lw ] = bw0;w[Mw0 ] bw0;w = ( 1)`(w) `(w0) ; for some integer coe cients bw0;w. In the case of the algebra so(2; 3), these coe cients are particularly simple | and it is mostly here that we exploit that we restrict to the example of so(3; 2) in our warm-up section. The coe cients bw0;w for the so(2; 3) algebra are given by where `(w) is the length of the Weyl group element w, which can be read from the Bruhat graph [4] (see gures 1 and 2). The dotted Weyl group action is still given by the formula (2.7) (2.8) { 4 { re ections si, their Bruhat order, the simple roots i and the integral weight lattice. The red lines correspond to singular weights, and delimitate the shifted Weyl chambers. The intersections of gray lines correspond to integral weights. w . We have restricted to integral weights in the interior of a Weyl chamber | those are called regular. We turn to an example. Firstly, let us introduce the parameterization of roots and weights in terms of an orthonormal basis ei (described in detail in appendix A). The simple roots are 1 = e1 weight e2 and 2 = e2, see gure 1. The so(3; 2) weights are denoted ( 1; 2) for a = 1e1 + 2e2. Let us then consider the weight = ( 1; 2). It sits inside the Weyl chamber labeled by the Weyl group element w = s2s1s2 of length three. For this example, the formula (2.7) gives rise to the character [L( 1;2)] = [M( 1;2)] [M( 1; 3)] [M( 2;2)] + [M( 2; 3)] + [M( 4;0)] [M( 4; 1)] : (2.9) Integral singular weights. The formula (2.7) provides the character of any irreducible highest-weight module with highest weight in the interior of a Weyl chamber, i.e. away from the red lines in gure 1. Now we focus on singular integral weights, which are the { 5 { s2 ` = 4 to the s2 re ection, and we use the same color for elements of W connected by this re ection: they contribute to the same module M c, see equation (2.10). integral weights such that h + ; _i = 0 for at least one root . They lie on a red line in gure 1. The rule here is as follows: consider all the Weyl group elements that label the Weyl chambers of which the closure contains , and pick the smallest such group element w according to the Bruhat order. We can then write again = w for an antidominant weight , and the character formula (2.7) remains true. Example. Let us consider the weight = ( 32 ; 12 ). This is an integral weight, but it is singular. It belongs to the closure of the Weyl chambers labeled by the Weyl group elements s2s1 and s2s1s2 of length two and three respectively. The smallest of these two elements is w = s2s1, and therefore one writes = ( 32 ; 12 ) = w ( 52 ; 12 ). We then compute [L( 32 ; 12 )] = [M( 32 ; 12 )] [M( 52 ; 12 )] [M( 32 ; 32 )] + [M( 52 ; 12 )] = [M( 32 ; 12 )] [M( 32 ; 32 )] : The cancellation between Verma module characters occurs because we are studying a representation with singular highest weight. Non-integral weights. Finally, we extend our computation to non-integral weights. For an arbitrary weight , we construct the set [ ] of roots that satisfy h ; _ i 2 Z. To get a grasp on [ ], we compute this scalar product for all positive roots, with as before = 1e1 + 2e2. See table 1. A priori, since there are four positive roots we have 24 = 16 con gurations to consider, but consistency restricts this number to 7 con gurations, which are listed in table 1. One observes that [ ] is a root system, and its Weyl group W[ ] will play the role that the Weyl group W of the whole algebra played in the integral case. The root system [ ] determines the integrality class of . In this low rank case, the integer coe cients bw;w0 again simplify to a sign depending on the length of the elements in the group W[ ]. The character formula takes the form (2.7), but where the sum is restricted to the Weyl group elements W[ ] and the length function is inside this group. In this manner, we have found the characters of all irreducible highest weight representations of the B2 algebra. { 6 { HJEP04(218)5 1 2 2 1 of the B2 algebra, the scalar product of the roots with the weights = 1e1 + 2e2 as well as the root systems [ ] they give rise to. = ( 12 ; 0). The integrality class is D2, and the associated Weyl group has four elements, W[ ] = f1; s2; s1s2s1; s1s2s1s2g, using the notations of gure 1. The weight lies in the chamber of the longest element s1s2s1s2, so the irreducible character with highest weight is [L( 12 ;0)] = [M( 12 ;0)] [M( 12 ; 1)] [M( 52 ;0)] + [M( 52 ; 1)] : 2.3 The unitary representations As we review in full generality in section 4, only a subset of the irreducible modules L are unitarizable. We say that a weight is unitary if the corresponding irreducible module L is unitarizable. In this context, it is important that we consider the real form so(2; 3) of the complex B2 algebra. Manifestly, this is a non-compact real form, and therefore non-trivial unitary representations will be in nite-dimensional. As we recall in section 4, in the case of the algebra so(2; 3), the result of the identi cation of unitary weights is as represented in gure 3, where the unitary weights are painted in blue. A second observation is that for all unitary weights we have that 2 2 2 N. Thus, from the point of view of table 1, the unitary weights correspond to the third, sixth or seventh cases, i.e. with root systems [ ] = A1; D2 or B2. This corresponds to the fact that the compact subalgebra su(2) = so(3) so(2; 3) is nitely represented in a unitary highest weight representation. In other words, for unitary irreducible modules the only source of in nitedimensionality is the non-compact part of the algebra. We exploit this fact to write more compact formulas for the characters. Firstly, we introduce notations that re ect this desire. For a unitary weight, let us de ne a module M c which is the quotient of two Verma M c = M =Ms2 : [M c] = [M ] [Ms2 ] : { 7 { modules: of the module M c is This is sensible because of the restriction on unitary weights. Accordingly, the character Thus, we have already divided out a Verma module that is guaranteed to be a submodule because of the fact that the compact algebra is nitely represented. Using this notation, we can write down the characters of all irreducible unitary representations of so(2; 3) as follows: (2.10) (2.11) algebra so(2; 3). The green circles correspond to dominant weights. 0) The highest weight = 0 corresponds to the trivial representation, and we simply have we nd that the compact subtraction is the end of the story 1) For highest weights which fall in one of the following categories: 2 1 2= Z (the A1 case) 2 1 2 Z and 1 2 2= Z (the D2 case) and 1 North-West chamber of D2) 32 (the weight is in the is integral (the B2 case) and 1 2 2 (the West-North chamber of B2) = [L ] = [M c] : in one of the two following categories, we nd a further subtraction: 2a) If 2 1 2 Z and the North-East chamber of D2) | this category contains only two weights, 2b) If is integral (the B2 case) and 1 > 2 2 and 1 < 32 (the North-West chamber of B2) [L ] = [M c] [M(cs1s2s1) ] : [L ] = [M c] [M(cs2s1s2) ] : These results comprise all characters of unitary irreducible highest weight representations of the conformal algebra so(3; 2). In the next subsection, we render more manifest the physical content of these results. 2.4 In physics conventions Early physics references classifying the unitary representations of the so(3; 2) algebra and their characters are [6{8] and [9]. The algebra so(2; 3) admits a basis made of three so(3) spins J1;2;3, three translations P1;2;3, three special conformal transformations K1;2;3 and the dilatation operator D. In order to de ne the Verma modules, we declare two operators to be in the Cartan subalgebra, which we choose to be the spin component J3 and the dilatation operator D which are in a compact subalgebra. We pick four raising operators (J+ and K1;2;3) and four lowering operators (J and P1;2;3). We consider highest-weight modules, so the weights will consist of eigenvalues ( E; j) of ( D; J3). In these conventions, closer to traditions in physics, the above generic Verma module characters translate into where the fugacity x keeps track of the conformal dimension of the states, while the fugacity s codes (twice) the 3-component of the spin. The characters with respect to the su(2) compact subalgebra read (2.14) (2.15) HJEP04(218)5 (2.16) (2.17) (2.18) k=0;1;::: with the usual spin j character [Ls2uj(2)] of the representation of the su(2) subalgebra de ned by On the lower blue line in gure 3, we nd the trivial representation with ground state energy and spin ( E = 1; j = 2) = (0; 0), the singleton ( E; j) = ( 1=2; 0) as well as the other scalars ( E < 1=2; 0). On the second line, we have the singleton ( 1; 1=2), { 9 { as well as the other spinors ( E < 1; 1=2). The other representations are the generic ( E j 1; j) representations. See e.g. [9] for an early summary. For the weights of type 1) in subsection 2.3, which include the generic scalar, spinor and higher spin representations we nd the characters For the weights of type 2), we have for the singletons (type 2a)) and for the other extremal representations (type 2b) | note that for those, j 1), These calculations exhaust the characters of unitary highest weight representations of so(3; 2), and are in agreement with the physics literature. Summary remarks. The warm-up example of the three-dimensional conformal algebra is illuminating in multiple respects. It identi es the crucial role of the Weyl vector and the Weyl group for all irreducible characters, as well as the role of the compact subalgebra in the simpli cation of the unitary characters. It also motivated that we need to come to terms with at least two more advanced mathematical concepts: the rst is the multiplicity of the Verma modules in the characters of irreducible modules, and the second is the generic classi cation of unitary highest weight representations. The generic treatment of these points requires further levels of abstraction. 3 The characters of irreducible representations In this section, we explain how to write the characters of irreducible modules in terms of the characters of Verma modules for an arbitrary semisimple complex Lie algebra g. Since the full mathematical solution to this problem is available, but may be hard to read, or even identify, we provide a very brief guide to the history and literature. Important early contributions to the understanding of the category of modules with highest weight are [10] and [11]. The generic solution to the character calculation is based on the Kazhdan-Luzstig conjecture [12] proven in [13, 14]. The book [5] makes the mathematics signi cantly more accessible. Furthermore, to understand the unitary characters the parabolic Kazhdan-Lusztig polynomials [15] are instrumental, in particular as pertaining to Hermitian symmetric spaces [16]. The parabolic polynomials were computed in [17] and in more technical detail in [18]. The nal step in summarizing the literature requires the use of translation functors [5], and the resulting nal formulation is most easily read in [19] and [20]. We refer to the book [5] as well as to the summary [20] for further history. 3.1 In this subsection, we brie y remind the reader of basic concepts in Lie algebra and representation theory. See e.g. [21{23] for gentler introductions. Let g be a semisimple complex Lie algebra, with Cartan subalgebra h. We denote the set of roots of g by , the subset of positive roots by + and by s is the subset of simple roots. The Weyl group is W , the Weyl vector , and we de ne the dot action w group is denoted W[ ]. The Bruhat order on W[ ] is consistent with the Bruhat order on W, and the parity of the length functions agree [4]. We will use a handy parameterization for the weights [5]. A weight is called3 antidominant if for all dominant if for all 2 + + , h + ; _i 2= Z>0; 2 , h + ; _i 2 Z>0. Both of these subsets of weights are highly restrictive, and in particular, their union does not include all weights. Note also that our de nition of dominant makes all dominant weights integral. For any weight 2 h , there is a unique antidominant weight in the dot . Therefore, any weight can be written in a unique way as with antidominant and w 2 W[ ] of minimal length. The minimal length requirement ensures that the decomposition (3.2) is unique. Given a weight 2 h , we focus our attention on two modules, which are both highestweight modules with highest weight . The rst one is the Verma module M . It is de ned as the module generated from a highest weight state by the action of all lowering operators.4 Its character [M ] follows from the de nition, We introduce the simple module L (also called the irreducible module), which is the unique simple quotient of M . Writing down the character of the module L is a central task in this paper. Given an antidominant weight [Lw ] of the irreducible module Lw 2 h , our goal is to understand how the character decomposes into characters of Verma modules [M ]. 3We warn the reader that some authors use di erent de nitions for these concepts. 4More precisely, the relevant object here is the universal enveloping algebra U(g), which can be thought as g with an associative product such that the Lie bracket is given by the commutator. We start with the one-dimensional (h according to the linear form n+)-module C (where the raising operators n + give zero and the Cartan h acts ), and form the tensor product with U(g), M ( ) = U(g) U(h n+) C . = w (3.2) (3.3) Only weights of the form where PwW0;[w] (1) are coe cients, and we have factored out the sign contribution of the length di erence `(w; w0) = `(w) Lusztig polynomials PwW0;[w] (q) associated to the Weyl group W[ ] and two elements w0 and w of the group W[ ], evaluated at q = 1. In the next subsection, we give an algorithm to compute these polynomials. Note that we have presented a crucial property of the theory of representations and characters, namely that the coe cients only depend on the relevant Weyl group [5]. This property was surmised early and proven late in the development of the theory. It implies that extensive manipulations of Lie algebra generators can be summarized in the more e cient combinatorics of the Weyl group only. `(w0). The coe cients PwW0;[w] (1) are the Kazhdan The Kazhdan-Lusztig polynomials We review one algorithm to compute the Kazhdan-Lusztig polynomials for Coxeter groups (which includes all Weyl groups that we encounter) [4]. Firstly, one computes the Bruhat partial order, that we denote by . Secondly, we proceed as follows. Let x; w be two elements of the Coxeter group W . We are ultimately interested in the Kazhdan-Lusztig polynomial Px;w(q), but the algorithm requires to compute as well an auxiliary integer denoted (x; w). If x If x If x = w, set Px;w(q) = 1 and (x; w) = 0. w, set Px;w(q) = 0 and (x; w) = 0. w and x 6= w, then let s be a simple re ection such that `(sw) < `(w). Let c = 0 if x sx, and c = 1 otherwise. Then set (see the core of the existence proof provided in [4], section 7.11) Px;w(q) = q1 cPsx;sw(q) + qcPx;sw(q) (z; sw)q(`(w) `(z))=2Px;z(q) (3.5) X where the sum runs over those z 2 W such that z sw and sz z. Finally, de ne5 (x; w) = Coe cient of q(`(w) `(x) 1)=2 in Px;w(q) : (3.6) Using the algorithm, we can compute all the Kazhdan-Lusztig polynomials for the Weyl groups W appearing in the character formula (3.4). Thus, the proof [13, 14] of the KazhdanLusztig conjecture [12] solves the problem of determining all characters of highest weight representations of semisimple Lie algebras. 5In particular, note that if the degree of the polynomial Px;w(q) is strictly less than (`(w) `(x) 1)=2, then (x; w) = 0. HJEP04(218)5 The reader may nd comfort in recovering the Weyl character formula for nite-dimensional irreducible representations as a particular case of the vast generalization (3.4). The irreducible representation L of the simple Lie algebra g is nite-dimensional if and only if its highest weight is dominant (see subsection 3.1). Let the weight be dominant. Then we can write the weight in the form = w with the weight antidominant and w the longest element of the Weyl group. For all elements x in the Weyl group W , the Kazhdan-Lusztig polynomial Px;w (q) trivializes to Px;w (q) = 1 [5]. Therefore, for nite dimensional representations, the generic character formula (3.4) simpli es to which includes a sum over the whole Weyl group. Intuitively, the further the highest weight is from antidominance, the bigger the character sum. For nite representations, the sum has the maximal number of terms. A remark on some singular integral weights. According to our de nition, a dominant weight can not be singular. In fact, the integral weights located in the dominant shifted Weyl chamber (those that satisfy h + ; _ i 2 Z 0 for all positive roots ) are split into two categories: the dominant weights and the singular weights. An interesting consequence of the general formula (3.4) is that the character of an irreducible module L( ) where belongs to the second category vanishes. This property is useful in simplifying character formulas. 3.4 Examples The generic character formula captures (among others) the character of all highest weight representations of the conformal algebras so(2; n). In the rest of the paper, we will mainly be interested in the unitary highest weight representations, which are a small subclass of all highest weight representations. These are the representations most evidently relevant in physical theories. Nevertheless, non-unitary representations can play a role in unitary theories with gauge symmetries, or in non-unitary theories of relevance to physics. Therefore, we want to make the point that the mathematical formalism that we reviewed also readily computes the characters of this much more general set of representations. To stress that point, we compute an example character which involves a slightly more complicated Kazhdan-Lusztig polynomial. A B3 example. The Weyl group of B3 has 48 elements. They are arranged in ten levels, depending on the number of simple Weyl re ections that occur in their reduced expression. See gure 4. Since there are 48 Weyl chambers, and a proliferation of walls and weights of various singular types, we do not provide a complete catalogue of characters. The results are straightforward to obtain, but unwieldy to present. We only provide a avour of what such a catalogue looks like. contribute to the same module M c (see section 5). The compact subgroup WJ is isomorphic to the Weyl group of B2, and one can check that the subset of elements in each given color is isomorphic to gure 2. To discern the features of the catalogue, it is su cient to analyze the geometry of the chambers, the walls, and the corners. The positive root system e3; e3; e2; e1; e1 e3; e1 + e2; e1 + e3; e2 + e3g of the algebra so(5; 2) can be divided into subsystems in various ways. If the set of roots orthogonal to the weight + is empty, we are in a chamber. If it is non-empty, we are on at least one wall. We have nine walls, given by the equations i = j , i = j and i = 0. We have weights living on a single wall, weights living in the corner of two walls, in the corner of three, in the corner of four or on the intersection of the nine walls. This provides us with a rst glimpse of the structure of + = fe1 e2; e2 the catalogue. Next, we want to clarify the di culty of computing the Kazhdan-Lusztig polynomials. While laborious, the di culty remains well within reach of ancient computers. The most complicated Kazhdan-Lusztig polynomial turns out to be P1;s2s3s2s1s2s3s2 (q) = q2 + q + 1 (and it arises for a single other combination of Weyl group elements as well). At q = 1, this will give rise to a triple multiplicity for a Verma module character in the character sum formula. An example weight for which we need this polynomial is produced by acting with w = s2s3s2s1s2s3s2 on an anti-dominant weight. Thus, we give the following example entry in the catalogue. Example. Consider the weight (s2s3s2s1s2s3s2) ( 2 ) = ( 1; 1; 1). We apply the general procedure outlined in this section using a symbolic manipulation program, and nd the character: + [M( 3;0;2)] + [M( 3;1; 2)] [M( 3;1;1)] + [M( 2; 4; 2)] [M( 2; 4;1)] [M( 2; 3; 3)] + [M( 2; 3;2)] + [M( 2;0; 3)] [M( 2;1; 2)] [M( 1; 4; 1)] + [M( 1; 4;0)] + [M( 1; 2; 3)] [M( 1; 2;2)] [M( 1; 1; 3)] + [M( 1;1; 1)] : (3.8) Note the multiplicities of the Verma modules, which go up to three, even in this low rank example. Proceeding in this fashion, one can imagine lling out systematically the thick catalogue of character formulas. The reader who is so inclined will surely nd the tables to be constructed shortly equally mesmerizing. 4 The unitary conformal multiplets In section 3 we exhibited how to compute the structure and character of any highest weight representation of the conformal algebra so(2; n). In this section, we determine which of the highest weight conformal multiplets are unitary. Those multiplets are the representation theoretic building blocks of unitary conformal eld theories. The mathematical analysis of the unitarizability of the representations of the conformal algebra ts into a more general framework, which we recall brie y. Firstly, let G be a simply connected and connected simple Lie group, and K a closed maximal subgroup. Then, the group G admits a non-trivial unitary highest weight module precisely when (G; K) is a Hermitian symmetric pair [24, 25]. Hermitian symmetric pairs have been classi ed [16]. See appendix B for a summary of the relevant structure theory of real simple Lie groups, and [26] for a complete treatment. The conformal group G = SO(2; n) satis es the condition, with the maximal compact subgroup Spin(n). The techniques used to classify the unitary highest weight representations for such groups include the identi cation of weights of null vectors and the degeneration of the contravariant form on the Verma module [27, 28]. The full classi cation of the unitary highest weight modules of the conformal algebras was obtained in [27]. It is based on an exploitation of necessary and su cient inequalities satis ed by unitary representations. These were derived in full generality in [29]. The analysis of physicists of level one and level two constraints on unitary representations can be viewed as a partial analysis of the necessary conditions. In this section, we demonstrate that it su ces to decipher the earlier and more complete mathematical classi cation results to recuperate in a uniform manner the results in the physics literature. We provide a glimpse of the concepts that underlie the classi cation result, illustrate the general analysis in the example of B2 = so(2; 3), and then recall the full classi cation of the unitary highest weight multiplets for the Bk = so(2; 2k 1) and Dk = so(2; 2k 2) algebras. A physics reference in the same vein is [1]. We again consider highest weight modules based on a highest weight state with respect to a Borel subalgebra b of the complexi ed Lie algebra. The elements h in the Cartan subalgebra h act as (h) where is the highest weight. The span of the compact root system c has co-dimension one in the dual h of the Cartan subalgebra [26]. We de ne to be the maximal non-compact root [26]. The classi cation theorem of [27] introduces a class of weights, which we generically write , which are compact subalgebra k is nitely represented) and which satisfy c+ dominant (because the where is the maximal non-compact positive root of the conformal algebra.6 We also introduce an element of the weight space which satis es that it is orthogonal to all compact roots as well as the normalization h + ; i = 0 ; h ; i 2h ; i = 1 : (4.1) (4.2) Then the highest weights corresponding to unitarizable representations all lie on the lines = + z where z is a real number. See gure 5. There is a half-line of unitary representations ending at a point which is generically at a positive value of z, depending on the algebra g and the weight . Then, there are further points where unitary representations can occur, taking values in an equally spaced set, with a spacing which depends on the algebra only. There is an endpoint to this discrete set. The calculation of the three constants (called A( ), B( ) and C( ) in gure 5) that determine this set proceeds via the introduction of auxiliary root systems. Indeed, we want to bring to the fore how singular the weight is with respect to the compact root system. To that end, we de ne the subset c( ) of compact roots orthogonal 6In [27], the weights are called 0. C( ) highest weights lie on lines of the form = + z , and the gure represents the values of z 2 R that give unitary weights. On a given line, there is a semi-in nite line of highest weights which is allowed, and then an equally spaced set of discrete allowed values, starting at the end of the half-line, and ending after a nite number of steps. to . We then de ne the new root system f g [ c( ) and decompose it into simple root subsystems. The simple root system which contains the maximal non-compact root is baptized Q( ). Exceptionally, we will make use of a second root system R( ), de ned as follows. If the root system has two root lengths and there is a short root not orthogonal to the system Q( ) and such that h ; i=h ; i = 1, then we adjoin the short root to Q( ). The simple component containing of the resulting root system is named R( ). These root systems can be algorithmically determined from the weight , and they allow for the calculations of the three constants, which in turn determine all the unitary highest weight conformal multiplets. The calculations are performed explicitly in [27]. We review the results of the calculations in subsections 4.2 and 4.3. 4.2 The algebras Bk = so(2; 2k 1) In this subsection, we generalize the example of the so(2; 3) algebra to include all conformal algebras so(2; 2k 1) associated to a space-times of odd dimension 2k 1. We list all unitary highest weight representations [27]. We again use the conventions of appendix A. Firstly, we review the set c+ of positive compact roots, as well as the set n+ of positive non-compact roots [26]: HJEP04(218)5 c+ = fei ej j2 ej j2 i < j j ng [ fe1g : ng [ fej j2 j ng The highest non-compact root coincides with the highest root of the algebra. When we include the highest root of the algebra in the Dynkin diagram, we obtain the a ne untwisted Dynkin diagram. The Weyl vector for the Bk algebra is = (k 1=2; k 3=2; : : : ; 1=2) : We parameterize the weights in terms of their components in the basis ei of orthonormal vectors and demand that the components corresponding to the compact subalgebra = e1 + e2 (4.3) (4.4) (4.5) so(2k 1) are dominant integral weights: i 2 j 2 Z 3 2 k 2 Z : i < j k We moreover need the 1 component of the weight to be tuned such that the weight is orthogonal to the maximal non-compact root , which implies We moreover parameterize the line on which the unitaries with highest weight terms of the normalized orthogonal vector HJEP04(218)5 The root systems Q( ) and R( ) that measure the degenerate nature of the anker will either be the full conformal algebra so(2; n) or su(1; p) with p smaller than the rank of the conformal algebra. We distinguish three cases for the root systems Q and R [27]. The rst case is labelled by an extra integer p that satis es 1 Case (I,p) corresponds to root system Q = su(1; p) = R with anker weights p < k. obeying 2 = 2 = 3 = = p+1 for p k 1. Case II corresponds to Q = so(2; 2k 1) = R and = 0. Case III is exceptional in that it has a root system Q = su(1; k 1) that di ers from the root system R = so(2; 2k 1). The weight satis es 2 = = 1=2. The theorem of [27] states that the highest weight irreducible module with highest weight = + z is unitarizable if the module is trivial, or if the highest weight obeys the inequalities z z p k for case (I,p) and 1=2 for cases II and III : Preparing for a physicist's energetic lowest weight perspective, we denote the rst compoby 1 = E. We summarize the unitarity conditions for so(2; 2k 1) nent of the weight in tables 2 and 3. 4.3 The algebras Dk = so(2; 2k 2) In this subsection, we list the unitary highest weight modules of the conformal algebra in even dimensions. We again present highlights of the classi cation theorem proven in detail in [27]. The nal result in all even dimensions can also be summarized very succinctly. The ground work is layed by noting that the set of compact positive roots c+ and the set of non-compact positive roots n+ is given for the Dk algebra by c+ = fei ej j2 ej j2 i < j j kg : kg (4.6) (4.7) lie in (4.8) (4.9) (4.10) (I,1) 2 (I,p) p k 2 (I,k 1) II III 1 0 1 0 1 0 1 0 1 2 2 2 2 2 3 3 3 3 3 p p p p p + 1 p + 1 p + 1 p + 1 p + 1 k k k k k 1 1 1 1 1 k k k k k non-compact root , equal to the a ne root. The non-circled black nodes are the roots that are orthogonal to . Because of the constraints on , the root 1 can never be orthogonal to . The small black dot means that the root satis es h ; i = 12 . The root system Q( ) is the one generated by the big black dots, and the root system R( ) is the one generated by all the black dots. Type (I,p) II III De nition Unitarity constraint 2 = 2 = = = p+1 > p+2 k 2= f0; 12 g (1 p k 2) (p = k 1) 2 = 2 = = = k = 0 k = 1=2 E E 2k k E 1). Here the ( 2; : : : ; k) are either all integers or all half-integers, and 2 k 0. We use interchangeably the notation E = 2 + 2 p 32 or E = 0 k 1 1 HJEP04(218)5 again coincides with the highest root for this algebra and the Weyl vector is We use the following parameterization of the anchor weight and the orthogonal pointer is dominant in its compact components: i 2 j 2 N 3 2 k 2 Z i < j The root systems Q( ) and R( ) are always equal since all roots have equal length. There are again three cases to distinguish, but two of them are related by the outer automorphism 2). The latter acts on the weight components by ipping the sign of the nal component k. This symmetry of our classi cation problem reduces the number of cases to two, namely the root systems su(1; p) with p k 1, and the root system so(2; 2k 2). The nal statement is that the representation is unitarizable if and only if z z p k for case (I,p) 1 for case II ; with the exception of z = 2k 3 in case II, which corresponds to the trivial representation 2). We now determine in which case we are, depending on the weight . The root system Q( ) is simple, and contains at least the maximal non-compact root = e1 + e2. Thus, we consider rst whether the compact roots containing an e2 term belong to the root system Q( ). Given the constraints on the weights i of the nite dimensional representation of so(2k 2), this is the case if and only if 2 = 3. If these entries are not equal, then the root system Q( ) corresponds to the rank one non-compact algebra su(1; 1). When 2 = 3, we attach one further node. We continue in this manner, and nd that when we have consecutive components 2 = 3 = = p = p+1 equal, then the non-compact algebra is su(1; p). When we reach the end of the chain, we have the case 2 = j kj 6= 0 with algebra su(1; k 1). Finally, we have the exceptional case 2 = 0 for which the root system Q( ) corresponds to the full algebra so(2; 2k 2). Thus, for each weight , we have found the root system Q( ). We can then summarize all unitary highest weight representations. We again declare 1 = E, and we run through all possible cases. We list the results for so(2; 2k 2) in tables 4 and 5. (4.11) (4.12) (4.13) (4.14) (4.15) k 2 (I,k 1) (I,k 1)' II 0 1 0 1 0 1 0 1 0 1 2 2 2 2 2 ... ... ... ... ... p p p p p + 1 p + 1 p + 1 p + 1 p + 1 ... ... ... ... ... k k k k k k 2 k k 2 k k 2 k k 2 k 1 1 1 1 The non-circled black nodes are the roots that are orthogonal to . Because of the conventions for , the root 1 can never be orthogonal to . Type (I,p) II 2 = = p+1 > j p+2j 2 = 2 = = = k 6= 0 k 6= 0 (1 p k 2) for (I,k for (I,k 1) 1)' 2 = = k = 0 De nition Unitarity constraint E E k 1 2k k 3 + 2 p 2 or E = 0 j kj, all the di erences i+1 2 Z for i = 2; : : : ; k 1 and 2 k 2 Z. We use interchangeably the notation E = 1 HJEP04(218)5 The Weyl group cosets In this section, we combine the results of sections 3 and 4 to compute the characters of the unitary irreducible representations of the conformal algebras in various dimensions. We already saw how the generic representation theory boils down to Weyl group theory and the calculation of Kazhdan-Lusztig polynomials (evaluated at one). We will further show that in the case of unitary representations, the Weyl group combinatorics can be simpli ed by performing an e cient (parabolic) decomposition. The paper [2] summarizes some of the features that we exhibit in detail. See also [30, 31] for a related approach, based on the exact sequences of [32]. Consider the conformal algebra so(2; n) in n dimensions and its Weyl group W . The this Weyl group WJ . Finally, we construct the set W J Weyl group is generated by the simple re ections S = fs1; : : : ; sn+1g. The Weyl group of the compact subalgebra so(n) is generated by the re ections J = fs2; : : : ; sn+1g. We call W by taking, in each equivalence class of WJ nW , the element of minimal length. Then, each w 2 W can be written in a unique way as w = wJ wJ ; wJ 2 WJ ; wJ 2 W J : PwJ;i;w1j = 1 i j 0 otherwise: wi n + 2 wj otherwise : 7In the last step, following Proposition 3.4 in [15], we have introduced the standard notation for parabolic KL polynomials. In the context of our paper, we can take the last equation as their de nition. Since the notation matches the mathematics literature, comparison is easier. 8 >0 > > >:1 PwJ;i;w1j = < 1 + qj n 1 j 2n 1 ; 1 i 2n j In particular, the longest element w of W decomposes as w ;J wJ , where w ;J is the longest element of WJ . The highest weights of unitary irreducible representations of the conformal algebras are dominant in the compact direction. Let = w be such a weight, with w 2 W and antidominant. Because is dominant in the compact direction, the parabolic decomposition of w reads w = w ;J wJ . Let v = vJ vJ that [15]7 2 W . We want to evaluate Pv;w. We have Pv;w = PvJ vJ ;w ;J wJ = Pw ;J vJ ;w ;J wJ = PvJJ; ;w1J : We see that for an element w of the form w = w ;J wJ , the polynomial Pv;w depends only on the representatives of v and w in W J . For that reason, it is necessary to study the structure of W J . The Bruhat order in W J is given in Using the notations of the gure, we have that [17, 18] gure 6 for the conformal algebras. (5.1) (5.2) (5.3) (5.4) . . . wk+2 wk 1 . . . w1 w2k wk+1 wk . . . We note the drastic simpli cation in the complexity of the Kazhdan-Lusztig polynomials. They can be explicitly computed, and when evaluated at one, they equal no more than two. These preliminaries allow us to simplify the character formula. Let be a unitary weight. It can be written = ( 1; 2; : : : ; k) | {z c } where c is a dominant integral weight of the compact subalgebra. As such, it is the highest weight of a nite dimensional representation of the compact algebra k, whose character is denoted by [Lk ]. This generalizes equation (2.18). Then we can introduce the generalized Verma modules M c, de ned from those nite-dimensional representations of the compact subalgebra by induction to the full algebra. The characters are related by (5.5) (5.6) (5.7) and the Weyl character formula gives 2 n+ z [Lk ] X ( 1)`(w)[Mw ] : w2WJ Our goal is now to express the irreducible characters [L ] in terms of the induced characters (5.8) ( 1)`(w ;J wJ ;wJ0 w0J ) PwJ0 w0J ;w ;J wJ (1) MwJ0 w0J ( 1)`(w ;J )+`(wJ ) `(wJ0 ) `(w0J )Pw ;J w0J ;w ;J wJ (1) MwJ0 w0J ( 1)`(w ;J )+`(wJ ) `(w0J )Pw ;J w0J ;w ;J wJ (1) M wc ;J w0J h : [M c]. We have, for integral and unitary, X X X X wJ0 2WJ w0J 2W J wJ0 2WJ w0J 2W J X w0J 2W J We have reduced the sum over the Weyl group, which contains respectively 2 2k k! elements for Dk and Bk, to a sum over the 2k elements of W J . Thus, unitary highest weight conformal representation theory has been reduced to the analysis of Weyl group k 1 k! and parabolic cosets, and their associated Kazhdan-Lusztig polynomials. All the ingredients in the character formula can be explicitly computed. 6 The unitary conformal characters In this section, we apply the schemes of sections 3 and 4 to systematically calculate all characters of unitary highest weight representations of the conformal algebras so(2; n). To facilitate the calculations, we give in tables 7 and 6 the values of the parabolic coset representatives wi in terms of simple re ections si. We number the simple roots as in the Dynkin diagrams of tables 2 and 4, and denote by si the re ection through the simple root i. Finally, we turn to the longest element8 of WJ , which is constructed from the longest element of the subgroups Bk 1 and Dk 1. We give them as k k matrices acting on the orthonormal basis ei of the dual h of the Cartan subalgebra, introduced in appendix A:9 w ;J = (Diag(+1; 1; : : : ; 1; 1) Diag(+1; 1; : : : ; 1; +1) for Bk and Dk odd for Dk even : (6.1) We rst treat odd space-time dimensions, and then even space-time dimensions. 6.1 In odd space-time dimension Consider the so(2; 2k 1) conformal algebra. We know that the unitary weights fall into k + 1 categories (see table 2). For these unitary weights, we compute the possible Weyl groups W[ ]. 8The longest element of the Weyl group of a simple Lie algebra can be obtained in terms of simple re ections as follows [4, 21, 33]. Color the nodes of the Dynkin diagrams in white and black in such a way that no two dots of the same color are connected. Let wblack (respectively wwhite) be the product of the simple re ections associated to the simple roots painted in black (respectively white). The longest element of the Weyl group is w = (wblackwwhite)h=2 where h is the Coxeter number. Since w induces an automorphism of the Dynkin diagram, we have w = 1 for algebras other than Ak, Dk and E6. In the case of Dk, one nds that w 1 if k is even, and for k odd w exchanges the last two simple roots. 9The dots represent minus ones. wi (1 2 2 3 3 Inequality i 1 1 w2k+1 i = s1 : : : sk 1sksk 1 : : : si k = w ;J wi with antidominant. We recall that in Bk is antidominant if and only if 1 HJEP04(218)5 There are k2 positive roots: k short roots of the form = ei, for which h ; _i = 2 i, and k2 k long roots ei weight, we saw that ( 2; : : : ; k) 2 Z for 2 i < j k and ei for 2 i < j k for which h ; _i = i [ (Z + 12 )k 1. Therefore the (k j . For a unitary 1)2 roots ei ej k satisfy h ; _ i 2 Z. We have to examine the remaining roots e1 and e1 ei. This leads to the three following possibilities, which de ne what we call the integrality class of the weight : i+1 i+1 1 1 Condition 2 2= 21 Z 2 2 Z 2 2 21 + Z Bk 1 A1 (half-integral) W[ ] (Integrality class) Bk 1 (non-integral) Bk (integral) (6.2) We see that in addition to the integral case (where W[ ] = Bk), there are two other integrality classes to consider, which have W[ ] = Bk 1 A1 and W[ ] = Bk 1. We examine them in turn in the following paragraphs. In the non-integral case, W[ ] reduces to the parabolic Weyl group WJ , so we just have In the half-integral case, we have to take into account a possible re ection with respect to the A1 root, which in our notation is e1. The two A1 Weyl chambers are delimited by the wall 1 = 12 k. From this, we deduce that If 1 If 1 > 12 1 2 k, then [L ] = [M c]. k, we have to remove a correction, corresponding to the dot image of under the e1 re ection. Using the notation (5.5), this gives the character formula [L ] = [M(c 1; c)] [M(c1 2k 1; c)] : (6.3) Finally, the integral case is the most complicated one. We apply the Kazhdan-Lusztig formula (3.4), where the antidominant weight is needed. If the weight is singular, several di erent pairs (w; ) can a priori be used in the parameterization (3.2), but for equation (3.4) to be valid, we need to choose the Weyl group element w of minimal length. Moreover we know that w 2 w ;J W J . In table 6, we have gathered for each such element wi (1 1) wk wk+1 k 1) 2 2 = w ;J wj inant. We recall that a weight in Dk is antidominant if and only if 1 2 i Inequality k + k w2k+1 i = s1 : : : sk 1sksk 2sk 3 : : : si 1 i+1 w the inequalities that w satis es, with antidominant. This means that we select the lowest value of i such that satis es the inequality associated to the coset representative wi in table 6. In the table, in order to write the inequalities in a more compact way, we use the shifted notation := + : Then, combining equations (5.8) and (5.3) gives the result. In even space-time dimension Secondly, we perform the same analysis for even space-time dimension. The algebra i < j k. For a root ej , we have h ; _i = i j . A unitary weight has i j 2 Z for 2 i < j k, so only two con gurations are possible: Condition W[ ] (Integrality class) 1 1 2 2= Z 2 2 Z Dk 1 (non integral) Dk (integral) In the non-integral case, we have [L ] = [M c]. In the integral case, one has to examine the inequalities satis ed by . Again, we pick the smallest wi such that the corresponding inequality in table 7 is satis ed,10 and use formula (5.8). The compact notation := is also used. Let us study a few explicit examples. Consider the weight = ( 1; 0; 0) in so(2; 4). It is unitary of type II. Moreover, we have = (1; 1; 0), which satis es the inequalities for w5 and w6 but not the other 10Since the order is only partial, this could be ill de ned if the inequalities of wk and wk+1 could both be satis ed, and not those for wi, i k 1. However, one checks that the inequalities of wk and wk+1 imply 1 = k = 0, and therefore the inequality of wk 1 would be satis ed. Thus, there really is no ambiguity. j kj j 1j j 1j HJEP04(218)5 with j kj j kj. antidom(6.4) (6.5) wi. So we must write = w ;J w5 with = ( 3; 2; 0). Using the polynomials (5.4), we obtain ( 3; 0; 0). The compact part of the weights ( 2; 0; 1) is singular, so the corresponding module is trivial and disappears in the character formula, as per the remark at the end of section 3.3. We conclude [L( 1;0;0)] = [M(c 1;0;0)] [M(c 3;0;0)] : (6.7) HJEP04(218)5 Note that this result arises from cancelling terms that contain multiplicities larger than one. Example. Similarly, the weight = ( 2; 0; 0; 0) in so(2; 6) is associated to the coset representative w6, and we have [L ] = [M wc ;J w6 ] [M wc ;J w5 ] [M wc ;J w4 ] + [M wc ;J w3 ] 2[M wc ;J w2 ] + 2[M wc ;J w1 ] : This reduces to [L( 2;0;0;0)] = [M(c 2;0;0;0)] [M(c 4;0;0;0)] : (6.8) Again, multiplicities larger than one (and therefore non-trivial Kazhdan-Lusztig polynomials) play an intermediate role. Conclusion. We conclude that the calculation of the characters of all highest weight unitary representations of the conformal algebra in any dimension is straightforward using the mathematical technology. Deciphering su ces. 7 dimension E = A large physics literature exploring the representation theory of the conformal algebras so(2; n) is available. The literature concentrates on unitary representations. These were classi ed in three dimensions [6, 9] and in four dimensions [34]. See also the more general treatment in [35]. The paper [1] identi es the unitary representations in arbitrary dimensions, based on the earlier mathematical treatment in [27] which we reviewed in section 4 and which we summarized in tables 3 and 5. Character formulas were computed in many instances. The most general treatment across dimensions is [36].11 In this section, we translate the uniform mathematical results of section 6 into a notation more frequently used by physicists in order to make both the mathematics and the physics literature more accessible. We again identify the energy E, equal to the conformal of the ground state, with minus the rst component of the highest weight, 1. Moreover, the compact subalgebra so(n) describes space rotations, and we switch to spin labels (j1; : : : ; j[n=2]) to describe the highest weights of the rotation algebra,12 = ( 1; 2; : : : ; 1+[n=2]) = ( E; j1; : : : ; j[n=2]) : | M{azth Ph{yzsics } (7.1) See the remark in footnote 15. Our convention is uniform across dimensions. 11However, the physics literature has not always been entirely accurate, even in the better of resources. 12The spin labels are closer but not yet identical to the most common spin labels in the physics literature. 7.1 Firstly though, we summarize the results of sections 4, 5 and 6 in an e ective algorithm that can be used to compute the conformal character | and indeed the module decomposition | for any unitary weight. The irreducible conformal character with highest weight is denoted [L ], and we will obtain an expression in terms of the Verma modules characters [M ] de ned in equation (3.3). The procedure runs as follows. Let = ( 1; : : : ; k) be a weight in Bk or Dk. 1. Determine whether it is unitary or not using tables 3 and 5. If the weight is not HJEP04(218)5 unitary, then the character is given by the general Kazhdan-Lusztig formula (3.4). To obtain the character, one needs to compute generic Kazhdan-Lusztig polynomials. If the weight is unitary, then a simpli cation of the generic formula occurs, as explained in step two. is unitary, determine its integrality class using (6.2) for a conformal algebra of type B, and (6.5) for an algebra of type D. If the integrality class is Bk 1 or Dk 1, then [L ] = [M c]. (See equation (5.6) for the character [M c] of the Verma module induced from an irreducible representation of the compact subalgebra.) If the integrality class is Bk 1 when 1 > 12 k, and [L ] = [M c] otherwise. A1, then [L ] = [M(c 1; c)] [M(c1 2k 1; c)] If the integrality class is Bk or Dk, then look for the lowest wi in gure 6 such that de ne satis es the corresponding inequality in tables 7 and 6,13 and = (w ;J wi) 1 . The irreducible character is then given by 2k j=1 X( 1)`(wi) `(wj)bji[M wc ;J wj ] ; where the length function ` is the height in gure 6, and the multiplicities bji are obtained by evaluating expressions (5.3) and (5.4) at q = 1, i.e. for Bk, bji = ( 1 j i 0 otherwise; bji = < 8 >0 > wj wi > >:1 otherwise : 2 k + 2 i 2k 1 ; 1 j 2k i (7.2) (7.3) (7.4) and for Dk, 13We recall that = (k action is de ned by w given in equation (6.1). 12 ; k 32 ; : : : ; 12 ) for Bk and = (k 1; k 2; : : : ; 0) for Dk. Moreover, the dot . Finally, the longest element of the parabolic Weyl group w ;J is Example. Before we delve into the exhaustive treatment of the low dimensions, we illustrate how the algorithm allows to e ectively compute the character of any highest-weight irreducible representation in any dimension. Consider the weight = ( 8; 2; 2; 2; 2; 1) in so(2; 10). This algebra is of type Dk with k = 6. First, we check that this weight is unitary. We have E = 1 = 8, and we observe in table 5 that the unitary constraint is of type (I,4) and reads E 12 3 + 2 4 = 7, which is satis ed. To determine the integrality class, we look at table (6.5), and since 1 2 = is integer, we are in the integral case, called Dk. Hence we are instructed to look in table 7 for the smallest wi, in the order given by gure 6, such that the corresponding inequality holds. For that, we rst compute = ( 3; 6; 5; 4; 3; 1). The inequalities for w4 and w5 are both satis ed, but because w4 is smaller than w5, we pick w4. Now we compute . First, note that w ;J = Diag(1; 1; 1; 1; 1; 1), and w4 = s1s2s3 is a cyclic permutation of the four rst entries of a weight. So = ( 11; 9; 7; 5; 4; 1). Finally, the coe cients bji are non-vanishing only for j = 1; 2; 3; 4, so we compute the action of w ;J wj for these values of j on : w ;J w1 ( 9; 2; 2; 1; 2; 1), w ;J w4 conclude that = ( 11; 1; 1; 1; 2; 1), w ;J w2 = ( 10; 2; 1; 1; 2; 1), w ;J w1 = = ( 8; 2; 2; 2; 2; 1). Then, reading the lengths on gure 6, we [L( 8;2;2;2;2;1)] = [M(c 8;2;2;2;2;1)] [M(c 9;2;2;1;2;1)]+[M(c 10;2;1;1;2;1)] [M(c 11;1;1;1;2;1)] : (7.5) In appendix C, we execute the procedure, and explicitly write down the results of formula (7.2) for dimensions up to and including seven, for all integral unitary weights. A brief comparison to the physics literature While the formalism we presented is e cient, it may be bene cial to make an explicit comparison to results in the literature. We kick o the brief comparison in three dimensions. Three dimensions: so(2; 3). We write the highest weights = ( E; j) in terms of the energy E and spin j of the representation. The unitarity condition of table 3 becomes, 1 with j 2 2 Z 0: Type (I,1) is E Type II is E Type III is E j + 1 for j 6= 0; 12 ; 12 or E = 0 for j = 0; 1 for j = 12 . If 2E 2= Z, [L ] = [M c]. For a unitary weight, we then look at the integrality classes: [L( E;j)] = [M(c E;j)] otherwise. If E + j 2 21 + Z, then [L( E;j)] = [M(c E;j)] [M(c (3 E);j)] if E < 32 , and w1 w2 w4 j j E E 2 2 E + 1 j E + 1 1 2 1 2 (7.6) We look for the smallest i such that the wi condition is satis ed by , and read the character in gure 6. For reference, the results are listed in table (C.1), for each possible value of i. In this way we recover the results of section 2. HJEP04(218)5 Four dimensions: so(2; 4). In four dimensions, there are three types of unitary weights = ( E; j1; j2). Table 5 gives, for j1 jj2j and j1 j2 2 Z:14 Type (I,1) is j1 > jj2j and E j1 + 2; Type (I,2) is j1 = j2 6= 0 and E j1 + 1; Type II is j1 = j2 = 0 and E 1 or E = 0. There are two integrality classes of unitary weights, namely D3 and D2, depending on whether E j1 2 Z, or not. In the non integral case, we have [L ] = [M c]. In the integral case, we look in table 7 for the smallest i such that the appriopriate inequality is satis ed, with 1 = E + 2, 2 = j1 + 1 and 2 = j2. For each value of i, the character [L ] can then be read in table (C.3). Thus, we recover the results of [36, 37]. Five dimensions: so(2; 5). We distinguish the generic representations with E 3 + 2 and 2 > 3 (case I,1), the representations E 2 + 2 where 2 = 3 (case I,2) and the representations with E 3=2, which are scalar (case II), or E = 0, and E for the spinor (case III). The analysis runs along the lines of the analysis of the conformal algebra so(3; 2) in three dimensions. We provide the explicit results for the integral unitary weights in table (C.4). When the results can easily be compared, they coincide with [36]. The remark on singular weights in subsection 3.3 plays a role in interpreting the results of [36] correctly. Six dimensions: so(2; 6). The analysis is as for the four dimensional conformal algebra. We provide the explicit results for the integral unitary weights in table (C.5). When the results of [38] can be unambiguously compared, they agree with ours. Remark on the generic case. Our treatment is generic, as is [36], but we carefully keep track of possible multiple subtractions of Verma modules.15 As in [1], our analysis has the advantage of being proven necessary and su cient in arbitrary dimension in regards to unitarity. 14For comparison with most of the physics literature, one rede nes ~j1 = (j1 + j2)=2 and ~j2 = (j1 j2)=2. 15Historically, in the mathematical literature, this was not analyzed correctly [5]. In particular, the otherwise important contribution by Verma [39] was mistaken on the possibly larger than one multiplicity of Verma modules to be added or subtracted in the character formula. This has led to wrong claims in the mathematics literature, which unfortunately have propagated to the physics literature (see appendix A of [36]). It will be interesting to attempt to prove the character formulas of [36], using the techniques we explained. Our main aim was to provide physicists with an overview of the representation theory of conformal multiplets. Highest weight representations make up a large category of representations that is well understood. The minimal data to compute character formulas for irreducible representations is coded in the Weyl group and the Kazhdan-Lusztig polynomials. Mathematicians have also provided a complete analysis of the necessary and su cient conditions for unitarity, using a more powerful version of the inequalities derived in the physics literature. Moreover, unitarity restricts the highest weights such that the combinatorial Kazhdan-Lusztig calculations drastically simplify. Secondly, by translating mathematics, we have added to the physics literature. We explained how to systematically compute the characters of irreducible highest weight representations even when they are not unitary. We have stressed that the conditions for unitarity are necessary and su cient, and that they can be formulated at arbitrary rank. In our analysis, we have dealt systematically with both non-integral as well as singular weights. Moreover, we have provided a clear classi cation of all cases of unitary characters in terms of coset representatives of a Weyl subgroup of the Weyl group of the conformal algebra. Using our systematic insight, we provided look-up tables for unitary highest weight representation characters for conformal algebras up to and including rank four. They are guaranteed to be complete. Mostly, we hope these tables have gained in transparency. Thirdly, these techniques can be re ned to apply to superconformal characters. We plan to discuss the necessary extensions elsewhere. Finally, we wrote this paper because we would have liked to read it. Acknowledgments We thank James Humphreys for his writing in general, and his correspondence in particular. A.B. acknowledges support from the EU CIG grant UE-14-GT5LD2013-618459, the Asturias Government grant FC-15-GRUPIN14-108 and Spanish Government grant MINECO16-FPA2015-63667-P. J.T. thanks the High Energy Physics Theory Group of the University of Oviedo for their warm hospitality and acknowledges support from the grant ANR-13BS05-0001. We thank Xavier Bekaert, Nicolas Boulanger and Hugh Osborn for comments on the rst version of this paper. A The Lie algebra conventions We use the parameterization of [26] for the roots and weights of the Bk and Dk simple Lie algebras. We describe these conventions in detail. In both cases, the dual of the Cartan subalgebra h is spanned by an orthonormal basis (e1; : : : ; ek). When we write a weight in components, it is always understood that the coordinates are with respect to this basis. A.1 The Lie algebra Bk We have the set of roots = f e i ej ; eig and a choice of set of simple roots s = f i<k = i ei+1; k = ekg. The fundamental weights can then be written as $i<k = e1 + +ei and $k = (e1 + + ek)=2. The Weyl vector equals 1)e1 + (2k 3)e2 + The Weyl group is WBn = Z2k o Sk and acts by permutations and sign changes of the orthonormal vectors ei. The conformal algebra so(2; n) with n odd corresponds to a Bk algebra of rank k = (n + 1)=2. A.2 The Lie algebra Dk i (e1 + = (k The set of roots is = f e i ej g while simple roots are collected in the set s = f i<k = ei+1; k = ek 1 + ekg. The fundamental weights are $i<k 1 = e1 + + ei; $k 1 = + ek 1 1)e1 + (k ek)=2; $k = (e1 + + ek)=2. The Weyl vector comes out to be HJEP04(218)5 2)e2 + + ek 1. The Weyl group is WDn = Z2 k 1 o Sk and acts by permuting the vectors ei and an even number of sign changes. For the conformal algebra so(2; n) with n even, we have a Dk algebra of rank k = (n + 2)=2. B The structure of real simple Lie algebras We summarize results of the structure theory of semisimple real Lie algebras. We follow the notation of [26] to which we must refer the reader for a complete exposition.16 B.1 The structure theory Every complex semisimple algebra g has a compact real form. We denote the compact real forms by su(n); so(n); sp(n) and e6;7;8; f4 and g2. The Killing form on a compact semisimple Lie algebra is negative semi-de nite and non-degenerate. Every real semisimple Lie algebra g0 has a Cartan involution , unique up to conjugation. It is such that B (X; Y ) = B(X; (Y )) is positive de nite, where B is the Killing form. This involution gives rise to an eigenspace decomposition g0 = k0 p0 h0 = t0 a0 into eigenspaces of eigenvalues +1 and 1 respectively. In matrix realizations of Lie algebras, the Cartan involution can be de ned by (X) = Xy, where the dagger stands for the conjugate transpose. The Killing form on g0 is negative semi-de nite on k0 and positive semi-de nite on p0. Every Cartan subalgebra h0 of g0 is conjugate to a -stable subalgebra, and we will assume that we have picked a Cartan subalgebra h0 that is -stable. We can then decompose the Cartan subalgebra into subalgebras with t0 k0 and a0 p0. The dimension of t0 is called the compact dimension of h0, and the dimension of a0 is called the non-compact dimension. We say that a Cartan subalgebra is maximally (non-)compact if its (non-)compact dimension is maximal. 16Our summary is mainly based on chapters VI on the Structure Theory of Semisimple Groups, and chapter VII on the Advanced Structure Theory. (B.1) (B.2) Given a -stable Cartan subalgebra h0 = t0 a0, the roots of (g; h) are imaginary on t0 and real on a0. As a consequence, we say that a root 2 h0 is real if it vanishes on t0, and that it is imaginary if it vanishes on a0. Otherwise, the root is said to be complex. We say that an imaginary root is compact if the associated root space is included in k, and that it is non-compact if it is included in p. To a real semi-simple Lie algebra g0, we associate a Vogan diagram which is the Dynkin diagram of its complexi cation g, adorned with additional data. For a maximally compact choice of h0, there are no real roots. Since there are no real roots, we can pick a set of positive roots such that ( +) = . The Vogan diagram of the triple (g0; h0; +) is the Dynkin diagram of + with 2-element orbits of made manifest, and with the 1-element orbits painted when corresponding to a non-compact simple root, and unpainted when compact [26]. B.2 The classi cation of real simple Lie algebras Firstly, there are the complex simple Lie algebras, considered as an algebra over the real numbers. Secondly, there are the Lie algebras whose complexi cation is simple over the complex numbers. These algebras always have a Vogan diagram with at most one simple root painted. Amongst these diagrams, one can remove further equivalences. The resulting classi cation of simple real Lie algebras is summarized e.g. in Theorem 6.105 in [26]. It includes the non-compact forms so(p; q) of the special orthogonal algebras. The Vogan 1) is 1 (B.3) (B.4) They summarize all of the Lie algebra data of the real simple algebra. B.3 The classi cation of Hermitian symmetric pairs Unitary discrete highest weight representations only exist for algebras g0 that are part of a Hermitian symmetric pair. This is because the Cartan subalgebra should be entirely within the compact subalgebra k0 (as follows from analyzing unitarity within a Cartan subgroup and the matrix realization of the Cartan involution ), which is equivalent to the Hermitian symmetric pair condition. Hermitian symmetric spaces are coset spaces G=K (with G a real group and K its maximal compact subgroup) which are Riemannian manifolds with a compatible complex structure and on which the group G acts by holomorphic transformations. A manifold X = G=K is Hermitian if and only if the center of K is a one-dimensional central torus. They were classi ed by Cartan [16], and fall into the list recorded in table 8.17 Crucial to us is the entry so(2; n). 17Reference [26] table (7.147). su(p; q) so(2; n) sp(n; R) so (2n) E III E VII su(p) R R R R R The character tables for integral unitary weights We collect the tables of characters of integral unitary highest weight representations, classi ed by their parabolic coset representative wi. See section 6. Some wi are not associated with any unitary weight. In the following tables, they are signalled by an asterisk. Moreover, the brackets around M c are omitted. As always, we use the notation (7.1) for the weights. B2 = so(2; 3) w1 w2 w3 w4 M c ( E;j) M c M c ( E;j) ( j 2;E 2) M(cE 3;j) + M c ( E;j) + M c ( j 2;1 E) M(cE 3;j) + M c ( E;j) + M c ( j 2;1 E) M(cj 1;1 E) To illustrate how these tables can be used, let us recover the character of the trivial representation L(0;0) of so(2; 3). This corresponds to the coset representative w4, and we read in the table [L(0;0)] = [M(c 3;0)] + [M(c0;0)] + [M(c 2;1)] [M(c 1;1)] : Using the explicit expression (2.17), we obtain [L(0;0)] = 1, as expected. D3 = so(2; 4) w1 w2 w3 w4 w5 w6 M c ( E;j1;j2) M c ( j1 3; j2 1;2 E) M c ( E;j1;j2) ( j1 3;E 3;j2) M c ( E;j1;j2) + M c M c ( E;j1;j2) + M c M(cE 4;j1; j2) + M c ( E;j1;j2) M(cE 4;j1; j2) + M c ( E;j1;j2) M(cj2 2;j1;2 E) M c 2M c M c ( j1 3;1 E; j2) ( j1 3;1 E; j2) ( j1 3;j2 1;E 2) ( j2 2;j1;E 2) M c ( j2 2;j1;E 2) M(cj2 2;j1;2 E) M(cj1 1;1 E;j2) + M c ( j2 2;1 E; j1 1) + M(cj2 2;1 E;j1+1) M c M c + M c +M c + M c + M c M c M c M c M c + M c + M c + M c M c M c M c M c M c (E 5;j1;j2) +M c (E 5;j1;j2) ( j2 3;j1;2 E) ( j1 4;1 E;j2) ( j2 3;j1;2 E) (j2 2;j1;2 E) (E 5;j1;j2) ( j1 4;1 E;j2) (j1 1;1 E;j2) ( j2 3;1 E;j1+1) (j2 2;1 E;j1+1) M c + M c ( j1 4;E 4;j2) ( j1 4;j2 1;E 3) ( j2 3;j1;E 3) M c ( E;j1;j2;j3) M c ( E;j1;j2;j3) ( j1 5;E 5;j2;j3) ( E;j1;j2;j3) ( j1 5;j2 1;E 4;j3) ( j2 4;j1;E 4;j3) ( E;j1;j2;j3) ( j1 5;j2 1;j3 1;E 3) ( j2 4;j1;j3 1;E 3) ( E;j1;j2;j3) ( j2 4;j1; j3 1;3 E) w1 w2 w3 w4 w? 5 w6 D4 = so(2; 6) w1 w2 w3 w4 w5 w6 w? 7 w8 M c M c M c M c 2M c M c M c M c (C.4) (C.5) M c + M c M c M c M c M c M c + M c + M c M c +M c + M c M c + M c + M c 2M c + M c M c + M c M c ( j3 3;j1;j2;E 3) (j3 3;j1;j2;3 E) + 2M c ( j1 5;j2 1;2 E; j3) ( j2 4;j1;2 E; j3) ( j3 3;j1;j2;E 3) (j3 3;j1;j2;3 E) M c (E 6;j1;j2; j3) ( E;j1;j2;j3) ( j1 5;1 E;j2; j3) ( j2 4;j1;2 E; j3) (j2 2;j1;2 E;j3) ( j3 3;j1;2 E; j2 1) (E 6;j1;j2; j3) ( E;j1;j2;j3) ( j1 5;1 E;j2; j3) M c (j1 1;1 E;j2;j3) + M c M c ( j2 4;1 E;j1+1; j3) (j2 2;1 E;j1+1;j3) ( j3 3;1 E;j1+1; j2 1) (j3 3;1 E;j1+1;j2+1) w1 w2 w3 w4 w5 w? 6 w? 7 w8 M c M c M c ( j1 6;E 6;j2;j3) M c ( E;j1;j2;j3) + M c ( j1 6;j2 1;E 5;j3) ( j2 5;j1;E 5;j3) M c M c M c ( j1 6;j2 1;j3 1;E 4) + M c ( j2 5;j1;j3 1;E 4) M c ( j3 4;j1;j2;E 4) M(cE 7;j1;j2;j3) + M c ( E;j1;j2;j3) + M c ( j1 6;j2 1;j3 1;3 E) M c ( j2 5;j1;j3 1;3 E) + M c ( j3 4;j1;j2;3 E) M(cE 7;j1;j2;j3) + M c M c ( E;j1;j2;j3) ( j1 6;j2 1;2 E;j3) +M c ( j2 5;j1;2 E;j3) + M c ( j3 4;j1;j2;3 E) M(cj3 3;j1;j2;3 E) M(cE 7;j1;j2;j3) + M c ( E;j1;j2;j3) + M c +M c ( j2 5;j1;2 E;j3) M(cj2 2;j1;2 E;j3) +M(cj3 3;j1;2 E;j2+1) ( j1 6;1 E;j2;j3) M c ( j3 4;j1;2 E;j2+1) M(cE 7;j1;j2;j3) + M c ( E;j1;j2;j3) + M c ( j1 6;1 E;j2;j3) M(cj1 1;1 E;j2;j3) M c ( j2 5;1 E;j1+1;j3) + M(cj2 2;1 E;j1+1;j3) +M c ( j3 4;1 E;j1+1;j2+1) M(cj3 3;1 E;j1+1;j2+1) Open Access. 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Antoine Bourget, Jan Troost. The conformal characters, Journal of High Energy Physics, 2018, 55, DOI: 10.1007/JHEP04(2018)055