PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 94 (108) (2013), 169–180 DOI: 10.2298/PIM1308169B

FIXED POINTS AND D-BRANES

Elaine Beltaos

Abstract. The affine Kac–Moody algebras give rise to rational conformal field theories (RCFTs) called the Wess–Zumino–Witten (WZW) models. An important component of an RCFT is its fusion ring, whose structure constants are given by the associated S-matrix. We apply a fixed point property possessed by the WZW models (“fixed point factorization") to calculate nonnegative integer matrix representations of the fusion ring, allowing for the calculation of D-brane charges in string theory.

1. Introduction

To each quantum field theory is associated an S-matrix. This matrix expresses amplitudes and thus is a fundamental component of the theory. In string theory, a particle is a finite curve of length approximately 10^{−33} cm. Modern string theories contain both open (e.g., photon) and closed (e.g. graviton) strings. Topologically, a closed string is a circle S^{1} and an open string is the interval [0, 1]. As a string evolves through time, it traces out a surface called a worldsheet. For example, an incoming closed string travelling from t = −∞ to t = 0 traces out a semi-infinite cylinder. An important discovery, in 1989, was of higher dimensional objects (‘membranes’) called Dirichlet-branes, or D-branes for short (co-discoverers were Polchinski, Dai, Leigh, and Hořava). Physically, D-branes are the membranes where the endpoints of open strings reside. In 1995, Polchinski proved that a consistent theory requires branes. D-branes are physical entities having tension and charge. During physical processes, these charges are conserved – thus D-brane charges in string theory are analogous to electrical charges in particle physics. Unlike regular electrical charges however, D-brane charges are usually preserved only modulo some integer M.

In this paper, we are specifically interested in the Wess–Zumino–Witten (WZW models). The WZW models are a well-studied class of two dimensional rational conformal field theories (RCFTs). A conformal field theory (CFT) is a quantum field theory whose symmetries include the conformal transformations. Since ℝ^{2} ≅ ℂ, in two dimensions, the space of conformal maps is infinitely dimensional. Let f(z) be a holomorphic map such that f′(z_{0}) ≠ 0 for some z_{0} ∈ ℂ. Then f is conformal in a neighbourhood of z_{0}. A rational conformal field theory obeys a further finiteness condition. The WZW models correspond to strings propagating on compact Lie groups (e.g., SU(n), the group of n × n unitary matrices with determinant 1).

2010 Mathematics Subject Classification: Primary: 17B81.

Our main interest is in calculating D-brane charges for the WZW models. In a given model, the D-brane charges form a discrete abelian group, the charge group. One approach to determining the charge group is through K-theory [43, 44, 48]; however, while this yields the charge group, it does not give the actual charges themselves. To find the charges, we adopt the conformal field theory approach, discussed in Section 4.

For the simply connected groups (e.g., SU(n)), the D-brane charges have been determined using both methods [1, 6, 9, 19, 29, 42]. Also see [20, 37] for results on twisted charges and [17, 18] for results on semi-simple groups. The K-theory calculation for the non-simply connected group SO(3) = SU(2)/ℤ_{2} has been done in [10]. Other than this, for the non-simply connected groups, the only way known to this author to find D-brane charges requires knowing the NIM-reps. The NIMrep is explained in Section 3. It is important to note that not every NIM-rep corresponds to a consistent boundary condition of a CFT – e.g., the tadpoles of SU(2) [13].

The rest of this paper is structured as follows: in Section 2, we give the relevant mathematical background on the WZW models, including modular data and simple-currents; Section 3 discusses the NIM-rep and fixed point factorization in detail, and in Section 4, we give some details about the conformal field theory approach to finding D-brane charges. This is a work in progress, and the results of solving the charge equation (4.1) for the non-simply connected groups will follow in [4]. We conclude with some questions and further work in this direction.

2. The WZW models

The WZW models have the property that most of their quantities have a natural interpretation in terms of the underlying affine Lie algebra 𝔤. We denote the algebra 𝔤 at level k by 𝔤_{k}. More precisely, the modes of the current algebra generate an algebraic structure called a vertex operator algebra (VOA), whose representations are precisely the integrable representations of 𝔤_{k} (for an introduction to VOAs, see e.g., [41]). These representations form a modular tensor category, and the fusion ring of 𝔤_{k} (see (2.2)) is the Grothendieck ring of that tensor category. For an introduction to these matters, see e.g., [12, 23, 46]. In the following sections, we describe the WZW models from a Lie-theoretical point of view. For an introduction to Lie theory, see [26, 27, 39].

2.1. Affine algebras. Let \overline{𝔤} be a simple, finite dimensional Lie algebra. We construct the affine algebra 𝔤 := \overline{𝔤}^{(1)} as follows. Let L(\overline{𝔤}) be the set of all Laurent polynomials in \overline{𝔤}. That is, L(\overline{𝔤}) = \{∑ _{n∈ℤ} a_{n}t^{n} | a_{n} ∈ \overline{𝔤}\}, where all but finitely many a_{n} are zero. This is an infinite dimensional Lie algebra with bracket [at^{n}, bt^{m}] = [a, b]t^{n+m} . To have true, rather than projective, representations, we centrally extend by an element C;^{1} i.e., [C, x] = [x, C] = 0 for all x. This algebra has bracket [at^{n}, bt^{m}] = [ab]t^{n+m} + nδ_{n+m,0} (a|b)C, where (a|b) is the Killing form of \overline{𝔤}. Finally, we extend one more time (non-centrally) by a derivation ℓ_{0} := t\frac{d}{dt},^{2} and denote this algebra by \overline{𝔤}^{(1)}=: 𝔤. We call \overline{𝔤} the horizontal subalgebra of 𝔤. The affine algebras behave similarly to their horizontal subalgebras – they have highest weight representations, and (extended) Dynkin diagrams. The extended Dynkin diagrams are obtained from the Dynkin diagrams of the horizontal subalgebra by the addition of an extra node, the 0 node. The superscript ‘1’ in the notation of the affine algebra means ‘nontwisted’. The above construction can be twisted by an automorphism of the Dynkin diagram of \overline{𝔤}, yielding affine algebras A_{2}^{(2)}, A_{2r−1}^{(2)} , A_{2r}^{(2)}, D_{r+1}^{(2)} , E_{6}^{(2)}, and D_{4}^{(3)}, where the superscript denotes the order of the automorphism.^{3} Though there are subtle differences, these twisted algebras behave mostly analogously to the nontwisted ones.

The algebra 𝔤 constructed above has irreducible highest weight modules, indexed by highest weights (λ_{0}; λ_{1}, …, λ_{r}). In any highest weight representation, the central element C is mapped to a scalar multiple k of the identity – we call k the level of the representation. We are interested in the integrable highest weight modules, i.e., those for which λ_{i} ∈ ℤ_{⩾0} for all i; they are indexed by the set

(2.1)  P_{+}^{k}(𝔤) = { λ = (λ_{0}; λ_{1}, …, λ_{r}) ∈ ℤ_{⩾0}^{r+1} | ∑_{i=0}^{r} a_{i}^{∨} λ_{i} = k}

of integrable highest weights. In (2.1), the λ_{i} are the Dynkin labels, r is the rank of the underlying horizontal subalgebra, and a_{i}^{∨} are the colabels (The zeroth colabel a_{0}^{∨} is always 1). For example, for 𝔤 = A_{r}^{(1)}, the Lie algebra corresponding to SU(r + 1), a_{i}^{∨} = 1 for all 0 ⩽ i ⩽ r. Physically, the set P_{+}^{k} indexes the primaries of the theory, with the vacuum primary (the state of lowest energy) corresponding to the weight kΛ_{0} = (k; 0, …, 0) =: 0.

2.2. Modular data. In a rational CFT, the chiral algebra V has finitely many irreducible modules M.^{4} Their characters are the one-point functions ch_{A}(τ, u) = tr_{A} o(u)e^{2πi(L_{0} −c/24)} , where c is the central charge, L_{0} is the energy operator, τ, the modular parameter, is in the upper half plane, and o(u) is the zero-mode. We usually specialize to the variable τ to obtain ch_{A}(τ) = tr_{A} e^{2πi(L_{0} −c/24)} , though we lose linear independence of the characters by doing so. These characters satisfy the modularity property

ch_{A}(−1/τ) = ∑_{B} S_{AB} ch_{B}(τ), ch_{A}(τ + 1) = ∑_{B} T_{AB} ch_{B}(τ)

where the sum is over all irreducible modules B. The S and T matrix are called the modular data of the theory and generate an SL_{2}(ℤ) representation ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ↦ S and ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ↦ T.^{5} For an introduction to modular data, see [36]. The entries of S and T lie in a cyclotomic field and S and T satisfy the following properties:

1Up to isomorphism, there is a unique nontrivial one dimensional central extension.

2This extension is taken so that the simple roots are linearly independent.

3We use the notation of [39] for the twisted algebras.

4We assume the the theory is non-heterotic, i.e., that the two chiral algebras are isomorphic.

• T is diagonal and of finite order,

• S is unitary and symmetric (i.e., SS^{∗} = I),

• (ST)^{3} = S^{2} =: C, an order two permutation matrix called charge-conjugation. For the WZW models, the modules M are labelled by the set P_{+}^{k} of level k highest weights λ, and the characters coincide with the characters of the underlying affine algebra, specialized to τ.

The coefficients defined by Verlinde’s formula

(2.2)  N_{λμ}^{ν} = ∑_{α} \frac{S_{λα} S_{μα} S_{να}^{∗}}{S_{0α}}

are nonnegative integers. They are structure constants for a commutative, associative ring called the fusion ring. That is, the ring multiplication is given by x_{λ} ∗ x_{μ} = ∑ _{ν} N_{λμ}^{ν} x_{ν}, where the sum is over P_{+}^{k}. For each λ ∈ P_{+}^{k}, the fusion matrix is the matrix indexed by P_{+}^{k} and given by entries (N_{λ})_{μν} := N_{λμ}^{ν} .

The Kac–Peterson formula gives an explicit formula for the S-matrix [40]:

(2.3)  S_{λμ} = κ^{−r/2} s ∑_{w∈\overline{W}} (detw) exp[−2πi\frac{w(\overline{λ + ρ} ) · (\overline{μ + ρ} )}{κ}] ,

where \overline{W} is the \overline{𝔤} Weyl group, \overline{ρ} = (1, …, 1) is the Weyl vector, and κ and s are constants depending on the rank r and level k.

A simple example of modular data is for A_{1}^{(1)}. The group manifolds SU(2) and SO(3)=SU(2)/ℤ_{2} are based on this algebra. The S and T matrices are

S_{λμ} = \sqrt{\frac{2}{k + 2}} sin[π \frac{(λ + 1)(μ + 1)}{k + 2} ] ,
T_{λλ} = exp[−\frac{πi}{4} ] exp[πi\frac{(λ + 1)^{2}}{2(k + 2)} ] ,

where 0 ⩽ λ, μ ⩽ k, and the fusion coefficients are

N_{λμ}^{ν} = \{ \begin{matrix}1, & if ν ≡_{2} λ + μ and |λ − μ| ⩽ ν ⩽ min\{λ + μ, 2k − λ − μ\} \\ 0, & otherwise \end{matrix}

2.3. Simple-currents. The S-matrix has a strictly positive column and satisfies the inequality

S_{λ0} ⩾ S_{00} > 0.

Equality occurs for weights called simple-currents. These weights are given by some permutation J of the vacuum–we also refer to the permutation J as a simplecurrent. In all cases except E_{8}^{(1)}, level 2, simple-currents correspond to extended diagram automorphisms [21]. This is realized by labelling each node of the extended diagram with a Dynkin label of λ– the automorphism of the graph permutes the Dynkin labels yielding Jλ. The set of all simple-currents forms an abelian group J, isomorphic to the centre of the universal cover of the corresponding Lie group. For example, take A_{r}^{(1)}, the affine algebra for the models SU(r + 1)/H, where H is any subgroup of ℤ_{r+1} , the centre of SU(r + 1). Its graph is an (r + 1)-gon. Let λ ∈ P_{+}^{k} and label each node consecutively by the Dynkin labels of λ. Let J be the rotation of 2π/(r + 1). Then J acts on λ by the rule (λ_{0}; λ_{1}, …, λ_{r}) ↦ (λ_{r}; λ_{0}, …, λ_{r−1} ). This simple-current generates the group J, isomorphic (as it must be) to ℤ_{r+1} . For B_{r}^{(1)}, C_{r}^{(1)}, A_{2r−1}^{(2)} , and D_{r+1}^{(2)} , there is one nontrivial diagram automophism, of order two, so J ≅ ℤ_{2}. For D_{r}^{(1)}, J ≅ ℤ_{2} × ℤ_{2} if r is even and ℤ_{4} if r is odd. If r is even, the generators of the simple-current group are J_{v} which exchanges the zeroth and first nodes, and J_{s} : (λ_{0}; …, λ_{r}) ↦ (λ_{r}; …, λ_{0}), and if r is odd, J is generated by the order four simple-current J_{v} : (λ_{0}; …, λ_{r}) ↦ (λ_{r−1} ; λ_{r}, λ_{r−2} , …, λ_{0}).

5The group is SL_{2}(ℤ) rather than PSL_{2}(ℤ) because of the u variable in the original definition of character.

2.4. Modular invariants. The modular invariant partition function of a WZW model is the function

(2.4)  Z(τ) = ∑_{λ,μ} M_{λμ} χ_{λ}(τ)χ_{μ}^{∗}(τ)

where the sum is over all λ, μ ∈ P_{+}^{k}, and where the M_{λμ} ∈ ℤ_{⩾0} are the multiplicities in the decomposition H = ⊕M_{λμ} λ ⊗ μ of state-space H into V ⊗ V′-modules, which are labelled by P_{+}^{k}.^{6} The function (2.4) satisfies Z(A.τ) = Z(τ) for any A ∈ SL_{2}(ℤ), where A.τ = \frac{aτ+b}{cτ+d} for A = ( \begin{matrix}a & b \\ c & d \end{matrix}), and the numbers M_{λμ} are nonnegative integers with M_{00} = 1. The matrix M of coefficients is called a modular invariant. It satisfies the following three properties

• M_{λμ} ∈ ℤ_{⩾0} for all λ, μ ∈ P_{+}^{k} (positivity)

• M_{00} = 1 (uniqueness of vacuum)

• MS = SM and MT = T M (modular invariance). Modular invariants fall into series–the A-series (this includes the identity and

charge-conjugation C := S^{2}), the D-series (simple-current extensions–see (2.5)) and the E-series (exceptionals that do not fit into the previous two categories– these occur at low levels). This notation is used because the modular invariants in the first classification, for A_{1}^{(1)} fell into an A-D-E pattern [11] (see [34] for another approach to this classification, which has been generalized to other cases).^{7} Classifying modular invariants is an important step in classifying RCFTS. A precise mathematical formulation of this problem (simultaneously the classification of NIMreps) as the classification of Frobenius algebras in their representation categories is given in [15]. Also see the review [23].

6Physically, state-space is the set of all possible states of the theory– it carries a representation of V ⊗ V′, where V and V′ are the vertex operator algebras of holomorphic and antiholomorphic fields resp. (a typical quantum field is neither holomorphic nor antiholomorphic, but a sesquilinear combination of both). A vertex operator algebra is the natural algebraic structure formed by the holomorphic (or antiholomorphic) fields.

7Because the exceptionals must be classified separately each time, complete modular invariant classifications are rare and difficult–see e.g., [32, 33] for the A_{2}^{(1)} classification.

Every (non-heterotic) RCFT has a modular invariant and a NIM-rep [14] (we describe the NIM-rep in Section 3 below). Our interest is in the simple-current modular invariants, as they correspond to non-simply connected Lie groups. Let G be a compact simply connected Lie group corresponding to the affine algebra 𝔤_{k} (this denotes the algebra 𝔤 at level k). Let J be an order n simple-current for 𝔤_{k}, and define the matrix [45]

(2.5)  M[J]_{λμ} = ∑_{i=1}^{ord(J)} δ_{J^{i} λ,μ} δ^{ℤ}(Q_{J}(λ) + ir_{J})

where δ^{ℤ}(x) = 1 if x ∈ ℤ and 0 otherwise, and Q_{J}(λ) and r_{J} are rational numbers. The matrix M[J] is a modular invariant if and only if T_{J0,J0} T_{00}^{∗} is an nth root of unity. In this case, it corresponds to the model with group G/⟨J⟩. For example, for A_{1}^{(1)} (with Lie group SU(2)) there is one order two simple-current J(λ_{0}; λ_{1}) = (λ_{1}; λ_{0}). At level 4, the D-series modular invariant (2.5) for J is

D_{4} = ( \begin{matrix}1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{matrix}) .

The group ⟨J⟩ is isomorphic to ℤ_{2}. This modular invariant corresponds to the model with Lie group SU(2)/ℤ_{2}=SO(3).

Because of the similarity of (2.3) to the Weyl character formula (see e.g., [39]), we have the relationship

(2.6)  χ_{λ}(μ) := ch\overline{λ} ( −2πi \frac{(\overline{μ + ρ} )}{κ}) = \frac{S_{λμ}}{S_{0μ}}

between the ratios of the S-matrix and characters at elements of finite order of the underlying horizontal subalgebra \overline{𝔤}. Equation (2.6) is the key to Theorem 3.1 below.

3. The NIM-rep

In this section, we discuss the NIM-rep and a property of the S-matrix – fixed point factorization – that allows for the calculation of NIM-rep coefficients. This in turn allows for the calculation of D-brane charges.

3.1. Description of the NIM-rep. A NIM-rep [2] N is a nonnegative integer representation of the fusion ring. More precisely, to each λ ∈ P_{+}^{k}, associate a nonnegative integer matrix N_{λ} such that N_{λ}N_{μ} = ∑ _{ν} N_{λμ}^{ν} N_{ν}, with N_{0} = I, the identity matrix, and N_{Cλ} = N_{λ}^{t}, where ^{t} denotes transpose. Two NIM-reps N and N′are called equivalent if there is a permutation matrix P such that N′_{λ} = P^{−1} N_{λ}P for all λ ∈ P_{+}^{k}. The easiest example of a NIM-rep is the assignment λ ↦ N_{λ}, the fusion matrices. These are indexed by P_{+}^{k}, but a general NIM-rep matrix N_{λ} is indexed by boundary states–these were given a Lie theoretic interpretation in [28,30] and will be described below.

In a given NIM-rep, all matrices are normal and commute, so are simultaneously diagonalized by a unitary matrix Ψ. The eigenvalues of N_{λ} are S_{λμ} /S_{0μ}, where μ lies in a multi-set, with multiplicities as eigenvalues. We call the multi-set of all such μ the exponents of the NIM-rep and denote it by E(N). We then have the Verlinde-like formula (recall eqn. (2.2))

(3.1)  N_{λx}^{y} = ∑_{μ} \frac{Ψ_{xμ} S_{λμ} Ψ_{yμ}^{†}}{S_{0μ}} ,

where the sum is over all exponents of the NIM-rep, x and y are boundary states, and † denotes complex conjugate transpose.

Consider a simple-current J of order n and its associated modular invariant M[J]. Let λ ∈ P_{+}^{k}. Define the order of λ, written ord(λ), to be the order of the stabilizer of λ in ⟨J⟩. By [λ] we mean the J-orbit [λ] = {J^{a}λ | 0 ⩽ a ⩽ n − 1}. Then the set of boundary states for the model is the set of all pairs [[λ], j] such that 1 ⩽ j ⩽ ord(λ). The exponents of a modular invariant M are elements of the multi-set E(M) consisting of all λ with M_{λλ} ≠ 0, appearing with multiplicity M_{λλ} . We will associate this with the set {(λ, i) | 1 ⩽ i ⩽ M_{λλ} }. If E(M) = E(N), then we say that N and M correspond to each other. Recall the NIM-rep N_{λ} = N_{λ} for all λ ∈ P_{+}^{k}. In this case, the boundary states coincide with the set P_{+}^{k}, each weight occurring with multiplicity 1.

3.2. Fixed points and NIM-reps. Here, we describe fixed point factorization in detail. Fixed point factorization has been used successfully in the case of SU(n) to find D-brane charges for non-simply connected groups [30, 31]. As mentioned in the introduction, fixed point factorization refers to a simplification of the S-matrix at entries where at least one index is a fixed point of a simple-current. More precisely, the S-matrix ratio χ_{Λ_{ℓ}} (φ) (see (2.6)) reduces to some simple polynomial in S-matrix ratios for a smaller rank algebra (also at the fundamental weights). This is remarkable because a priori, there would have been no reason to expect the modular data of the two theories to be linked. The theorem below, adapted from Theorem 3.1 of [3], makes this precise. By a fixed point, we mean a weight φ such that Jφ = φ, and by a truncated fixed point φ̃, we mean the part of φ that contains the non-redundant Dynkin labels. These are given for each algebra in [3], and below for the examples relevant to us. For example, the extended diagram for A_{5}^{(1)} is a regular hexagon. Consider the simple-current rotation by 2π/3 radians–it has fixed points φ = (φ_{0}; φ_{1}, φ_{0}, φ_{1}, φ_{0}, φ_{1}) with φ̃ = (φ_{0}; φ_{1}).

Theorem 3.1. Let X_{r}^{(n)} be one of the algebras in the first column of Table 1 or 2, with fixed level k, and let χ_{λ}(μ) be defined as in (2.6). Let J be a simple-current, φ a fixed point of J, and φ̃ the truncated fixed point associated to φ. Then

(1) For any fundamental weight Λ_{i} of X_{r}^{(n)} , the character χ_{Λ_{i}} (φ) can be expressed as a linear combination, with all coefficients in the set {−1, 0, 1}, in the characters χ′_{Λ′_{j}} (φ̃), 0 ⩽ j ⩽ s, for the algebra X_{s}^{′(m)} given in the third column, where the level is given in the fourth column; (2) For any λ ∈ P_{+}^{k}(X_{r}^{(n)} ), there exists a polynomial P_{λ} such that χ_{λ}(φ) =

P_{λ}(χ′_{Λ′_{1}} (φ̃), …, χ′_{Λ′_{s}} (φ̃)).

Explicit formulas for part (1) are given in [38] (the A-series) and [3] (all other nonexceptional algebras); we show two below. Part (2) follows from Part (1), equation (2.6), and the classical result that the character ring of \overline{𝔤} is generated by the characters at the fundamental weights (see e.g., [5, Chapitre IV–VI]).

The A-series fixed point factorization is as follows: 𝔤=A_{r}^{(1)}. Suppose d | (r +1), and fix a level k, divisible by \frac{r+1}{d} . The simple-current J^{d}, of order (r + 1)/d, has fixed points φ = (φ_{0}; …, φ_{d−1} , …, φ_{0}, …, φ_{d−1} ). Let φ̃ = (φ_{0}; …, φ_{d−1} ). Then

χ_{Λ_{ℓ}} (φ) = χ′_{Λ′_{ℓd/(r+1)}} (φ̃)

if \frac{r+1}{d} | ℓ, where primes denote A_{d−1}^{(1)} level kd/(r + 1) quantities. If \frac{r+1}{d} ∤ ℓ, then χ_{Λ_{ℓ}} (φ) = 0. This is a very clean example of how the character χ_{Λ_{ℓ}} (φ) reduces to that of a smaller rank algebra. The paper [38] also gives a formula for general λ. However, the existence of such a nice formula for general λ turned out to be special to the A-series and did not generalize to the other cases.

To illustrate the most complicated scenario, consider the algebra C_{r}^{(1)}, where r is even, and let J be the order two simple-current. Fixed points of J are φ = (φ_{0}; …, φ_{r/2} , …, φ_{0}). Let φ̃ = (φ_{r/2} ; …, φ_{0}). Then

χ_{Λ_{2m}} (φ) = (−1)^{m} ∑_{ℓ=0}^{m} χ′_{Λ′_{ℓ}} (φ̃),

where primes denote A_{2( \frac{r}{2} )}^{(2)} level k quantities, and χ_{Λ_{ℓ}} (φ) = 0 if ℓ is odd.

The formulas are similar in the other cases.^{8} Tables 1 and 2 below show the smaller rank algebras involved in each case. We call these the ‘fixed point factorization’ (FPF) algebras. It is interesting to note that the fixed point factorization algebras are the orbit Lie algebras of [24] (the orbit Lie algebra ğ of a symmetrizable Kac–Moody algebra 𝔤 is obtained by a matrix-folding, or diagram-folding technique).

Recall the NIM-rep coefficients N_{λ,[[ν],i]}^{[[κ],j]} of (3.1). If at least one of ν, κ is not a fixed point, then these reduce to fusions of 𝔤_{k} (we give an example below), which can be calculated easily–for example using the Kac–Walton formula [39,47]. However, when both ν and κ are fixed points, the NIM-rep coefficients are more difficult to calculate, and the only way known to this author is to use the formulas given by fixed point factorization. In this case, the NIM-rep also reduces to fusions, but of both the original algebra and the fixed point factorization algebra.

8The algebras E_{6}^{(1)} and E_{7}^{(1)} have nontrivial simple-currents; the current work suggests that fixed point factorization formulas exist for these algebras as well, but these have only to be worked out.

Table 1. Fixed point factorization algebras for the nontwisted algebras

X_{r}^{(1)}, level k Simple-current FPF algebra Level
A_{r}^{(1)}  J^{d}  A_{d−1}^{(1)}  \frac{kd}{r+1}
B_{r}^{(1)}  J  A_{2(r−1)}^{(2)}  k
C_{r}^{(1)}, r odd  J  C(1)_{\frac{r−1}{2}}  \frac{k}{2}
C_{r}^{(1)}, r even  J  A_{2( \frac{r}{2} )}^{(2)}  k
D_{r}^{(1)}  J_{v}  C_{r−2}^{(1)}  \frac{k}{2}
D_{r}^{(1)}, r odd  J_{s}  C(1)_{\frac{r−3}{2}}  \frac{k}{4}
D_{r}^{(1)}, r even  J_{s}  B(1)_{\frac{r}{2}}  \frac{k}{2}

Table 2. Fixed point factorization algebras for the twisted algebras

X_{r}^{(1)}, level k Simple-current FPF algebra Level
A_{2r−1}^{(2)}  J  C_{r−1}^{(1)}  \frac{k}{2}
D_{r+1}^{(2)} , r odd  J  A_{2( \frac{r−1}{2} )}^{(2)}  \frac{k}{2}
D_{r+1}^{(2)} , r even  J  D_{\frac{r}{2} +1}^{(2)}  \frac{k}{2}

For example, for C_{r}^{(1)}, r even, we have the NIM-rep coefficients

N_{λ[ν]}^{[κ]} = N_{λν}^{κ} + N_{λν}^{Jκ} , N_{λ[ν]}^{([κ],j)} = N_{λν}^{κ}

N_{Λ_{n} (φ,i)}^{(ψ,j)} = \{ \begin{matrix}\frac{1}{2}(N_{Λ_{2m} φ}^{ψ} + (−1)^{i+j+m} ∑ _{ℓ=0}^{m} Ñ_{ψ} ; _{Λ_{ℓ}} _{φ}) & if n = 2m \\ \frac{1}{2}N_{Λ_{n} φ}^{ψ} & if n = 2m + 1 \end{matrix}

where tildes indicate A_{2( \frac{r}{2} )}^{(2)} level k quantities. Interestingly, our formulas above look similar to formulas found in [7] for the fusion rules of orbifolds involving twist fields (see e.g., eqn. (2.55)).^{9}

4. Charges of WZW D-branes

In this section, we introduce the conformal field theory description of D-brane charges.

9We thank one of the anonymous referees for pointing this out to us.

Consider a WZW model that corresponds to the modular invariant M. The number of maximally symmetric, untwisted D-branes (our interest in this paper) is the trace of M; they are indexed by the boundary states from Section 3.1. Denote by q_{x} the charge of the D-brane corresponding to the boundary state x. These satisfy the charge equation [19]

(4.1)  dim(λ)q_{x} = ∑_{y} N_{λx}^{y} q_{y}

where dim(λ) is the Weyl dimension of λ in \overline{𝔤}; N_{λx}^{y} are the NIM-rep coefficients, and the sum is over all boundary states.

The simply connected group G corresponds to the charge-conjugation modular invariant C. In this case, the D-branes are parametrized by P_{+}^{k}; the NIM-rep coefficients are precisely the fusion coefficients, and modulo an integer M(𝔤, k), (4.1) has the unique solution q_{λ} =dimλ. The integer M(𝔤, k) has been found for all algebras and levels [1, 6, 19], and the charge groups are then ℤ_{M(𝔤,k)} [43, 44, 48].

In the case of the non-simply connected groups, not as much is known. The K-theory calculation for SO(3)=SU(2)/ℤ_{2} was done in [10]. This corresponds to the A_{1}^{(1)} modular invariant for the order two simple-current. The paper [30] adopts the CFT approach, finding the charge groups to be ℤ_{2} × ℤ_{2} if 4|k and ℤ_{4} if 4 ∤ k. On the other hand, the charge group for SU(2) is ℤ_{k+2} , which grows with the level k, showing that the behaviour of the non-simply connected groups may be quite different from the simply connected ones. The charge groups for SO(3) were also studied, using the CFT approach, in [16], where different groups were found, corresponding to another supersymmetric CFT. Using the NIM-rep coefficients for A_{r}^{(1)}, obtained via fixed point factorization, the charges and charge groups were found for the non-simply connected groups SU(n)/ℤ_{d} in [30, 31]. Our goal is to use the NIM-rep coefficients found for the remaining algebras [4] to solve (4.1) for all WZW models listed in Tables 1 and 2.

5. Summary and future work

In this paper, we described the NIM-reps of WZW models and how fixed point factorization can be used to find them. We then discussed how this will lead to solving the charge equation (4.1). A full treatment of D-brane charges requires finding the actual charges themselves, which in turn requires using the conformal field theory description of D-branes. Thus the only way we know to do this in the non-simply connected case is to use Theorem 3.1. Once this is completed, we are interested in comparing our results with K-theory calculations. Further work based on fixed point factorization is to find more physical applications for, in addition to calculating D-brane charges, find a conceptual explanation for this phenomenon, and explain the link between fixed point factorization and the orbit Lie algebras.

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Department of Mathematics and Statistics Grant MacEwan University Edmonton Canada

    beltaose@macewan.ca