hep-th/yymmnnn CERN-TH/98-145

May 1998

THE ACTION OF OUTER AUTOMORPHISMS ON BUNDLES OF CHIRAL BLOCKS

Jürgen Fuchs

Max-Planck-Institut für Mathematik Gottfried-Claren-Str. 26, D – 53225 Bonn

Christoph Schweigert

CERN CH – 1211 Genève 23

Abstract On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik--Zamolodchikov connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case, the bundle of chiral blocks is shown to be a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented, both for genus zero and for higher genera; the proposed relation generalizes the Verlinde formula and is compatible with factorization. Our results have applications to conformal field theories based on non-simply connected groups and to the classification of boundary conditions in such theories.

1 Introduction and summary

1.1 Chiral blocks

Spaces of chiral blocks [1] are of considerable interest both in physics and in mathematics. In this paper we construct a natural class of linear maps on the spaces of chiral blocks of WZW conformal field theories and investigate their properties. Recall that a WZW theory is a twodimensional conformal field theory which is characterized by an untwisted affine Lie algebra 𝔤 and a positive integer, the level.

The chiral block spaces of a rational conformal field theory are finite-dimensional complex vector spaces. In the WZW case, one associates to each complex curve C of genus g with m distinct smooth marked points and to each m-tuple Λ⃗ of integrable weights of 𝔤 at level k^{∨} a finite-dimensional vector space BΛ⃗ of chiral blocks. These spaces are the building blocks for the correlation functions of WZW theories both on surfaces with and without boundaries, and on orientable as well as unorientable surfaces (see e.g. [2]). Moreover, since WZW theories underly the construction of various other classes of two-dimensional conformal field theories, the spaces of WZW chiral blocks enter in the description of the correlation functions of those theories as well. They also form the space of physical states in certain three-dimensional topological field theories [3]. Finally they are of interest to algebraic geometers because via the Borel--Weil--Bott theorem they are closely related to spaces of holomorphic sections in line bundles over moduli spaces of flat connections (for a review, see e.g. [4]).

It is necessary (and rather instructive) not to restrict ones attention to the case of fixed insertion points, but rather to vary both the moduli of the curve C and the position of the marked points. In conformal field theory this accounts e.g. for the dependence of correlation functions on the positions of the fields.) This way one obtains a complex vector space BΛ⃗ over each point in the moduli space M_{g,m} of complex curves of genus g with m marked smooth points. It is known (see e.g. [5, 6]) that these vector spaces fit together into a vector bundle BΛ⃗ over M_{g,m}, and that this vector bundle is endowed with a projectively flat connection, the Knizhnik--Zamolodchikov connection. (Sometimes the term ‘chiral block’ is reserved for sections in the bundle BΛ⃗ that are flat with respect to the Knizhnik--Zamolodchikov connection.) In this paper, we will restrict ourselves mainly to curves of genus g = 0, but allow for arbitrary number m ≥ 2 of marked points. For brevity, we will denote the moduli space of m distinct points on ℙ^{1} simply by M_{m}.

1.2 Automorphisms

It has been known for a long time that certain outer automorphisms of the affine Lie algebra 𝔤 that underlies a WZW theory play a crucial role in the construction of several classes of conformal field theories. The automorphisms in question are in one-to-one correspondence with the elements of the center Z of the relevant compact Lie group, i.e. of the real compact connected and simply connected Lie group G whose Lie algebra is the compact real form of the horizontal subalgebra 𝔤̄ of the affine Lie algebra. These automorphisms underly the existence of non-trivial modular invariants, both of extension and of automorphism type, which describe the ‘non-diagonal’ WZW theories that are based [7] on non-simply connected quotients of the covering group G. (For the modular matrices of such theories see [8].) The same class of automorphisms also plays a crucial role in the construction of ‘gauged’ WZW models [9–11], i.e. coset conformal field theories [12].

Let us remark that the structures implied by such automorphisms have been generalized for arbitrary rational conformal field theories in the theory of so-called simple currents (for a review, see [14]).

It is also worth mentioning that in the representation theoretic approach to WZW chiral blocks [6, 13] one heavily relies on the loop construction of the affine algebra 𝔤. That description treats the simple roots of the horizontal subalgebra 𝔤̄ and the additional simple root of 𝔤 on a rather different footing and accordingly does not reflect the full structure of 𝔤 in a symmetric way. In particular the symmetries of the Dynkin diagram of 𝔤 which are associated to the (classes of) outer automorphisms of 𝔤 that correspond to the center Z do not possess a particularly nice realization in the loop construction.

1.3 Main results

The basic purpose of this paper is to merge the two issues of chiral block spaces and outer automorphisms of affine Lie algebras. Thus our goal is to implement and describe the action of the outer automorphisms corresponding to the center Z of G on the spaces of chiral blocks of WZW theories. This constitutes in particular a necessary input for the definition of chiral blocks for the theories based on non-simply connected quotient groups of G. It also turns out to be an important ingredient in the description of the possible boundary conditions of WZW theories with non-diagonal partition function [15]. Moreover, our results are also relevant for the quest of establishing a Verlinde formula (in the sense of algebraic geometry) for moduli spaces of flat connections with non-simply connected structure group. (This might also pave the way to a rigorous proof of the Verlinde formula for more general conformal field theory models, such as coset conformal field theories.)

Let us now summarize informally the main results of this paper. Consider the direct sum of m copies of the coweight lattice of 𝔤̄ and denote by Γ_{w} the subgroup consisting of those m-tuples μ̄ = (μ̄_{i}) which sum to zero, ∑_{i=1}^{m} μ̄_{i} = 0. To each μ̄ ∈ Γ_{w} we associate a map ω^{⋆} on the set of integrable weights at level k^{∨} and, for each m-tuple Λ⃗ of integrable weights, a map

Θ⃗⋆ ; μ̄ : BΛ⃗ → B_{ω^{⋆}Λ⃗}

which is an isomorphism of the bundles BΛ⃗ of chiral blocks over the moduli space M_{m} and which projects to the identity on M_{m}. We show that this map respects the Knizhnik--Za-molodchikov connections on BΛ⃗ and B_{ω^{⋆}Λ⃗}. Finally, we show that the map Θ⃗⋆ ; μ̄ is the identity map on BΛ⃗ whenever the collection μ̄ of coweight lattice elements contains only elements of the coroot lattice of 𝔤̄. We denote the sublattice of Γ_{w} that consists of such m-tuples μ̄ by Γ. (With these properties the maps Θ⃗⋆ ; μ̄ constitute prototypical examples for so-called implementable automorphisms of the fusion rules. For the role of such automorphisms in the determination of possible boundary conditions for the conformal field theory, see [2,16].)

Let us now discuss a few implications of the bundle-isomorphism Θ⃗⋆ ; μ̄. First, when ω^{⋆}Λ⃗ ≠ Λ⃗, then Θ⃗⋆ ; μ̄ can be used to identify the two bundles BΛ⃗ and B_{ω^{⋆}Λ⃗}. On the other hand, suppose that the stabilizer Γ_{Λ⃗}, i.e. the subgroup of Γ_{w} that consists of all elements which satisfy ω^{⋆}Λ⃗ = Λ⃗, is strictly larger than Γ. In this case the finite abelian group Γ_{Λ⃗}/Γ acts by linear endomorphisms on each fiber. We then decompose each fiber into the direct sum of eigenspaces under this action. It is now crucial to note that the action of this finite group commutes with the Knizhnik--Za-molodchikov connection; this property implies that the eigenspaces fit together into sub-vector bundles of BΛ⃗. In other words, the bundle of chiral blocks is in this case a reducible vector bundle. Moreover, each of these subbundles comes equipped with its own Knizhnik--Zamo-lodchikov connection, which is just the restriction of the Knizhnik--Zamolodchikov connection of BΛ⃗ to the subbundle. This implies in particular that the chiral conformal field theory that underlies the WZW model with a modular invariant of extension type possesses a Knizhnik--Za-molodchikov connection as well.

This pattern – identification, respectively splitting of fixed points into eigenspaces – is strongly reminiscent of what has been found in coset conformal field theories [12]. It is the deeper reason why similar formulæ hold [8,12] for the so-called resolution of fixed points in the definition of coset conformal field theories and in simple current modular invariants.

The rank of the subbundles is the same for all values of the moduli. This rank is related via Fourier transformation over the finite abelian group Γ_{Λ⃗}/Γ to the traces of Θ⃗⋆ ; μ̄ on any fiber. Thus we can conclude that these traces do not depend on the moduli either. We present a conjecture for a formula for these traces and support it by several consistency checks. Most features of our construction can be generalized to curves of arbitrary genus. We conjecture a formula for the traces at arbitrary genus (see equation (7.10)) and check that it is consistent with factorization. So far we do not know how to prove our trace formulæ. Some traces on spaces of chiral blocks on higher genus surfaces with only the vacuum module involved have, however, been computed [17] in some special cases, and in these cases they agree with our formula.

1.4 Organization of the paper

The rest of this paper is organized as follows. In section 2 we introduce (see equation (2.6)) a class of automorphisms of an untwisted affine Lie algebra 𝔤, which we call multi-shift automorphisms. These automorphisms can be extended uniquely to the semidirect sum of 𝔤 with the Virasoro algebra; they depend on a collection of m elements μ̄_{s} of the coweight lattice, and also on a collection of pairwise distinct complex numbers z_{s}. We explain how such automorphisms are implemented on irreducible highest weight modules over the affine Lie algebra and determine their class modulo inner automorphisms.

In section 3 we introduce the algebra 𝔤̄⊗F of Lie algebra valued meromorphic functions over ℙ^{1} with possible finite order poles at m prescribed different points, the so-called block algebra. The Lie algebra 𝔤̄⊗F can be regarded as a subalgebra of 𝔤^{m}, the direct sum of m copies of the affine Lie algebra 𝔤 with all centers identified. We observe that when the complex numbers z_{s} that enter in the definition of the multi-shift automorphisms are interpreted as the coordinates of points on ℙ^{1}, then a tensor product of the automorphisms of section 2 restricts to an automorphism of the block algebra, see formula (3.9). One should regard this latter automorphism as a global object from which the former automorphisms are obtained by local expansions. In section 4 we study the special case of multi-shift automorphisms that correspond to elements in the coroot lattice. We show that they are inner automorphisms with respect to the block algebra.

In section 5 we combine the implementation of these automorphisms on the modules of the affine Lie algebra at all insertion points, so as to obtain an implementation on the modules over the block algebra that arise from tensor products of the affine modules. We show that for fixed values of the insertion points these maps induce isomorphisms on the spaces of chiral blocks and prove that all these isomorphisms of chiral block spaces have finite order. We finally indicate in subsection 5.4 how most of our results can be generalized to curves of genus g > 0.

Up to this point the moduli of the problem, i.e. the positions of the punctures, are kept fixed in all our considerations. In section 6 we proceed to consider the dependence of our constructions on these moduli. Our guiding principle will be that the isomorphisms on the spaces of chiral blocks are compatible with the Knizhnik--Zamolodchikov connection. We show that the implementation on the modules can indeed be chosen in a such way that the Knizhnik--Za-molodchikov connection is preserved and that for each value of the moduli the implementation on the module respects the group law in the group Γ_{w}. We then show that for all values of the moduli the action of a multi-shift automorphism on the chiral blocks is the identity map when all coweights are actually coroots, so that in this case the implementation Θ⃗⋆ ; μ̄ of the automorphism on the bundle B of chiral blocks has finite order.

Finally, in section 7 we exploit these results to formulate a conjecture for a general formula (equation (7.4)) for the traces of the implementing maps Θ⃗⋆ ; μ̄. We extend this conjecture also to curves of higher genus (see (7.10)) and check that it is consistent with factorization. The expression (7.10) for the traces constitutes a generalization of the Verlinde formula. Surprisingly enough we also observe that in all examples that have been checked numerically, the traces turn out to be integers.

Details of various calculations are deferred to appendices.

2 Action of Γ_{w} on affine Lie algebras and their modules

We start our investigations by introducing a certain family of automorphisms of affine Lie algebras which we call multi-shift automorphisms. To determine the class of a multi-shift automorphism modulo inner automorphisms, we study its implementation on irreducible highest weight modules.

2.1 Multi-shift automorphisms of affine Lie algebras

In this subsection we introduce a distinguished class of automorphisms of untwisted affine Lie algebras 𝔤. Such a Lie algebra 𝔤 can be regarded as a centrally extended loop algebra, i.e.^{1}

𝔤 := 𝔤̄ ⊗ ℂ((t)) ⊕ ℂK ,  (2.1)

where 𝔤̄ is a finite-dimensional simple Lie algebra and t an indeterminate. We identify 𝔤̄ with the zero mode subalgebra of 𝔤, i.e. 𝔤̄ ∋ x̄ ≡ x̄^{⊗}t^{0} ∈ 𝔤, and refer to it as the horizontal subalgebra of 𝔤; the rank of 𝔤̄ will be denoted by r. We may already remark at this point that our main interest in the multi-shift automorphisms of the affine Lie algebra 𝔤 stems from the fact that collections of m such automorphisms combine in a natural way to automorphisms of m copies of 𝔤 with identified centers. The latter automorphisms, in turn, restrict to automorphisms of an important class of infinite-dimensional subalgebras, the so-called block algebras 𝔤̄⊗F; those automorphisms of 𝔤̄⊗F will be introduced in section 3. In the present section, however, we study the automorphisms of 𝔤 in their own right.

1 Often it is the subalgebra 𝔤^{∘} = 𝔤̄ ⊗ ℂ[t, t^{−1}] ⊕ ℂK of 𝔤 that is generated when one allows only for Laurent polynomials rather than arbitrary Laurent series that one refers to as the affine Lie algebra. However, in conformal field theory one deals with 𝔤-representations for which every vector of the associated module (representation space) is annihilated by all but finitely many generators of the subalgebra 𝔤^{∘}_{+} of 𝔤^{∘} that corresponds to the positive roots. Any such representation of 𝔤^{∘} can be naturally promoted to a representation of the larger algebra 𝔤.

We will describe the automorphisms of the algebra 𝔤 by their action on the canonical central element K and on the elements H^{i}^{⊗}f and Eᾱ^{⊗}f of 𝔤, where f ∈ ℂ((t)) is arbitrary and {H^{i} | i = 1, 2, ... , r} ∪ {Eᾱ | ᾱ a 𝔤̄-root} is a Cartan--Weyl basis of 𝔤̄. (By choosing f = t^{n} with n ∈ ℤ one would obtain a (topological) basis of 𝔤, but for the present purposes it is more convenient to write the formulæ for general f ∈ ℂ((t)).) In terms of these elements, the relations of the Lie algebra 𝔤 read [K, ·] = 0 and

[H^{i}^{⊗}f, H^{j}^{⊗}g] = G^{ij} Res(df g) K ,

[H^{i}^{⊗}f, Eᾱ^{⊗}g] = ᾱ^{i} Eᾱ ^{⊗} fg ,

[Eᾱ^{⊗}f, Eβ̄^{⊗}g] = e_{ᾱ,β̄} E^{ᾱ+β̄} ^{⊗} fg for ᾱ + β̄ a 𝔤̄-root ,

[Eᾱ^{⊗}f, E^{−ᾱ}^{⊗}g] = ∑_{i=1}^{r}ᾱ_{i} H^{i} ^{⊗} fg + \frac{2}{(ᾱ,ᾱ)} Res(df g) K .

(2.2)

Here d ≡ ∂/∂t, and the residue Res ≡ Res_{0} of a formal Laurent series in t is defined by

Res( ∑_{n=n_{0}}^{∞} c_{n} t^{n}) = c_{−1} ;  (2.3)

G^{ij} is the symmetrized Cartan matrix of the horizontal subalgebra 𝔤̄, and e_{ᾱ,β̄} is the two-cocycle that furnishes structure constants of 𝔤̄ via [Eᾱ, Eβ̄] = e_{ᾱ,β̄}E^{ᾱ+β̄} when ᾱ+β̄ is a 𝔤̄-root.

The automorphisms to be constructed are characterized by a sequence of elements μ̄_{s}, for s = 1, 2, ... , m with m an integer m ≥ 2, of the coweight lattice Lw^{∨} of 𝔤̄ (i.e. the lattice dual to the root lattice of 𝔤̄) that add up to zero. The set

Γ_{w} := {μ̄ ≡ (μ̄_{1}, μ̄_{2}, ... , μ̄_{m}) | μ̄_{s} ∈ Lw^{∨} for all s = 1, 2, ... , m; ∑_{s=1}^{m}μ̄_{s} = 0}  (2.4)

of such sequences forms a free abelian group with respect to elementwise addition. As an abstract group, we have the isomorphism

Γ_{w} ≅ (Lw^{∨})^{r−1} .  (2.5)

In addition, the automorphisms of our interest also depend on a sequence of pairwise distinct complex numbers z_{s} (s = 1, 2, ... , m), one of which is singled out. However, in contrast to the elements of Γ_{w}, in this section we regard these numbers as fixed once and for all, and accordingly we do not refer to them in our notation for the automorphisms. Thus we will deal with precisely one automorphism of 𝔤 for each μ̄ ∈ Γ_{w}; we denote this map by σ_{μ̄}. Only in case we wish to stress the dependence of the automorphism on the label s_{∘} ∈ {1, 2, ... , m} that is singled out, we employ instead the notation σ_{μ̄;s_{∘}}.

For any μ̄ ∈ Γ_{w}, the automorphism σ_{μ̄} ≡ σ_{μ̄;s_{∘}} is defined by

\begin{matrix}σ_{μ̄}(K) := K , \\ σ_{μ̄}(H^{i}^{⊗}f) := H^{i}^{⊗}f + K ∑_{s=1}^{m} μ̄_{s}^{i} Res(φ_{s_{∘},s} f) , \\ σ_{μ̄}(Eβ̄^{⊗}f) := Eβ̄^{⊗}f · ∏_{s=1}^{m}(φ_{s_{∘},s})^{−(μ̄_{s},β̄)} \end{matrix}  (2.6)

with

φ_{s_{∘},s}(t) := (t + (z_{s_{∘}} − z_{s}))^{−1} .  (2.7)

Here for q ∈ ℚ we employ the notation f^{q} for the function with values

f^{q}(z) = (f(z))^{q}  (2.8)

for all z ∈ ℂ. In the sequel we will also use the short-hand

Φᾱ := ∏_{s=1}^{m}(φ_{s_{∘},s})^{−(μ̄_{s},ᾱ)}  (2.9)

for ᾱ a 𝔤̄-root.

It is readily checked that the mapping (2.6) indeed constitutes an automorphism of the affine Lie algebra 𝔤. Indeed, except for the bracket [Eᾱ^{⊗}f, E^{−ᾱ}^{⊗}g], all the relations (2.2) are trivially left invariant. As for the latter, invariance follows by the identity

Res(d[f · Φᾱ] · g · Φ_{−ᾱ}) K = Res(df g + fg · ∑_{s=1}^{m}(μ̄_{s}, ᾱ) φ_{s_{∘},s}) K

= Res(df g) K + ∑_{i=1}^{r}ᾱ_{i} ∑_{s=1}^{m} μ̄_{s}^{i} Res(φ_{s_{∘},s} fg) K ,

(2.10)

which follows with the help of

dΦᾱ = Φᾱ · ∑_{s=1}^{m}(μ̄_{s}, ᾱ) φ_{s_{∘},s} .  (2.11)

Moreover, it follows immediately from the definition that

σμ̄_{′} ∘ σ_{μ̄} = σ_{μ̄+μ̄′} = σ_{μ̄} ∘ σμ̄_{′} .  (2.12)

Thus the automorphisms (2.6) respect the group law in Γ_{w}, i.e. they furnish a representation of the abelian group Γ_{w} in Aut(𝔤). This representation provides in fact a group isomorphism; accordingly we will henceforth identify the group of automorphisms σ_{μ̄} (2.6) with the group Γ_{w} (2.4).

As already mentioned, we refer to the automorphisms (2.6) as multi-shift automorphisms of the affine Lie algebra 𝔤. The origin of this terminology is that they are close relatives of the ordinary shift automorphisms σμ̄^{(0)} of 𝔤 that appear in the literature, e.g. in [18–20]. In the conventional notation

H_{n}^{i}:= H^{i}^{⊗}t^{n} ,  Eᾱ ; n:= Eᾱ^{⊗}t^{n} ,  (2.13)

the action of such a ‘single-shift’ automorphism σμ̄^{(0)} is given by

\begin{matrix}σμ̄^{(0)}(K) = K , \\ σμ̄^{(0)}(H_{n}^{i}) = H_{n}^{i} + μ̄^{i} δ_{n,0} K , \\ σμ̄^{(0)}(Eᾱ ; n) = Eᾱ ; n+(μ̄,ᾱ) , \end{matrix}  (2.14)

where μ̄ ∈ Lw^{∨}. These ordinary shift automorphisms are recovered from the formula (2.6) as the special case where μ̄_{s} = 0 for s ≠ s_{∘}. In the case of (2.14), the coweight lattice element μ̄ is known as a shift vector; we will use this term also for the collection μ̄ of coweight lattice elements that characterizes the multi-shift automorphisms (2.6).

Note that the ordinary shift automorphisms do not satisfy the constraint ∑_{s=1}^{m} μ̄_{s} = 0, i.e. do not belong to the class of automorphisms of 𝔤 that we consider here. In this context we should point out that when we dispense of the restriction ∑_{s=1}^{m} μ̄_{s} = 0, the formula (2.6) still provides us with an automorphism of 𝔤. The reason why we nevertheless impose that constraint will become clear in the next section, where we glue together several of the ‘local’ objects (2.6) to a ‘global’ object that only exists when the constraint is satisfied.

2.2 Implementation on modules

A natural question is to which class of outer modulo inner automorphisms a multi-shift automorphism σ_{μ̄} belongs. Before we can address this issue, we first have to study the implementation of σ_{μ̄} on irreducible highest weight modules.

We start by observing that for any automorphism σ of an affine Lie algebra 𝔤 and any irreducible highest weight module (R, H) over 𝔤,

(R̃, H) := (R ∘ σ, H)  (2.15)

furnishes again an irreducible highest weight module. Since this fact is interesting in itself, we pause to describe how it can be established. First, irreducibility of (R̃, H) is immediate: if a subspace M of H is a non-trivial submodule under R̃, then it is a non-trivial submodule under R as well; but by the irreducibility of H such submodules do not exist. To check the highest weight property, it is sufficient to show that the module (R̃, H) possesses at least one singular vector; this vector is necessarily unique (up to a scalar), since otherwise the module would not be irreducible. Now consider any Borel subalgebra 𝔟 of 𝔤; then σ(𝔟) is a Borel subalgebra again. Since any two Borel subalgebras are conjugated under an inner automorphism of 𝔤 [21], it follows that there exists an inner automorphism σ̃ of 𝔤 such that σ(𝔟) = σ̃(𝔟). This implies that the two automorphisms σ and σ̃ just differ by an automorphism that does not affect the triangular decomposition, i.e. by a so-called diagram automorphism ω (see e.g. [22]). Let us write

σ̃ = ∏_{i}^{→} exp(ad_{x_{i}}) ,  (2.16)

where a fixed ordering of the product is understood. We consider the vector

ṽ := ∏_{i}^{←} exp(R(x_{i})) v ,  (2.17)

where the opposite ordering of the product is understood and where v is the highest weight vector of (R, V). (One can think of ṽ as a Bogoliubov transform of v.) One can check that

R(σ(x)) ṽ = R(σ̃ω(x)) ṽ = ∏_{i}^{→} exp(R(x_{i}))R(ω(x)) v  (2.18)

(compare also the relation (5.16) below). This tells us that the highest weight properties of v imply analogous highest weight properties for ṽ. This concludes the argument.

In addition, by similar arguments one can show that when σ is an inner automorphism, then the 𝔤-modules (R, H) and (R̃, H) are isomorphic, and hence share the same highest weight. On the other hand, they do not necessarily share the same highest weight vector, of course. Thus in general the weight of the highest weight vector v of (R, H) is different from its weight as a vector in (R̃, H); but the previous statement tells us that the difference must be an element of the root lattice of 𝔤.

We also note that any diagram automorphism ω restricts to an automorphism of the Cartan subalgebra 𝔤_{∘} of 𝔤 and hence by duality induces an automorphism ω^{⋆} of the weight space of 𝔤, which constitutes a permutation Λ ↦ ω^{⋆}Λ of the finitely many integrable weights at fixed level. It follows that for each integrable 𝔤-weight Λ any multi-shift automorphism σ_{μ̄} of 𝔤 induces a map

Θμ̄ ≡ Θ_{μ̄;s_{∘}} : H_{Λ} → H_{ω^{⋆}Λ}  (2.19)

between the irreducible highest weight modules H_{Λ} and H_{ω^{⋆}Λ} over 𝔤 which obeys the twisted intertwining property

Θμ̄ x = σ_{μ̄}(x) Θμ̄  (2.20)

for all x ∈ 𝔤; the map Θμ̄ is characterized by this property together with the prescription that the (normalized) highest weight vector of H_{Λ} (which is unique up to a phase) gets mapped to the highest weight vector of H_{ω^{⋆}Λ}. In the special case where σ_{μ̄} is an inner automorphism of 𝔤

and hence of the form σ_{μ̄} = \overrightarrow{∏i} exp(ad_{x_{i}}), the map Θμ̄ can be implemented by

Θ̃μ̄ = ∏_{i}^{→} exp(x_{i}) ,  (2.21)

where the right hand side is to be understood as an element of the universal enveloping algebra of 𝔤. Notice, though, that the implementation Θμ̄ is only determined up to a phase. Indeed, owing to the fact that K is central so that exp(ad_{ξK}) = id, we can always modify Θμ̄ by central terms,

Θμ̄ = exp(f K) ∏_{i}^{→} exp(x_{i}) ,  (2.22)

where f is an arbitrary Laurent series. We will make use of this freedom later on. Moreover, notice that even if all automorphisms in σ_{μ̄} = ∏_{i} exp(ad_{x_{i}}) commute so that this product is well-defined without specifying the ordering, this does not imply that the same holds for the product that appears in the implementation (2.21). We will therefore choose some specific ordering of the insertion points and keep this ordering fixed in the sequel.

2.3 Outer automorphism classes

We are now in a position to determine in which outer automorphism class [σ_{μ̄}] a multi-shift automorphism σ_{μ̄} lies. To this end we take for the Laurent series f the constant function id and exploit the transformation property of the elements H^{i}^{⊗}id. Because of the identity

Res(φ_{s_{∘},s} t^{n}) = \{ \begin{matrix}1 & for s = s_{∘}, n = 0 , \\ −(z_{s}−z_{s_{∘}})^{n} & for s ≠ s_{∘}, n < 0 , \\ 0 & else \end{matrix}  (2.23)

for n ∈ ℤ, we have in particular

Res(φ_{s_{∘},s}) = δ_{s,s_{∘}} ,  (2.24)

so that

σ_{μ̄;s}(H^{i}^{⊗}id) = H^{i}^{⊗}id + μ̄_{s}^{i} K .  (2.25)

Expressed in terms of the notation (2.13), the transformation (2.25) reads

H_{0}^{i} ↦ H_{0}^{i} + μ̄_{s}^{i} K .  (2.26)

When combined with the results of subsection 2.2 it follows that modulo the root lattice L of 𝔤̄ the highest weight Λ of any irreducible highest weight module (R, H) over 𝔤 and the highest weight Λ̃ of (R̃, H) ≡ (R ∘ σ_{μ̄;s}, H) are related by

˜Λ̄ − Λ̄ = k μ̄_{s} mod L ,  (2.27)

where k is the eigenvalue of the central element K. Note that this is a statement about the horizontal parts of the highest weights; the transformation of the rest of the weight is then fixed by the fact that because of σ_{μ̄;s}(K) = K the level does not change, while the change in the grade is prescribed by the formula for σ_{μ̄;s}(L_{0}) that is given in (2.32) below.^{2}

Now the quantity λ̄ mod L is nothing but the conjugacy class of the 𝔤̄-weight λ̄. Taking also into account the relation k = k^{∨} · (θ̄, θ̄)/2 (with θ̄ the highest root of 𝔤̄) between the eigenvalue k of K and the level k^{∨}, we learn that the automorphism σ_{μ̄;s} leads to a change δ_{s} in the conjugacy class of level-1 modules that is given by (θ̄, θ̄)/2 times the shift vector μ̄_{s}, taken modulo the root lattice L, or what is the same,

δ_{s} = \frac{(θ̄,θ̄)}{2} (μ̄_{s} mod L^{∨}) .  (2.28)

Furthermore, there is a natural one-to-one correspondence between conjugacy classes of 𝔤̄- weights and classes of certain outer automorphisms of 𝔤. Namely, as abelian groups we have the isomorphisms

L_{w}/L ≅ Lw^{∨}/L^{∨} ≅ Z(𝔤) ,  (2.29)

2 Of course the property that, modulo L, for fixed level there is a universal shift in the horizontal part of the highest weight is not shared by arbitrary automorphisms of 𝔤.

where Z(𝔤) is the unique maximal abelian normal subgroup of the group Out(𝔤) = Aut(𝔤)/Int(𝔤) of outer automorphism classes of 𝔤. It follows in particular that [σ_{μ̄;s}] is the same outer automorphism class as the class σμ̄_{s}^{(0)} of the ordinary shift automorphism σμ̄^{(0)} (compare formula (2.14)) that is characterized by the same shift vector μ̄ = μ̄_{s}. It is important to note that in general this way one cannot obtain all outer automorphism classes of 𝔤, but rather only those which belong to the subgroup Z(𝔤).

Let us also mention that the abelian subgroup Z(𝔤) of the group of diagram automorphisms is naturally isomorphic to the center of the compact, connected and simply connected real Lie group whose Lie algebra is the compact real form of 𝔤̄. Furthermore, the elements of Z(𝔤) are in one-to-one correspondence with the simple currents [23, 24] of the WZW theory, which are those primary fields which correspond to the units of the fusion ring of the theory (see e.g. [14]).

2.4 Extension the Virasoro algebra

Given the untwisted affine Lie algebra 𝔤, the affine Sugawara construction provides us with a Virasoro algebra Vir. Recall that Vir is the Lie algebra with generators L_{n} for n ∈ ℤ and C and relations

[L_{n}, L_{m}] = (n − m) L_{n+m} + \frac{1}{24} (n^{3} − n) δ_{n+m,0} C ,  [C, L_{n}] = 0 .  (2.30)

The algebra Vir combines with 𝔤 as a semidirect sum Vir ⊕ 𝔤, with relations

[L_{n}, x̄^{⊗}f] = −x̄^{⊗} t^{n+1}df  (2.31)

for n ∈ ℤ.

As we show in appendix A, the multi-shift automorphisms σ_{μ̄;s_{∘}} of 𝔤 can be extended in a unique manner to automorphisms of Vir ⊕ 𝔤. The action of such an automorphism on the generators of Vir reads σ_{μ̄;s_{∘}}(C) = C and

σ_{μ̄;s_{∘}}(L_{n}) = L_{n} + ∑_{ℓ∈ℤ} ∑_{s}(μ̄_{s},H_{n−ℓ}) Res(t^{ℓ}φ_{s_{∘},s}) + \frac{1}{2} K ∑_{s,s′}(μ̄_{s},μ̄_{s′}) Res(t^{n+1}φ_{s_{∘},s}φ_{s_{∘},s′}) (2.32)

(since the extension is unique, we employ the same symbol for the automorphism of Vir ⊕ 𝔤 as for the automorphism of 𝔤). Note that by putting μ̄_{s} = 0 for s ≠ s_{∘}, this reduces to

σμ̄^{(0)}(L_{n}) = L_{n} + (μ̄, H_{n}) + \frac{1}{2} K (μ̄, μ̄) δ_{n,0} ;  (2.33)

this is precisely the unique extension to the Virasoro algebra of the ordinary shift automorphism (2.14) of 𝔤.

For later reference we also mention that via the formula (2.20) and the affine Sugawara construction, the maps Θμ̄: H_{Λ} → H_{ω^{⋆}Λ} defined in (2.19) obey the twisted intertwining relations

Θ_{μ̄;s_{∘}}L_{n} = σ_{μ̄;s_{∘}}(L_{n}) Θ_{μ̄;s_{∘}}  (2.34)

with regard to the Virasoro algebra.

3 Action of Γ_{w} on the block algebra

The automorphisms constructed in section 2 also provide us with corresponding automorphisms of direct sums of affine Lie algebras with identified centers. In this section we show that such an automorphism preserves a distinguished subalgebra of the latter algebra, namely the so-called block algebra which plays a crucial role in the description of chiral blocks.

3.1 The block algebra

In the previous section we have established that given a shift vector μ̄ ∈ Γ_{w} together with a definite choice of distinguished label s_{∘} ∈ {1, 2, ... , m}, the prescription (2.6) provides us with an automorphism of the affine Lie algebra 𝔤. But the formula (2.6) of course makes sense for every choice s_{∘} = 1, 2, ... , m of this distinguished label, so that we are in fact even given a collection {σ_{μ̄;s_{∘}}} of such automorphisms, one for each value of s_{∘}, and all with the same shift vector μ̄ ∈ Γ_{w}. By a slight change of perspective we can rephrase this by saying that we are given an automorphism of the direct sum ⨁_{s=1}^{m}𝔤_{s} of m copies 𝔤_{s} ≅ 𝔤, namely the one that on the sth summand 𝔤_{s} acts as σ_{μ̄;s}. Moreover, the so obtained automorphism restricts to an automorphism of the quotient

𝔤^{m} := ( ⨁_{s=1}^{m} 𝔤_{s})/_{⟨K_{s}=K for all s=1,2,...,m⟩}  (3.1)

of ⨁_{s=1}^{m}𝔤_{s} that is obtained by identifying the centers of the algebras 𝔤_{s}, because the ideal ⟨K_{s} = K for all s = 1, 2, ... , m⟩ is σ_{μ̄}-invariant.

While the specific form of the automorphisms is rather irrelevant for this simple observation, it will become crucial in the considerations that follow now, where we focus our attention to a different infinite-dimensional Lie algebra that can be embedded into (3.1). A convenient starting point for the description of this algebra is to recall that the automorphisms (2.6) also depend on a chosen sequence of m pairwise distinct numbers z_{s} ∈ ℂ. We now regard these numbers as the coordinates of m pairwise distinct points p_{s} on the complex projective line ℙ^{1}, to which we refer as the insertion points or punctures. More precisely, we regard the complex curve ℙ^{1} as ℂ ∪ {∞} and denote by z the standard global coordinate on ℂ; then we write

z_{s} = z(p_{s}) .  (3.2)

For the present purposes the points p_{s} are kept fixed; later on we will allow them to vary over the whole moduli space M_{m} of m pairwise distinct points on ℙ^{1}.

Given this collection of points, we can consider the space F(ℙ_{m}^{1}) of algebraic functions on the punctured Riemann sphere ℙ_{m}^{1} ≡ ℙ^{1}\{p_{1}, p_{2}, ... , p_{m}}, i.e. of meromorphic functions on ℙ^{1} whose poles are of finite order and lie at most at the punctures {p_{1}, p_{2}, ... , p_{m}}. The space F(ℙ_{m}^{1}) is an associative algebra, the product being given by pointwise multiplication. As a consequence, the tensor product

𝔤̄⊗F ≡ 𝔤̄ ⊗ F(ℙ_{m}^{1})  (3.3)

of the horizontal subalgebra 𝔤̄ of 𝔤 with the space F(ℙ_{m}^{1}) inherits a natural Lie algebra structure from the one of 𝔤̄, with Lie bracket

[x̄^{⊗}f, ȳ^{⊗}g] = [x̄, ȳ]^{⊗} fg  (3.4)

for x̄, ȳ∈ 𝔤̄ and f, g ∈ F(ℙ_{m}^{1}). In terms of the 𝔤̄-generators H^{i} and Eᾱ, this yields the same relations as in (2.2), but with the residue terms removed, and with f, g elements of F(ℙ_{m}^{1}) rather than formal power series.

The algebra 𝔤̄⊗F provides a global realization of the symmetries of a WZW conformal field theory, which are locally realized through the affine Lie algebra 𝔤. As will be described in more detail later on, it constitutes an important ingredient in a representation theoretic description of the chiral blocks of WZW theories; for brevity, we will therefore refer to 𝔤̄⊗F as the block algebra. Clearly, the block algebra is spanned by the elements H^{i}^{⊗}f and Eᾱ^{⊗}f with i ∈ {1, 2, ... , r}, ᾱ a 𝔤̄-root and f ∈ F(ℙ_{m}^{1}). (By allowing for arbitrary f ∈ F(ℙ_{m}^{1}) of course we do not obtain a basis of the block algebra; while a basis can easily be given, we will not need it here. Note that an automorphism of 𝔤̄⊗F is uniquely defined by its action on a basis, and hence a fortiori by its action on the elements H^{i}^{⊗}f and Eβ̄^{⊗}f.)

Around each of the punctures p_{s} we can choose the local coordinate

ζ_{s} = z − z_{s} .  (3.5)

By identifying these local coordinates with the indeterminate t of the loop construction of the affine Lie algebra 𝔤_{s} ≅ 𝔤, one obtains an embedding of the block algebra 𝔤̄⊗F in the direct sum of m copies of the loop algebra 𝔤̄_{loop} ≅ 𝔤̄ ⊗ ℂ((t)), and thereby also an embedding ı in the algebra 𝔤^{m} that was introduced in (3.1). This is seen as follows. For any function f ∈ F(ℙ_{m}^{1}) we denote its expansion in local coordinates around the point p_{s}, considered as a Laurent series in the variable t ≡ ζ_{s}, by f_{|s}. With this notation, the image of x̄^{⊗}f ∈ 𝔤̄⊗F under the embedding ı is the sequence

ı(x̄^{⊗}f) = (x̄^{⊗}f_{|1}, x̄^{⊗}f_{|2}, ... , x̄^{⊗}f_{|m}) ,  (3.6)

where x̄^{⊗}f_{|s} is regarded as an element of 𝔤_{s}. In short, the embedding in the sth summand is obtained by replacing the global function f by its local expansion f_{|s} at the puncture p_{s}, or in other words, by its germ at p_{s}. Note that f is already determined completely by its germ at any single puncture. In particular, the block algebra is a proper subalgebra of 𝔤^{m}.

3.2 Automorphisms of the block algebra

Next we observe that also the functions φ_{s_{∘},s} defined in (2.7) are the germs of globally defined functions in F(ℙ_{m}^{1}). Indeed, in view of (3.5) for each pair s_{∘}, s the function φ_{s_{∘},s} (2.7) can be recognized as the local expansion

φ_{s_{∘},s}(ζ_{s_{∘}}) = (ζ_{s_{∘}} + z_{s_{∘}} − z_{s})^{−1} = φ_{|s_{∘}}^{(s)}(ζ_{s_{∘}})  (3.7)

at p_{s_{∘}} of the function φ^{(s)} ∈ F(ℙ_{m}^{1}) defined by

φ^{(s)}(z) := (z − z_{s})^{−1} .  (3.8)

At this point the meaning of the requirement ∑_{s=1}^{m} μ̄_{s} = 0 becomes clear: it ensures that the function ∏_{s}(φ^{(s)})^{−(μ̄_{s},β̄)} possesses poles only at the punctures p_{s} and hence lies in F(ℙ_{m}^{1}). It therefore follows in particular that the functions f · ∏_{s}(φ_{s_{∘},s})^{−(μ̄_{s},β̄)} that appear in the automorphism (2.6) of 𝔤 are the local germs at p_{s_{∘}} of globally defined functions f · ∏_{s}(φ^{(s)})^{−(μ̄_{s},β̄)}.

Of course, each of the germs individually already contains all information on the global function; this is reflected by the fact that for all choices of s_{∘} the Laurent series (φ_{s_{∘},s})^{−(μ̄_{s},β̄)} all involve one and the same shift vector μ̄.

From these observations we finally deduce that to any shift vector μ̄ ∈ Γ_{w} we can associate a linear map of the block algebra 𝔤̄⊗F which acts as

σ⃗_{μ̄}(H^{i}^{⊗}f) = H^{i}^{⊗}f ,  σ⃗_{μ̄}(Eβ̄^{⊗}f) = Eβ̄^{⊗}f · ∏_{s=1}^{m}(φ^{(s)})^{−(μ̄_{s},β̄)} .  (3.9)

It is readily checked that this map constitutes an automorphism of 𝔤̄⊗F. Note that on purpose here we employ a similar symbol σ⃗_{μ̄} as for the automorphisms σ_{μ̄} of 𝔤 in (2.6); indeed these maps should be regarded as the global and local realizations, respectively, of one and the same basic structure. To be precise, the local expansions of the automorphism (3.9) of the block algebra reproduce the automorphisms (2.6) of 𝔤_{s} only up to the central terms. The latter are needed in order to really have an automorphism of 𝔤_{s} rather than only an automorphism of the corresponding loop algebra; however, because of the identification of the centers of the subalgebras 𝔤_{s} in 𝔤^{m} (3.1), upon summing over all insertion points these terms cancel owing to the residue theorem:

\begin{aligned}σ⃗_{μ̄}(H^{i}^{⊗}f) & ˆ= σ⃗_{μ̄}( ∑_{s=1}^{m} Hi_{(s)}^{⊗}f_{|s}) = ∑_{s=1}^{m} σ_{μ̄;s}(H^{i}^{⊗}f_{|s}) ˆ= H^{i}^{⊗}f + K ∑_{s,s′=1}^{m} μ̄_{s′}^{i} Res(φ_{s,s′} f_{|s}) \\ & = H^{i}^{⊗}f + K ∑_{s′=1}^{m} μ̄_{s′}^{i} · ∑_{s=1}^{m} Res((φ^{(s′)}f)_{|s}) = H^{i}^{⊗}f . \end{aligned} (3.10)

As the reader may already have noticed, for any 𝔤̄-root ᾱ not only the poles, but also the zeroes of the meromorphic function ∏_{s}(φ^{(s)})^{−(μ̄_{s},ᾱ)} occur at the punctures p_{s}. In other words, these functions belong to the subset

F^{∗}(ℙ_{m}^{1}) := {f ∈ F(ℙ_{m}^{1}) | f^{−1} ∈ F(ℙ_{m}^{1})}  (3.11)

of F(ℙ_{m}^{1}), where in accordance with the prescription (2.8) the symbol f^{−1} stands for the function that has values inverse to those of f, i.e. f^{−1}(p) = (f(p))^{−1} for all p ∈ ℙ^{1}. The elements of this subset F^{∗}(ℙ_{m}^{1}) ⊂ F(ℙ_{m}^{1}), which is in fact a subalgebra, are called the invertible elements or units of F(ℙ_{m}^{1}).

At this point it is appropriate to mention that the point at infinity of ℙ^{1} ≅ ℂ ∪{∞} is by no means distinguished geometrically, but acquires its special role only through the choice of coordinates. Choosing a different quasi-global coordinate z̃ on ℙ^{1} one would assign the coordinate value ∞ to a different geometrical point. Accordingly we should also allow for z = ∞ as the value of z at one of the insertion points, for definiteness say z_{m} = ∞. As it turns out, in this situation the discussion above indeed goes through, but with some modifications.

An obvious modification consists in the fact that in place of ζ_{s} = z − z_{s} (3.5) a good local coordinate at z_{m} = ∞ is provided by ζ_{m} = z^{−1}. Accordingly we must e.g. replace the formula (2.7) for φ_{s_{∘},s} by

φ_{s_{∘},s}(t) := \{ \begin{matrix}(t + z_{s_{∘}} − z_{s})^{−1} & for z_{s_{∘}} ≠ ∞ , \\ (t^{−1} − z_{s})^{−1} & for z_{s_{∘}} = ∞ . \end{matrix}  (3.12)

However, this is not the only change that needs to be made. One finds that in fact in the formula (2.6) for the automorphisms of 𝔤 one must now only include the contributions from s ∈ {1, 2, ... , m−1}, but not from s = m, i.e. one now has

σ_{μ̄;s_{∘}}(H^{i}^{⊗}f) := H^{i}^{⊗}f + K ∑_{s=1}^{m−1} μ̄_{s}^{i} Res(φ_{s_{∘},s} f) ,  σ_{μ̄;s_{∘}}(Eβ̄^{⊗}f) := Eβ̄^{⊗}f · Φβ̄ .  (3.13)

where the definition (2.9) of Φᾱ is to be replaced by

Φᾱ := ∏_{s=1}^{m−1}(φ_{s_{∘},s})^{−(μ̄_{s},ᾱ)} .  (3.14)

Moreover, the formula (3.13) applies only for s_{∘} ∈ {1, 2, ... , m−1}, while for s_{∘} = m it gets replaced by

σ_{μ̄;m}(H^{i}^{⊗}f) := H^{i}^{⊗}f − K ∑_{s=1}^{m−1} μ̄_{s}^{i} Res(t^{−2}φ_{s_{∘},s} f) ,  σ_{μ̄;m}(Eβ̄^{⊗}f) := Eβ̄^{⊗}f · Φβ̄ . (3.15)

Similarly, instead by (3.9), the associated automorphism of the block algebra 𝔤̄⊗F is now given by

σ⃗_{μ̄}(H^{i}^{⊗}f) = H^{i}^{⊗}f ,  σ⃗_{μ̄}(Eβ̄^{⊗}f) = Eβ̄^{⊗}f · ∏_{s=1}^{m−1}(φ^{(s)})^{−(μ̄_{s},β̄)} .  (3.16)

After these replacements all calculations go through as before. In particular, still the functions ∏_{s=1}^{m−1}(φ_{s_{∘},s})^{−(μ̄_{s},β̄)} are the local germs of the globally defined function ∏_{s=1}^{m−1}(φ^{(s)})^{−(μ̄_{s},β̄)} at p_{s_{∘}} for all s_{∘} ∈ {1, 2, ... , m}. Also, that function is still an element of F^{∗}(ℙ_{m}^{1}). (On the other hand, of course now one can no longer invoke the identity ∑_{s=1}^{m} μ̄_{s} = 0 to exclude poles at infinity; but this is also no longer needed, since ∞ is a puncture. Explicitly, now the divisor of the function ∏_{s}(φ^{(s)})^{−(μ̄_{s},β̄)} consists of the points p_{s}, s = 1, 2, ... , m−1, with finite value of z_{s}, which have multiplicity −(μ̄_{s}, β̄), and in addition of ∞ which has multiplicity ∑_{s=1}^{m−1}(μ̄_{s}, β̄) = −(μ̄_{m}, β̄).) Briefly, the function ∏_{s=1}^{m−1}(φ^{(s)})^{−(μ̄_{s},β̄)} has already by itself a pole respectively zero of the correct order at ∞.

Let us also note that at z_{m} = ∞ the formula (2.25) for σ_{μ̄;s_{∘}}(H^{i}^{⊗}id) still applies. Namely, in this case we have

σ_{μ̄;m}(H^{i}^{⊗}id) = H^{i}^{⊗}id − K ∑_{s=1}^{m−1} μ̄_{s}^{i} Res(t^{−2} (t^{−1} − z_{s})^{−1}) .  (3.17)

Using the fact that Res(t^{−2}(t^{−1} −z_{s})^{−1}) = 1 independently of s together with the condition that the shift vectors add up to zero, this is nothing but

σ_{μ̄;m}(H^{i}^{⊗}id) = H^{i}^{⊗}id + μ̄_{m}^{i} K .  (3.18)

It follows in particular that the statement that the outer automorphism class [σ_{μ̄;s}] of σ_{μ̄;s} can be read off the coweight lattice element μ̄_{s} is still true for s = m with z_{m} = ∞. We also mention that in the case z_{m} = ∞ the residue theorem still ensures the cancellation of central terms as in formula (3.10). Namely, while the summation over s′ in (3.10) is now only from 1 to m−1, the summation over s still ranges from 1 to m; the desired result then follows after taking into account that Res((φ^{(s′)}f)_{|m}) = Res_{∞}(φ^{(s′)}f) = −Res_{0}(t^{−2}φ^{(s′)}f).

Finally we remark that the Sugawara construction provides us at each of the insertion points p_{s}, with a Virasoro algebra Vir ≡ Vir_{s}. We write Ln^{(s)} for the generators of the Virasoro algebra Vir_{s} associated to 𝔤_{s}. Also note that in the formula (2.32) for the action of a multi-shift automorphism on Vir the sums over s and s′ extend over all punctures except, when present, the one at ∞.

4 Inner automorphisms

In subsection 2.2 we have seen how an automorphism of the affine Lie algebra 𝔤 can be implemented on 𝔤-modules. We can now implement the tensor product of m multi-shift automorphisms on the tensor product of m 𝔤-modules. We will see that this leads to a well-defined map on the spaces of chiral blocks because such a tensor product of multi-shift automorphisms leaves the block algebra 𝔤̄⊗F invariant. One convenient property that one would like to achieve for the maps on the chiral blocks is that they are of finite order. As it turns out, to this end we must find a suitable description for those multi-shift automorphisms which are inner automorphisms of 𝔤̄⊗F.

The subset

Γ := {μ̄ ≡ (μ̄_{1}, μ̄_{2}, ... , μ̄_{m}) | μ̄_{s} ∈ L^{∨} for all s = 1, 2, ... , m; ∑_{s=1}^{m}μ̄_{s} = 0}  (4.1)

of those elements of Γ_{w} for which the allowed range of the shift vectors μ̄_{s} is restricted to the coroot lattice L^{∨} ⊆ Lw^{∨} of 𝔤̄ clearly is a subgroup of Γ_{w}. In this section we show that the automorphisms (2.6) of 𝔤 and (3.9) of 𝔤̄⊗F associated to shift vectors that lie in Γ are inner automorphisms of the block algebra 𝔤̄⊗F, i.e. automorphisms that can be written as ∏_{i} exp(ad_{x_{i}}) with suitable elements x_{i} of 𝔤 respectively 𝔤̄⊗F;^{3} a fortiori they are then also inner automorphisms of the ambient algebra 𝔤^{m} ⊃ 𝔤̄⊗F.

For establishing this result we need to introduce some special inner automorphisms of the block algebra 𝔤̄⊗F, and for these we need some particular elements of 𝔤̄⊗F. For convenience, and without loss of generality, in this section on we assume that 0 and ∞ are among the insertion points p_{s}, and number the latter in such a way that z_{m} = ∞; this has the technical advantage that for each insertion point p_{s} with s ∈ {1, 2, ... , m−1}^{4} the function φ^{(s)} defined by (3.8) already lies in the subalgebra F^{∗}(ℙ_{m}^{1}) of units of F(ℙ_{m}^{1}). The relevant special elements of 𝔤̄⊗F are then of the form

X_{ᾱ,f} := Eᾱ^{⊗}f + E^{−ᾱ}^{⊗}f^{−1} ,  (4.2)

where Eᾱ ∈ 𝔤̄ is a step operator of the horizontal subalgebra 𝔤̄ and f ∈ F^{∗}(ℙ_{m}^{1}) is a unit in F(ℙ_{m}^{1}).

Furthermore, for any two elements X, Y ∈ 𝔤̄⊗F and any complex number ξ, we denote by Ad_{ξ;X,Y} the inner automorphism

Ad_{ξ;X,Y} := exp(ad_{−ξY}) ∘ exp(ad_{ξX})  (4.3)

3 In the present section we only show that μ̄ ∈ Γ is sufficient for having an inner automorphism. But as we shall see later on, this is also necessary.

4 Recall that there is no function φ^{(s)} associated to ∞.

of the block algebra. We are interested in automorphisms of the form Ad_{iπ/2;X_{ᾱ,f_{1}},X_{ᾱ,f_{2}}} with an arbitrary positive 𝔤̄-root ᾱ and some suitable functions f_{1}, f_{2} ∈ F^{∗}(ℙ_{m}^{1}). By direct computation we find that

Ad_{iπ/2;X_{ᾱ,f_{1}},X_{ᾱ,f_{2}}}(H^{i}^{⊗}f) = H^{i}^{⊗}f  (4.4)

for all i = 1, 2, ... , r and

Ad_{iπ/2;X_{ᾱ,f_{1}},X_{ᾱ,f_{2}}}(Eβ̄^{⊗}f) = Eβ̄ ^{⊗} (f_{1}/f_{2})^{−(ᾱ^{∨},β̄)} f  (4.5)

for all 𝔤̄-roots β̄; here ᾱ^{∨} = 2ᾱ/(ᾱ, ᾱ) ∈ L^{∨} is the coroot associated to the 𝔤̄-root ᾱ. (Though straightforward, this calculation is somewhat lengthy; some relevant formulæ are collected in appendix B.)

The results (4.4) and (4.5) imply in particular that any two automorphisms of this form commute. Also, only the quotient f_{1}/f_{2} appears in (4.5), so that without loss of generality we can put f_{2} = id. It follows in particular that when we compose inner automorphisms of the form Ad_{iπ/2;X_{ᾱ,f_{1}},X_{ᾱ,f_{2}}} with the special choice f_{1} = φ^{(s)} as defined by (3.8) for some s ∈ {1, 2, ... , m−1} and f_{2} = id and with arbitrary 𝔤̄-roots ᾱ, then we arrive at automorphisms of the form studied previously. Indeed,

∏_{s=1}^{m−1} ∏_{i_{s}=1}^{ℓ_{s}} Ad_{iπ/2;X_{ᾱ_{is} ,φ^{(s)}},X_{ᾱ_{is} ,id}} = σ⃗_{μ̄} ,  (4.6)

where σ_{μ̄} is the multi-shift automorphism of the block algebra 𝔤̄⊗F that acts as in (3.16), with

μ̄ = (μ̄_{1}, μ̄_{2}, ... , μ̄_{m}) ,  μ̄_{s} = ∑_{i_{s}=1}^{ℓ_{s}} ᾱi_{s}^{∨} ∈ L^{∨} for s = 1, 2, ... , m−1  (4.7)

(and where the ordering in the product is irrelevant). In short, for any choice of the 𝔤̄-roots ᾱ_{i_{s}} we obtain an inner automorphism of the form (3.16) with μ̄_{s} ∈ L^{∨} . Moreover, by appropriately choosing these roots we can obtain every automorphism σ_{μ̄} (3.16) for which μ̄ ∈ Γ. It follows in particular that any automorphism σ⃗_{μ̄} with μ̄ ∈ Γ is an inner automorphism of the block algebra 𝔤̄⊗F, as claimed.

When we replace the functions f etc. that appeared in the considerations above, which are elements of F(ℙ_{m}^{1}), by functions that have the same mapping prescription but are regarded as Laurent series in the relevant local coordinate ζ_{s}, respectively as formal Laurent series in the variable t, then we can immediately repeat all steps in the derivation of the formulæ (4.4) and (4.5); thereby for each summand 𝔤_{s}, s = 1, 2, ... , m−1, of the algebra ⨁_{s=1}^{m} 𝔤_{s} ⊃ 𝔤^{m} we construct a certain class of commuting inner automorphisms. The action of these automorphisms differs from the one of the automorphisms of the block algebra precisely in that there arise additional terms that are proportional to the central element K ≡ K_{s} of 𝔤_{s}. We find that (4.4) gets modified to (for some details see appendix B)

Ad_{iπ/2;X_{ᾱ,f_{1}},X_{ᾱ,f_{2}}}(H^{i}^{⊗}f) = H^{i}^{⊗}f − (ᾱ^{∨})^{i}K (Res(f_{1}^{−1}df_{1} f) − Res(f_{2}^{−1}df_{2} f)) ,  (4.8)

while the formula (4.5) applies without any change also to the present situation.

We now investigate the case where these Laurent series are precisely the local expansions of the functions φ^{(s)} studied previously. Thus for any s, s_{∘} = 1, 2, ... , m−1 we have to deal with the inner automorphism Ad_{iπ/2;x_{s},y_{s}} of 𝔤_{s} with

x_{s} = Eᾱ^{⊗}φ_{s_{∘},s} + E^{−ᾱ}^{⊗}(φ_{s_{∘},s})^{−1} and y_{s} = (Eᾱ + E^{−ᾱ})^{⊗} id ,  (4.9)

where φ_{s_{∘},s}(t) is the local expansion (2.7) at p_{s_{∘}} of the function φ^{(s)} as defined by (3.8). By inserting these functions in the general formulæ obtained above, we learn that the automorphism Ad_{iπ/2;x_{s},y_{s}} acts as

Ad_{iπ/2;x_{s},y_{s}}(H^{i}^{⊗}f_{|s_{∘}}) = H^{i}^{⊗}f_{|s_{∘}} + (ᾱ^{∨})^{i}K Res(φ_{s_{∘},s}f_{|s_{∘}}) ,
Ad_{iπ/2;x_{s},y_{s}}(Eβ̄^{⊗}f_{|s_{∘}}) = Eβ̄^{⊗}f_{|s_{∘}}(φ_{s_{∘},s})^{−(ᾱ^{∨},β̄)} .

(4.10)

Next we take the product of such automorphisms that is analogous to the product in (4.6). Thereby for each s_{∘} = 1, 2, ... , m−1 we arrive at automorphisms σ_{μ̄;s_{∘}} of 𝔤_{s_{∘}} which act as σ_{μ̄;s_{∘}}(K) = K and

\begin{matrix}σ_{μ̄;s_{∘}}(H^{i}^{⊗}f_{|s_{∘}}) = H^{i} ^{⊗} f_{|s_{∘}} + K ∑_{s=1}^{m−1} μ̄_{s}^{i} Res(φ_{s_{∘},s} f_{|s_{∘}}) , \\ σ_{μ̄;s_{∘}}(Eβ̄^{⊗}f_{|s_{∘}}) = Eβ̄ ^{⊗} f_{|s_{∘}} · ∏_{s=1}^{m−1}(φ_{s_{∘},s})^{−(μ̄_{s},β̄)} . \end{matrix}  (4.11)

For s_{∘} = m an analogous result holds, with the formula in the first line replaced by

σ_{μ̄;m}(H^{i}^{⊗}f_{|m}) = H^{i} ^{⊗} f_{|m} − K ∑_{s=1}^{m−1} μ̄_{s}^{i} Res(t^{−2}φ_{m,s} f_{|m})  (4.12)

(and with φ_{m,s} as given in (3.12)). Thus again we have succeeded in constructing all those multi-shift automorphisms σ_{μ̄} – this time the automorphisms (2.6) of the affine Lie algebra 𝔤 – for which the shift vector μ̄ lies in the subgroup Γ of Γ_{w}, and hence can conclude that all such automorphisms are inner automorphisms of 𝔤.

Of course, the function f_{|s_{∘}}(φ_{s_{∘},s})^{−(ᾱ^{∨},β̄)} = f_{|s_{∘}}(φ_{|s_{∘}}^{(s)})^{−(ᾱ^{∨},β̄)} is nothing but the local expansion of the function f(φ^{(s)})^{−(ᾱ^{∨},β̄)} at p_{s}. Hence in particular upon summation over s the central terms in the transformation (4.10) of H^{i}^{⊗}f cancel owing to the residue theorem. Thus the automorphisms (4.11) are the local realizations of the automorphism (4.6) of the block algebra.

One may also analyze the analogous inner multi-shift automorphisms of the Virasoro algebra. This is briefly mentioned at the end of appendix B.

5 Implementation on chiral blocks

5.1 Implementation on tensor products

The next step is now to implement a tensor product of multi-shift automorphisms on a tensor product of modules of the affine Lie algebra. This gives rise to an action of the group Γ_{w} on this tensor product and, of course, also to a dual action on the algebraic dual of the tensor product. As we will explain, the space B of chiral blocks can be identified with a subspace in this algebraic dual. We will see that the action of Γ_{w} can be restricted to B and that this action has finite order on the subspace B.

Let us first recall the existence of the maps Θμ̄ ≡ Θ_{μ̄;s_{∘}} (2.19). They act on the s_{∘}th factor of the tensor product

H⃗ ≡ HΛ⃗ = H_{Λ_{1}} ⊗ H_{Λ_{2}} ⊗ ⋯ ⊗ H_{Λ_{m}}  (5.1)

of irreducible highest weight modules over 𝔤, i.e. Θμ̄: H_{Λ_{s∘}} → H_{ω^{⋆}Λ_{s∘}}, with ω^{⋆} the permutation of integrable weights that was described before (2.19). The tensor product

Θ⃗μ̄ := Θ_{μ̄;1} ^{⊗} Θ_{μ̄;2} ^{⊗} ⋯ ^{⊗} Θ_{μ̄;m}  (5.2)

of these maps for all s_{∘} = 1, 2, ... , m provides us with an analogous map between tensor products,

Θ⃗μ̄ : HΛ⃗ → H_{ω^{⋆}Λ⃗} ,  (5.3)

where ω^{⋆}Λ⃗ = (ω^{⋆}Λ_{1}, ω^{⋆}Λ_{2}, ... , ω^{⋆}Λ_{m}). Now note that this map is defined for any μ̄ ∈ Γ_{w}. Moreover, we claim that the mapping

μ̄ ↦ Θ⃗μ̄  (5.4)

can be chosen in such a way that it becomes a group homomorphism, i.e. such that the group law of Γ_{w} is also realized by the action of the maps Θ⃗μ̄ on the modules. This can be accomplished as follows. First we realize that the highest weight vector of the 𝔤-module H_{Λ} can be required to be normalized and then is unique up to a phase. Accordingly the maps Θ_{μ̄;s_{∘}} are unique up to a phase, too. Our aim is to show that we can choose this phase as a function of μ̄ in such a manner that the group law is satisfied. Now the group Γ_{w} is a finitely generated free abelian group, so there is a finite set {μ̄_{(i)}}, i = 1, 2, ... , M, of independent generators. For these elements the phases can be chosen independently. Any arbitrary μ̄ ∈ Γ_{w} can be uniquely written as μ̄ = ∑_{i} n_{i} μ̄_{(i)}. We then define the image of μ̄ under the map (5.4) as

Θ⃗μ̄ := (Θ_{μ̄_{(M)}})^{n_{M}} (Θ_{μ̄_{(M−1)}})^{n_{M−1}} ⋯ (Θ_{μ̄_{(1)}})^{n_{1}} ,  (5.5)

which fixes their phases in terms of those of the generators μ̄_{(i)}. It is an easy exercise to check that the so defined maps Θμ̄ are indeed twisted intertwiners for σ_{μ̄}.

Next we show that all inner multi-shift automorphisms are implemented by the action of elements in the block algebra. To this end we note that the subgroup Γ of Γ_{w} is again a finitely generated free group. As a consequence we can again find a finite set {γ̄_{(j)}} of independent generators. For instance one can choose these generators to be all of the form

γ̄_{(j);s_{1}} = β̄^{∨} ,  γ̄_{(j);s_{2}} = −β̄^{∨} ,  γ̄_{(j);s} = 0 for s ≠ s_{1}, s_{2}  (5.6)

with suitable choices of s_{1}, s_{2} ∈ {1, 2, ... , m} with s_{1} < s_{2} and simple 𝔤̄-roots β̄. Furthermore, according to the results of subsection 2.2, each choice of a Laurent series f (which serves as the prefactor of the central terms) and ordering of the factors in the product provides us with some definite implementation ˜ Θ⃗_{γ̄_{(j)}} of the inner automorphism. Now for each γ̄_{(j)} we are given

a preferred implementation ˜ Θ⃗_{γ̄_{(j)}}, namely in the case of generators of the form (5.6)

˜Θ⃗_{γ̄_{(j)}} := id^{⊗} ⋯ ^{⊗} id^{⊗} exp (−\frac{iπ}{2}X_{β̄,id}) exp (\frac{iπ}{2}X_{β̄,φ_{s∘,s_{1}}})^{⊗} id^{⊗}
⋯ ^{⊗} id^{⊗} exp (−\frac{iπ}{2}X_{β̄,id}) exp (\frac{iπ}{2}X_{−β̄,φ_{s∘,s_{2}}})^{⊗} id^{⊗} ⋯ ^{⊗} id ,

(5.7) where the notation X_{ᾱ,f} is as introduced in (4.2) and where the non-trivial maps occur at the s_{1}th and s_{2}th factor of the tensor product. Via an analogous prescription as in (5.5) above, this definition then provides us with a preferred implementation ˜ Θ⃗μ̄ for any arbitrary μ̄ ∈ Γ, which in each tensor factor is given by a product (in prescribed order) of factors of the form exp ((−iπ/2)X_{β̄,id}) exp ((iπ/2)X_{β̄,φ_{s∘,s}}) (compare also the formulæ (2.21) and (4.6)).

Now ˜ Θ⃗_{γ̄_{(j)}} does not necessarily coincide with Θ⃗_{γ̄_{(j)}} as constructed according to the prescription (5.5); rather, the two implementations may differ by a phase. However, the so obtained phases extend uniquely to a group homomorphism from Γ to U(1), and we can extend this group homomorphism even further to a homomorphism Γ_{w} → U(1) (the latter extension is of course no longer unique, but we are free to pick any of the possible extensions). Given this homomorphism, we then correct the previous choice of the elements Θ⃗_{μ̄_{(i)}} by the appropriate phase. This way we finally achieve that the group law of Γ_{w} is preserved and that at the same time the elements ˜ Θ⃗_{γ̄_{(j)}} are correctly implemented.

Taken together, these results will allow us to implement the multi-shift automorphisms also on chiral blocks in such a way that we even obtain an action of the group Γ_{w} on the space of chiral blocks. To make this explicit, we need an appropriate description of the chiral blocks in terms of the action of the block algebra.

5.2 Chiral blocks from co-invariants

In a representation theoretic approach, the chiral blocks of a conformal field theory are constructed with the help of co-invariants ⌊H⃗⌋_{𝔅} of tensor products H⃗ ≡ HΛ⃗ (5.1) of irreducible modules over the chiral algebra with respect to a suitable block algebra 𝔅 (see e.g. [6, 2]). This statement involves two new ingredients that need to be explained. First, in general, by a co-invariant of a module V over some Lie algebra 𝔥 one means the quotient vector space

⌊V ⌋_{𝔥} := V / U^{+}(𝔥)V ,  (5.8)

where U^{+}(𝔥) = 𝔥U(𝔥) with U(𝔥) the universal enveloping algebra of 𝔥. When the 𝔥-module V is fully reducible, then (5.8) is just the submodule of 𝔥-singlets in V, but generically it is a genuine quotient which cannot be identified with a subspace of V. And second, the action of the block algebra 𝔅 = 𝔤̄⊗F on the tensor product vector space H is defined by its expansions in local coordinates, i.e. for X = x̄^{⊗}f and v = v_{1}^{⊗}v_{2}^{⊗} ⋯^{⊗}v_{m} one has

X v := ∑_{s=1}^{m} v_{1}^{⊗}v_{2}^{⊗} ⋯^{⊗}v_{s−1}^{⊗} (x̄^{⊗}f_{|s})v_{s} ^{⊗}v_{s+1}^{⊗} ⋯^{⊗}v_{m} ,  (5.9)

where x̄^{⊗}f_{|s} is regarded as an element of the loop algebra 𝔤̄_{loop} and hence of 𝔤, or more precisely, as the representation matrix of that element of 𝔤 in the 𝔤-representation R_{Λ_{s}}. (That this yields a 𝔤̄⊗F-representation follows by

((x̄^{⊗}f)(ȳ^{⊗}g) − (ȳ^{⊗}g)(x̄^{⊗}f))(v_{1}^{⊗}v_{2}^{⊗} ⋯^{⊗}v_{m}) = ∑_{s=1}^{m}v_{1}^{⊗}v_{2}^{⊗} ⋯^{⊗}[x̄^{⊗}f_{|s}, ȳ^{⊗}g_{|s}]v_{s}^{⊗} ⋯^{⊗}v_{m}

= ∑_{s=1}^{m} v_{1}^{⊗}v_{2}^{⊗} ⋯ ^{⊗} ([x̄, ȳ]^{⊗}(fg)_{|s})v_{s} ^{⊗} ⋯^{⊗}v_{m}

+ K κ(x̄, ȳ) ( ∑_{s=1}^{m}Res_{p_{s}}(df g)) v_{1}^{⊗}v_{2}^{⊗} ⋯^{⊗}v_{m} ,

(5.10)

where in the first equality one uses the fact that terms acting on different tensor factors cancel and in the second equality the bracket relations of 𝔤 are inserted. The terms in (5.10) that involve the central element K cancel as a consequence of the residue formula, while the other terms add up to [x̄^{⊗}f, ȳ^{⊗}g] v, where the Lie bracket is the one of 𝔤̄⊗F as defined in (3.4).)

The finite-dimensional vector spaces ⌊H⃗⌋_{𝔅} of co-invariants play the role of the dual spaces B^{⋆} of the chiral block spaces B, i.e.

BΛ⃗ = (⌊HΛ⃗⌋_{𝔅})^{⋆} .  (5.11)

By duality, the chiral blocks can then also be regarded as the invariants (or in other words, singlet submodules) in the algebraic dual H⃗^{⋆} of H⃗, i.e.

BΛ⃗ = (H⋆ ; Λ⃗)^{𝔅};  (5.12)

in this description, the blocks are linear forms β on H⃗ with the property

⟨β , Xv⟩ ≡ β(Xv) = 0  (5.13)

for all X ∈ 𝔤̄⊗F and all v ∈ H⃗.

5.3 Isomorphisms of chiral blocks

In (5.8) we followed the habit of suppressing the symbol R for the representation by which 𝔥 acts on the vector space V, e.g. ⌊V ⌋_{𝔥} is a shorthand for ⌊V ⌋_{R(𝔥)}. This cannot cause any confusion as long as we only deal with a single 𝔥-module (R, V) which is based on the vector space V. On the other hand, as is easily checked, given any automorphism σ of 𝔥, together with (R, V) also (R̃, V) := (R ∘ σ, V) furnishes a module over 𝔥. Accordingly we are then dealing with two different actions of 𝔥 on V, and hence with two different spaces (5.8) of co-invariants. It is then natural to try to associate to any automorphism σ of 𝔥 a corresponding mapping of the respective spaces of co-invariants. This is easily achieved once we are given a linear map Θ_{σ}: V → V with the property that Θ_{σ} R(x) v = R̃(x) Θ_{σ} v for all x ∈ 𝔥 and all v ∈ V, or in short (suppressing again the symbol R),

Θ_{σ} x = σ(x) Θ_{σ}  (5.14)

for all x ∈ 𝔥. Namely, we observe that via the prescription σ(1) := 1 and linearity, the automorphism σ of 𝔥 extends to an automorphism of the enveloping algebra U(𝔥) that respects the filtration, hence also to an automorphism of U^{+}(𝔥). Because of (5.14) the prescription

⌊V ⌋_{R(𝔥)} ∋ [v] ↦ [Θ_{σ}v] ∈ ⌊V ⌋_{R̃(𝔥)}  (5.15)

then supplies us with a well-defined mapping from ⌊V ⌋_{R(𝔥)} to ⌊V ⌋_{R̃(𝔥)}, and by construction this is in fact an isomorphism of vector spaces.

Now it is a general fact that inner automorphisms act trivially on co-invariants. Namely, an inner automorphism σ of 𝔥 can by definition be written as a product of finitely many automorphisms σ_{i} = exp(ad_{x_{i}}) for some elements x_{i} ∈ 𝔥. Moreover, for every y ∈ 𝔥 we have (compare the formulæ (2.18) and (2.21))^{5}

R ∘ σ_{i}(y) = R(exp(ad_{x_{i}})(y)) = exp(R(x_{i})) R(y) exp(−R(x_{i})) .  (5.16)

By expanding the exponentials it then follows in particular that

R̃(y) v = R ∘ σ(y) v = R(y) v mod U^{+}(𝔥)V  (5.17)

for all v ∈ V and all y ∈ 𝔥. By the definition of co-invariants, this means that the modules (R, V) and (R̃, V) possess the same co-invariants. By duality, this also means that the dual spaces to these modules possess the same invariants.

When applied to the situation of our interest, these general observations tell us that as far as the study of chiral blocks is concerned we need to regard the automorphisms σ⃗_{μ̄} of the block algebra 𝔤̄⊗F that we defined in (3.16) above only modulo inner automorphisms. Accordingly we should determine the outer automorphism class to which a multi-shift automorphism σ⃗_{μ̄} of 𝔤̄⊗F belongs. We have already addressed this question in section 2 for the case of multi-shift automorphisms of the affine Lie algebra 𝔤. Now we study the same issue for the block algebra 𝔤̄⊗F. We first note that the result of section 2 implies that whenever at least one of the vectors μ̄_{s} is not a coroot, then σ_{μ̄;s} is an outer automorphism of the corresponding affine Lie algebra 𝔤_{s}; it follows that in this case it is also an outer automorphism of 𝔤^{m}, and thereby a fortiori an outer automorphism of the block algebra 𝔤̄⊗F.

In other words, the subgroup Γ (4.1) already exhausts the set of those shift vectors μ̄ ∈ Γ_{w} for which σ_{μ̄} is an inner automorphism of 𝔤̄⊗F. Thus the group Γ_{out} of outer modulo inner multi-shift automorphisms of 𝔤̄⊗F is precisely the factor group

Γ_{out} = Γ_{w}/Γ .  (5.18)

According to the isomorphism (2.29) between Lw^{∨}/L^{∨} and the unique maximal abelian normal subgroup Z(𝔤) of Out(𝔤) = Aut(𝔤)/Int(𝔤), we thus have

Γ_{out} ≅ (Z(𝔤))^{r−1} .  (5.19)

In particular, Γ_{out} is a finite group of order ord(Γ_{out}) = |L_{w}/L|^{r−1}. For future reference we mention that a set of distinguished representatives of the elements of Out(𝔤) is provided by the diagram automorphisms ω_{μ̄;s} of 𝔤 (see subsection 2.2); thus elements of Γ_{out} may be regarded as collections of suitable diagram automorphisms whose product is the identity.

Explicitly, for every μ̄ ∈ Γ_{w} we have a map Θ⃗⋆ ; μ̄ from the chiral blocks BΛ⃗ = (H⋆ ; Λ⃗)^{𝔅} to H_{ω^{⋆}Λ⃗}^{⋆} which acts as

⟨Θ⃗⋆ ; μ̄(β) , v⟩ := ⟨β , Θ⃗−1 ; μ̄v⟩ .  (5.20)

5 This may be checked by replacing x_{i} by ξx_{i} and comparing both sides order by order in the dummy variable ξ. In the special case that 𝔥 integrates to a group, the formula is an immediate consequence of the fact that exp(ad_{x})(y) = γyγ^{−1}, where γ is a group element such that γ = exp(x).

The previous results imply, first, that the image of this map is in B_{ω^{⋆}Λ⃗}, i.e. that for any block β, Θ⃗⋆ ; μ̄(β) is again a chiral block, and second, that this map is of finite order and depends on μ̄ only via its class [μ̄] in Γ_{out}. In short, for each [μ̄] ∈ Γ_{out} we have constructed an isomorphism

Θ⃗⋆ ; μ̄ ≡ Θ⃗_{[μ̄]}^{⋆} :  BΛ⃗ ≅ B_{ω^{⋆}Λ⃗} .  (5.21)

(Analogously, we also have an isomorphism between the respective dual spaces, ⌊HΛ⃗⌋_{𝔤̄⊗F} ≅ ⌊H_{ω^{⋆}Λ⃗}⌋_{𝔤̄⊗F}.) Note that it already follows from the Verlinde formula and elementary properties of simple currents that the spaces BΛ⃗ and B_{ω^{⋆}Λ⃗} have the same dimension; being finitedimensional, it is then trivial that they are isomorphic as vector spaces. The virtue of the result (5.21) is, however, that it provides us with a canonical realization of this isomorphism. As we will soon see, this realization possesses the additional non-trivial property to be compatible with a variation of the moduli in the problem, i.e. with different choices of the insertion points. Moreover, knowing the isomorphism (5.21) one can also study the problem of fixed point resolution, which arises whenever the collection Λ⃗ of 𝔤-weights is left invariant by ω^{⋆}; this issue will be addressed in section 7 below.

5.4 Curves of higher genus

Most of our constructions allow for a direct generalization to complex curves of genus g ≥ 1. Indeed, let us consider some complex curve C of arbitrary genus g and m marked points p_{s} on C. Given an m-tuple μ̄ of coweights of 𝔤̄, we can associate to any 𝔤̄-root ᾱ the divisor

Dᾱ(μ̄) := ∑_{s=1}^{m}(μ̄_{s}, ᾱ) p_{s}  (5.22)

which, due to the condition ∑_{s} μ̄_{s} = 0, has degree zero.

Let us now recall a few properties of the prime form E(z, w). The line bundle over C × C that is associated to the diagonal divisor C has, up to normalization, a unique section, which is just the prime form E. It is antisymmetric, E(z, w) = −E(w, z), and has a simple zero at z = w. For the case of genus g = 0, the prime form is given by E(z, w) = z − w. For higher genera, E is not a function any more; it can be chosen to be single-valued around the a-cycles, but then it is not single-valued around the b-cycles. Still, given a divisor of degree zero, the prime form allows us to construct a meromorphic function with that divisor. Namely, to Dᾱ we associate the meromorphic function

Φg ; ᾱ(μ̄; z) = ∏_{s=1}^{m} (E(z, p_{s}))^{(μ̄_{s},ᾱ)} .  (5.23)

We then consider the automorphism of the block algebra 𝔤̄⊗F(C\{p_{1}, p_{2}, ... , p_{m}}) that is characterized by

σ⃗_{μ̄}(Eᾱ^{⊗}f) = Eᾱ^{⊗}f Φg ; ᾱ(μ̄) and σ⃗_{μ̄}(H^{i}^{⊗}f) = H^{i} ^{⊗} f .  (5.24)

It is now easy to generalize also the definition of the functions φ_{s_{∘},s}; we define φ_{s_{∘},s} as the expansion of the global section E(z, p_{s})^{−1} in a local coordinate t at the puncture p_{s_{∘}}. Using these functions of the local coordinates, we define multi-shift automorphisms of the affine Lie algebra 𝔤 as in the genus zero case. Obviously, they still combine to an automorphism of 𝔤^{m} that restricts to an automorphism of the block algebra. Moreover, for s_{∘} ≠ s the function φ_{s_{∘},s}(t) of the local coordinate t at p_{s_{∘}} is regular; on the other hand, for s_{∘} = s, we have a pole of first order. Hence we still have Res(φ_{s_{∘},s}) = δ_{s_{∘},s}. As a consequence, the class of the multi-shift automorphism modulo inner automorphisms can be determined as in subsection 2.3. Indeed, after this re-interpretation of φ_{s_{∘},s} all results of sections 2, 3 and 5 hold for arbitrary genus. In particular, the group law of Γ_{w} will be respected.

On the other hand, the results of section 4 cannot be expected to generalize in a straightforward way, because in those manipulations the prime form enters explicitly, so that multivaluedness around the b-cycles appears as a new feature. However, for higher genus it is anyway more natural to work with the projective spaces that are constructed from the vector spaces of chiral blocks.

6 Bundles of blocks

In all the considerations above we have regarded the punctures p_{s} as held fixed, i.e. we have analyzed block algebras and chiral blocks at a single point of the moduli space M ≡ M_{m} of m-punctured projective curves. We now address the issues that arise when the insertion points are allowed to vary over the whole moduli space M. Recall from the discussion in subsection 5.1 that in the implementation Θ⃗μ̄ of an automorphism of 𝔤^{m} a phase is still undetermined. Clearly, if we choose this phase at random for each point in the moduli space, we cannot expect to obtain quantities on the bundle of chiral blocks that vary smoothly with the moduli. On the other hand, the bundle of chiral blocks carries a natural projectively flat connection, the Knizhnik--Zamolodchikov connection [25–27, 6]. Our rationale will therefore be to implement the maps σ_{μ̄}(z⃗) for different values of the moduli z⃗ in such a way that they preserve the Knizhnik--Zamolodchikov connection.

6.1 Connections and bundles over the moduli space

The moduli space M is (ℙ^{1})^{m} minus the union of diagonals, i.e.

M = {(p_{1}, p_{2}, ... , p_{m}) | p_{s} ∈ ℙ^{1}, p_{s} ≠ p_{s′} for s ≠ s′} .  (6.1)

When considered as depending on the values of the punctures, the spaces B of chiral blocks combine to a vector bundle B over M [6]. Before we study that bundle, we introduce a few other bundles which in this context are of interest as well.

We first consider the trivial bundle over M with fiber given by the infinite-dimensional Lie algebra 𝔤^{m} defined in (3.1),

𝔤^{m} × M → M .  (6.2)

This bundle carries a fiberwise action of the group Γ_{w}, i.e. an action by automorphisms of the total space that cover the identity on the base space M. For every μ̄ ∈ Γ_{w} we write σ_{μ̄}(z⃗) for the map on the fiber over the point p⃗ ∈ M with coordinates z⃗ ≡ (z_{1}, z_{2}, ... , z_{m}). On the bundle (6.2) we have a flat connection D which acts on smooth sections g(z⃗) of (6.2) as

D_{s}g(z⃗) := ∂_{s}g(z⃗) + [L−1^{(s)}, g(z⃗)]  (6.3)

for s = 1, 2, ... , m, where ∂_{s} ≡ ∂/∂z_{s} and L−1^{(s)} ∈ Vir_{s} is a generator of the Virasoro algebra associated to 𝔤_{s}.

Moreover, the bundle (6.2) possesses a subbundle of block algebras, which is defined by taking for each p⃗ ∈ M the corresponding block algebra 𝔤̄⊗F = 𝔤̄ ⊗ F(ℙ^{1}\{p_{1}, p_{2}, ... , p_{m}}) ⊂ 𝔤^{m}. The smooth sections of this bundle will be denoted by X(z⃗). It is important to realize that while the block algebras at different points p⃗ are isomorphic as abstract Lie algebras, they are not naturally isomorphic, and the subbundle is not necessarily a trivial bundle. Still, by construction, the automorphisms σ_{μ̄} restrict to automorphisms on the subbundle of block algebras.

A crucial property of the connection (6.3) is that it preserves the subbundle of block algebras. Explicitly, this can be seen as follows. The algebra F(ℙ_{m}^{1}) is generated algebraically by the constant function φ^{(0)} := id and the functions φ^{(s)} with s = 1, 2, ... , m, where φ^{(s)}(z) = (z − z_{s})^{−1}.^{6} By induction with respect to the ‘length’ ℓ of f = φ^{(i_{1})}φ^{(i_{2})} ⋯ φ^{(i_{ℓ})} and using the Leibniz rule, it follows that this property is established as soon as it is shown to hold for each of the functions φ^{(s)}. Thus we consider the element X = x̄^{⊗}φ^{(s)} of the block algebra. For s = 0 we trivially have

∂_{s′}X = 0 and [L−1^{(s′)}, X] = 0 for all s′ = 1, 2, ... , m. For s ∈ {1, 2, ... , m} and s′ ≠ s both ∂_{s′}X and

[L−1^{(s′)}, X] have a component only in 𝔤_{s′}, namely

[L−1^{(s′)}, X]_{|s′} = x̄^{⊗} (t + z_{s′} − z_{s})^{−2} = −∂_{s′}X_{|s′}  (6.4)

so that again DX = 0. Finally, for s ∈ {1, 2, ... , m} and s′ = s, the commutator [L−1^{(s′)}, X] is as in (6.4), but now for ∂_{s′}X the component in 𝔤_{s′} vanishes while there are additional contributions in all 𝔤_{s′′} with s′′ ≠ s, namely ∂_{s′}X_{|s′′} = x̄^{⊗}(t + z_{s′′} − z_{s})^{−2}. Thus in this case D_{s′}X is not zero; however, it is still an element of the block algebra 𝔤̄⊗F, namely D_{s′}X = x̄^{⊗}(z − z_{s})^{−2}, since we encountered precisely the local expansions of this element of 𝔤̄⊗F.

Next we consider the trivial bundle over M with fiber given by the tensor product H⃗ ≡ HΛ⃗ (5.1) of irreducible highest weight modules H_{Λ_{s}} over the affine Lie algebra 𝔤,

H⃗ × M → M .  (6.5)

Again this bundle is endowed with a flat connection ∇, which acts on smooth sections v(z⃗) of (6.5) as

∇_{s}v(z⃗) := (∂_{s} + L−1^{(s)})v(z⃗)  (6.6)

for s = 1, 2, ... , m; we call this connection the Knizhnik--Zamolodchikov connection. And again on the bundle (6.5) there is a fiberwise action of a certain discrete group. This group is the (proper) subgroup Γ_{fix} of Γ_{w} that consists of all those elements μ̄ of Γ_{w} for which all associated maps Θ_{μ̄;s} of the tensor factors (see (2.19)) of H⃗ are endomorphisms. This group is given by

Γ_{fix} = Γ_{Λ⃗} := {μ̄ ∈ Γ_{w} | ω_{μ̄;s}^{⋆}(Λ_{s}) = Λ_{s} for all s = 1, 2, ... , m} .  (6.7)

Note that by the results of subsection 5.3, the definition of Γ_{Λ⃗} depends on the automorphisms σ⃗_{μ̄} only through the class [μ̄] in Γ_{out} (5.18), or what is the same, through the associated diagram automorphisms ω⃗_{μ̄}. We call Γ_{Λ⃗} the stabilizer subgroup of Γ_{w} associated to the weights Λ_{1}, ... , Λ_{m}. For every μ̄ ∈ Γ_{Λ⃗} we write Θ⃗μ̄(z⃗) for the map on the fiber over p⃗ ∈ M in the bundle (6.5).

6 When the puncture p_{m} is at z_{m} = ∞, for s = m we rather have φ^{(s)}(z) = z. The corresponding changes in the arguments below are obvious, and we refrain from writing them down explicitly.

6.2 Moduli dependence of the twisted intertwiners

To determine how the implementation of the automorphism depends on the moduli, we first study how the multi-shift automorphism σ⃗_{μ̄} changes when the punctures are varied (while the shift vector μ̄ is kept fixed). We denote by p^{[0]} a reference point on M and study how the automorphism σ⃗_{μ̄} ≡ σ⃗_{μ̄}(z⃗) differs from σ⃗_{μ̄}(z^{[0]}), or in other words, what the automorphism δ⃗_{μ̄} defined by

σ⃗_{μ̄}(z⃗) = δ⃗_{μ̄}(z⃗; z^{[0]}) ∘ σ⃗_{μ̄}(z^{[0]})  (6.8)

looks like. Note that we have to interpret δ⃗_{μ̄} as an automorphism of the algebra 𝔤^{m}; since the block algebra, regarded as a subalgebra of 𝔤^{m}, varies with the moduli, it does not make sense to look for an automorphism of the block algebra.

For every s_{∘} = 1, 2, ... , m the explicit form of the map δ_{μ̄} ≡ δ_{μ̄;s_{∘}} follows directly from the formula (2.6) for σ_{μ̄} ≡ σ_{μ̄;s_{∘}}. We get δ_{μ̄}(K) = K and

\begin{matrix}δ_{μ̄}(H^{i}^{⊗}f) = H^{i}^{⊗}f + K ∑_{s=1}^{m} μ̄_{s}^{i} Res((φ_{s_{∘},s} − φs_{∘},s^{[0]}) f) , \\ δ_{μ̄}(Eβ̄^{⊗}f) = Eβ̄^{⊗}f · ∏_{s=1}^{m}(φ_{s_{∘},s}/φs_{∘},s^{[0]})^{−(μ̄_{s},β̄)} \end{matrix}  (6.9)

with φ_{s_{∘},s} as defined in (2.7)^{7} and φs_{∘},s^{[0]}(t) := (t + zs_{∘}^{[0]} − zs^{[0]})^{−1}.

For simplicity, let us for the moment restrict our attention to the particular case where zs^{[0]} = z_{s} except for some fixed value u ∈ {1, 2, ... , m}. Then δ_{μ̄} acts on 𝔤^{m} as δ_{μ̄}(K) = K and

δ_{μ̄}(H^{i}^{⊗}f) = \{ \begin{matrix}H^{i}^{⊗}f + K μ̄_{u}^{i} Res((φ_{s_{∘},u} − φs_{∘},u^{[0]}) f) & for s_{∘} ≠ u , \\ H^{i}^{⊗}f + K ∑_{{s=1 \atop s≠u}}^{m} μ̄_{s}^{i} Res((φ_{u,s} − φu,s^{[0]}) f) & for s_{∘} = u , \end{matrix}

δ_{μ̄}(Eβ̄^{⊗}f) = \{ \begin{matrix}Eβ̄^{⊗}f · (φ_{s_{∘},u}/φs_{∘},u^{[0]})^{−(μ̄_{u},β̄)} & for s_{∘} ≠ u , \\ Eβ̄^{⊗}f · ∏_{{s=1 \atop s≠u}}^{m}(φ_{u,s}/φu,s^{[0]})^{−(μ̄_{s},β̄)} & for s_{∘} = u . \end{matrix}

(6.10)

Next we observe that a variation of the moduli does not change the class of the automorphism σ_{μ̄} modulo inner automorphisms. As a consequence, δ_{μ̄} is in fact an inner automorphism of 𝔤^{m}. As can be verified by direct calculation (see appendix C), we have in fact

δ_{μ̄} = \{ \begin{matrix}exp(ad_{(μ̄_{u},H)⊗g_{s∘,u}}) & for s_{∘} ≠ u , \\ exp(ad_{∑_{s≠u}(μ̄_{s},H)⊗g̃_{s}}) & for s_{∘} = u , \end{matrix}  (6.11)

where

g_{s_{∘},u}(t) := ln \frac{t + z_{s_{0}} − z_{u}}{t + z_{s_{0}} − zu^{[0]}}  and g̃_{s}(t) := ln \frac{t + z_{u} − z_{s}}{t + zu^{[0]} − z_{s}} .  (6.12)

7 Here and below we suppress again the obvious modifications that arise when the puncture p_{m} is at z_{m} = ∞.

(Here we choose some definite branch of the logarithm. It is readily checked that all the results below do not depend on this choice. For z_{u} ≠ zu^{[0]}, g_{s_{∘},u} can be considered as an element of ℂ((t)), for which the constant part is defined only modulo 2πiℤ.)

Now recall from section 5 that associated to the automorphism σ_{μ̄} of 𝔤^{m} there comes the map Θμ̄ that for each tensor factor of H⃗ is defined as in (2.19). This map is present for any z⃗ ∈ M; moreover, according to the relation (6.8) the maps Θμ̄(z⃗) and Θμ̄(z^{[0]}) are related by an analogous map that is associated to the inner automorphism δ_{μ̄}. Using the results (2.22) and (6.11) we learn that explicitly we have

Θ_{μ̄;s_{∘}}(z⃗) = exp ((μ̄_{u}, H)^{⊗}g_{s_{∘},u} + Kh̃_{s_{∘}}) ∘ Θ_{μ̄;s_{∘}}(z^{[0]}) for s_{∘} ≠ u ,  (6.13)

respectively

Θ_{μ̄;s_{∘}}(z⃗) = exp ( ∑_{{s=1 \atop s≠u}}^{m}(μ̄_{s}, H)^{⊗}g̃_{s} + Kh̃_{u}) ∘ Θ_{μ̄;s_{∘}}(z^{[0]}) for s_{∘} = u .  (6.14)

At this stage the functions h̃_{s_{∘}}, which are introduced here to take care of the fact that the implementation is only determined up to a phase, can still be chosen arbitrarily; we will make use of this freedom later on.

Combining these results with the twisted intertwining relations (2.34), we can show that the commutator [∇_{u}, Θ⃗μ̄(z⃗)] has the local expansions

[∇_{u}, Θ_{μ̄;s_{∘}}(z⃗)]|_{z_{u}=z_{u}^{[0]}} = (−(μ̄_{u}, H)^{⊗} φs_{∘},u^{[0]} + K ĝ_{s_{∘}}) Θ_{μ̄;s_{∘}}(z^{[0]}) ,  (6.15)

where

ĝ_{s_{∘}} = \frac{∂}{∂z_{u}} h̃_{u}|_{z_{u}=z_{u}^{[0]}} − ∑_{{s=1 \atop s≠u}}^{m} \frac{(μ̄_{s},μ̄_{u})}{z_{u}^{[0]} − z_{s}^{[0]}}  for s_{∘} = u ,  (6.16)

while ĝ_{s_{∘}} = \frac{∂}{∂z_{u}} h̃_{s_{∘}}|_{z_{u}=z_{u}^{[0]}} for s_{∘} ≠ u. (The derivation of (6.15) is explained in some detail in appendix C.) Collecting these local expansions, we learn that up to central terms we simply have

[∇_{u}, Θμ̄(z^{[0]})] = Y_{μ̄;u} Θμ̄(z^{[0]}) ,  (6.17)

where

Y_{μ̄;u} := − (μ̄_{u}, H)^{⊗} (z − zu^{[0]})^{−1}  (6.18)

is an element of the block algebra. Concerning the central terms, we claim that we can employ the freedom that is present in the choice of the functions h̃_{s_{∘}} so as to achieve

ĝ_{s_{∘}} ≡ 0 for all p⃗ ∈ M .  (6.19)

Clearly, this is possible at the point p⃗ = p^{[0]}, by just choosing ^{8}

h̃_{s_{∘}} ≡ 0 for s_{∘} ≠ u and h̃_{u} = ∑_{{s=1 \atop s≠u}}^{m}(μ̄_{s}, μ̄_{u}) ln \frac{z_{u} − z_{s}}{zu^{[0]} − zs^{[0]}} .  (6.20)

To discuss how the situation looks like globally, we first have to generalize the formula (6.15) to the case where all punctures may vary. By analogous considerations as above one can check that the generalization of (6.13) and (6.14) reads

Θ_{μ̄;s_{∘}}(z⃗) = exp ( ∑_{s=1}^{m}(μ̄_{s}, H)^{⊗} g_{s_{∘},s} + Kh_{s_{∘}}) ∘ Θ_{μ̄;s_{∘}}(z^{[0]})  (6.21)

for all s_{∘}, where

g_{s_{∘},s}(t) := ln \frac{t + z_{s_{0}} − z_{s}}{t + zs_{0}^{[0]} − zs^{[0]}} ,  (6.22)

which for all s_{∘} and all s is a sensible Laurent series. (Also note that the notation g_{s_{∘},u} is in agreement with the definition (6.12), and g_{u,s} = g̃_{s}; the functions h_{s_{∘}} are still to be determined.) The same arguments as in appendix C then lead to a formula analogous to (6.15),

[∇_{u}, Θ_{μ̄;s_{∘}}(z⃗)]|_{z⃗=z^{[0]}} = (−(μ̄_{u}, H)^{⊗} φs_{∘},u^{[0]} + K ĝ_{s_{∘}}) Θ_{μ̄;s_{∘}}(z^{[0]}) ,  (6.23)

with

ĝ_{s_{∘}} = \frac{∂}{∂z_{u}} h_{s_{∘}}|_{z⃗=z^{[0]}} − ∑_{{s=1 \atop s≠u}}^{m} \frac{(μ̄_{s},μ̄_{u})}{z_{u}^{[0]} − z_{s}^{[0]}}  (6.24)

for all s_{∘}. Our aim is now to choose the functions h_{s_{∘}} is such a way that in the tensor product Θ⃗μ̄ the terms proportional to K in the exponent cancel so that we are again left with an element of the block algebra. Thus we need again ∑_{s_{∘}} ĝ_{s_{∘}} ≡ 0, i.e.

∑_{s_{∘}=1}^{m} \frac{∂}{∂z_{u}} h_{s_{∘}} = ∑_{{s=1 \atop s≠u}}^{m} \frac{(μ̄_{s},μ̄_{u})}{z_{u} − z_{s}}  (6.25)

(note that the appearance of the summation is in agreement with the fact that 𝔤^{m} is obtained from a direct sum of affine Lie algebras by identifying the centers). The choice

h_{s_{∘}}(z⃗) = \frac{1}{2} ∑_{{s=1 \atop s≠s∘}}^{m}(μ̄_{s_{∘}}, μ̄_{s}) ln \frac{z_{s_{∘}} − z_{s}}{zs_{∘}^{[0]} − zs^{[0]}}  (6.26)

8 Here the freedom in the choice of the branch of the logarithm is not completely irrelevant. But since the level is always integral while the inner product (μ̄_{s}, μ̄_{u}) is rational, the twisted intertwiner Θ_{μ̄;s_{∘}} is determined only up to multiplication by a root of unity, and the presence of this root of unity does not destroy the important property of Θ_{μ̄;s_{∘}} to have finite order. Moreover, the precise value of the order really matters only in the case of fixed points, and closer inspection shows that in that case the number (μ̄_{s}, μ̄_{u}) is in fact always an integer so that the order is not changed at all.

for all s_{∘} = 1, 2, ... , m. indeed satisfies this requirement. Note that the requirement (6.24) determines h_{s_{∘}} only up to a z⃗-independent constant; the choice made here will be convenient later on.^{9} We conclude that we can choose the phase of Θ⃗μ̄(z⃗) in such a manner that the Knizhnik--Zamolodchikov connection is preserved at every point of the moduli space.

6.3 Transformation properties of flat sections

We are now in a position to investigate the bundle of chiral blocks and the transformation properties of those sections in the bundle which are flat with respect to the Knizhnik--Zamo-lodchikov connection. To this end we consider the trivial bundle

H⃗^{⋆} × M → M ,  (6.27)

where the fiber H⃗^{⋆} is the algebraic dual of H⃗, and in this bundle the subbundle B of chiral blocks which are the singlets under the (dual) action of the block algebra. This implies that the smooth sections β(z⃗) of B obey

⟨β(z⃗) , X(z⃗)v(z⃗)⟩ = 0  (6.28)

for all sections X(z⃗) of the bundle of block algebras and all sections v(z⃗) of the trivial bundle (6.5). This bundle B is of finite rank [6]. We endow the trivial bundle (6.27) with the connection that is dual to the connection (6.6); it restricts to a connection on the subbundle B. We will use the term Knizhnik--Zamolodchikov connection also for both the connection on the dual bundle and for its restriction to B. (Note, however, that a frequent convention in the literature is to reserve the term Knizhnik--Zamolodchikov connection only for the connection on B.)

An action of the group Γ_{fix} on B can be defined by

⟨Θ⃗⋆ ; μ̄(β(z⃗)) , v(z⃗)⟩ := ⟨β(z⃗) , Θ⃗−1 ; μ̄v(z⃗)⟩  (6.29)

for all sections v(z⃗) (recall that Θ⃗⋆ ; μ̄(β) is again a chiral block). We have

\begin{aligned}⟨∇_{s}(Θ⃗⋆ ; μ̄β(z⃗)) , v(z⃗)⟩ & = ∂_{s}⟨Θ⃗⋆ ; μ̄β(z⃗) , v(z⃗)⟩ − ⟨Θ⃗⋆ ; μ̄β(z⃗) , ∇_{s}v(z⃗)⟩ \\ & = ∂_{s}⟨β(z⃗) , Θ⃗−1 ; μ̄v(z⃗)⟩ − ⟨β(z⃗) , Θ⃗−1 ; μ̄∇_{s}v(z⃗)⟩ . \end{aligned}  (6.30)

Using the result (6.17), this can also be written as

\begin{aligned}⟨∇_{s}(Θ⃗⋆ ; μ̄β(z⃗)) , v(z⃗)⟩ & = ∂_{s}⟨β(z⃗) , Θ⃗−1 ; μ̄v(z⃗)⟩ − ⟨β(z⃗) , ∇_{s}Θ⃗−1 ; μ̄v(z⃗)⟩ + ⟨β(z⃗) , Y_{−μ̄;s} Θ⃗−1 ; μ̄v(z⃗)⟩ \\ & = ∂_{s}⟨β(z⃗) , Θ⃗−1 ; μ̄v(z⃗)⟩ − ⟨β(z⃗) , ∇_{s}Θ⃗−1 ; μ̄v(z⃗)⟩ , \end{aligned} (6.31)

where in the second line we used the fact that according to the definition (6.18), Y_{−μ̄;s} is an element of the block algebra. Note that Θ⃗μ̄(z⃗) depends smoothly on z⃗, so that Θ⃗−1 ; μ̄(z⃗)v(z⃗) is a smooth section as well. Let now β(z⃗) be a flat section in B,^{10} i.e. ∇β = 0, or more explicitly,

0 = ⟨∇_{s}β(z⃗) , v(z⃗)⟩ = ∂_{s}⟨β(z⃗) , v(z⃗)⟩ − ⟨β(z⃗) , ∇_{s}v(z⃗)⟩  (6.32)

9 Also note that in the general case considered here there is of course less freedom in the choice of these functions than we had for the analogous functions h̃_{s_{∘}} in the special case treated before. In particular, the simple choice made in (6.20) is no longer available.

10 Sometimes in the literature the term ‘chiral block’ is reserved for such flat sections.

for all smooth sections v(z⃗) and all s = 1, 2, ... , m. Applying this formula to the smooth section Θ⃗−1 ; μ̄(z⃗)v(z⃗), we see from (6.31) that

⟨∇_{s}(Θ⃗⋆ ; μ̄(z⃗)β(z⃗)) , v(z⃗)⟩ = 0 .  (6.33)

This means that together with β(z⃗) also the section Θ⃗⋆ ; μ̄(z⃗)β(z⃗) is flat.

6.4 Verification of the group law of Γ_{w}

In subsection 5.1 we have seen that for every given value z⃗ of the moduli the maps Θ⃗μ̄(z⃗) that implement the automorphisms σ⃗_{μ̄} on the 𝔤^{m}-modules can be chosen such that they respect the group law of Γ_{w}. Suppose we have made this choice for some point z^{[0]} ∈ M. To extend Θ⃗μ̄ to other values of the moduli, we have imposed the requirement that the extension should be compatible with the Knizhnik--Zamolodchikov connection. This requirement can only be satisfied when the implementation Θ⃗μ̄(z⃗) is chosen in a suitable manner. We will now show that the specific implementation Θ⃗μ̄(z⃗) that we have already chosen above for each μ̄ ∈ Γ_{w} still respects the group law of Γ_{w} at every point in M.

To start, we recall the formula (6.21), i.e. Θ_{μ̄;s_{∘}}(z⃗) = A_{μ̄;s_{∘}}(z⃗; z^{[0]}) Θ_{μ̄;s_{∘}}(z^{[0]}), with

A_{μ̄;s_{∘}}(z⃗; z^{[0]}) = exp ( ∑_{s=1}^{m}(μ̄_{s}, H)^{⊗}g_{s_{∘},s} + Kh_{s_{∘}}) ;  (6.34)

here g_{s_{∘},s}(t; z⃗) and h_{s_{∘}}(z⃗; μ̄) are defined by (6.22) and (6.26), respectively. It follows that

\begin{aligned}Θ_{μ̄+ν̄;s_{∘}}(z⃗) & = A_{μ̄+ν̄;s_{∘}}(z⃗; z^{[0]}) ∘ Θ_{μ̄+ν̄;s_{∘}}(z^{[0]}) \\ & = A_{μ̄;s_{∘}}(z⃗; z^{[0]}) ∘ exp ( ∑_{s=1}^{m}(ν̄_{s}, H)g_{s_{∘},s} + K [h_{s_{∘}}(z⃗; μ̄+ν̄) − h_{s_{∘}}(z⃗; μ̄)]) \\ & ∘ Θ_{μ̄;s_{∘}}(z^{[0]}) Θ_{ν̄;s_{∘}}(z^{[0]}) . \end{aligned} (6.35)

By commuting the exponential through Θ_{μ̄;s_{∘}}(z^{[0]}) we arrive at a similar exponential, but with (σμ̄^{[0]})^{−1} applied to the argument. With the help of the identity (D.28) we then get explicitly

Θ_{μ̄+ν̄;s_{∘}}(z⃗) = A_{μ̄;s_{∘}}(z⃗; z^{[0]}) ∘ Θ_{μ̄;s_{∘}}(z^{[0]}) ∘ exp(Ỹ_{s_{∘}}) ∘ Θ_{ν̄;s_{∘}}(z^{[0]})  (6.36)

with

Ỹ_{s_{∘}} := ∑_{s}(ν̄_{s}, H) g_{s_{∘},s} + K ( − ∑_{s,s′}(μ̄_{s′}, ν̄_{s}) Res(φ_{s_{∘},s′}g_{s_{∘},s})

+ \frac{1}{2} ∑_{{s \atop s≠s∘}} ln \frac{z_{s∘}−z_{s}}{z_{s∘}^{[0]}−z_{s}^{[0]}}[(μ̄_{s_{∘}}+ν̄_{s_{∘}}, μ̄_{s}+ν̄_{s}) − (μ̄_{s_{∘}}, μ̄_{s})])

= ∑_{s}(ν̄_{s}, H) g_{s_{∘},s} + \frac{1}{2} K ∑_{{s \atop s≠s∘}} ln \frac{z_{s∘}−z_{s}}{z_{s∘}^{[0]}−z_{s}^{[0]}} (ν̄_{s_{∘}}, ν̄_{s})

+ \frac{1}{2} K ∑_{{s \atop s≠s∘}} ln \frac{z_{s∘}−z_{s}}{z_{s∘}^{[0]}−z_{s}^{[0]}} [(ν̄_{s_{∘}}, μ̄_{s}) − (μ̄_{s_{∘}}, ν̄_{s})] .

(6.37)

Upon exponentiation, the terms in the first line of the last expression just yield the correct factor A_{ν̄;s_{∘}}(z⃗; z^{[0]}), while the rest of the terms amount to an additional phase. Now this is the situation at the puncture s_{∘}; taking into account that the centers of the affine algebras 𝔤_{s} are identified in 𝔤^{m}, we finally have to add up the prefactors of K for all insertion points s_{∘} = 1, 2, ... , m. Doing so, the two sums in the prefactor cancel each other,

∑_{{s∘,s \atop s≠s∘}} ln \frac{z_{s∘}−z_{s}}{z_{s∘}^{[0]}−z_{s}^{[0]}} [(μ̄_{s_{∘}}, ν̄_{s}) − (ν̄_{s_{∘}}, μ̄_{s})] = 0 .  (6.38)

As a consequence, we have

\begin{aligned}Θ⃗_{μ̄+ν̄}(z⃗) & = A_{μ̄+ν̄}(z⃗; z^{[0]}) Θ⃗ν̄(z^{[0]}) \\ & = Aμ̄(z⃗; z^{[0]})Θ⃗μ̄(z^{[0]}) ∘ Aν̄(z⃗; z^{[0]})Θ⃗ν̄(z^{[0]}) = Θ⃗μ̄(z⃗) ∘ Θ⃗ν̄(z⃗) . \end{aligned}  (6.39)

We conclude that, as claimed, with our chosen implementation the group law of Γ_{w} is respected at every point of the moduli space M.

Moreover, using the relation

exp (\frac{iπ}{2} (Eᾱ_{s}^{⊗}φ_{s_{∘},s} + E^{−ᾱ_{s}}^{⊗}φ_{s_{∘},s}^{−1}))

= exp (−\frac{1}{2} Hᾱ_{s}^{⊗}g_{s_{∘},s}) exp (\frac{iπ}{2} (Eᾱ_{s}^{⊗}φs_{∘},s^{[0]} + E^{−ᾱ_{s}}^{⊗}(φs_{∘},s^{[0]})^{−1})) exp (\frac{1}{2} Hᾱ_{s}^{⊗}g_{s_{∘},s}) ,

(6.40)

one can show that the preferred implementation of inner multi-shift automorphisms introduced in subsection 5.1 (see (5.7)) obeys

Θ̃_{μ̄;s_{∘}}(z⃗) = A_{μ̄;s_{∘}}(z⃗; z^{[0]}) Θ̃_{μ̄;s_{∘}}(z^{[0]})  (6.41)

precisely as in (6.21), which implies that at every point in the moduli space this implementation is compatible with the Knizhnik--Zamolodchikov connection. (For details, we refer to appendix D.) We conclude in particular that in each fiber of the bundle B the map Θ⃗⋆ ; μ̄ has finite order.

7 Fixed point resolution

7.1 Fixed points

In the previous section we have constructed an action of the group Γ_{w} on the tensor product H⃗ in such a way that for each μ̄ ∈ Γ the twisted intertwiner Θ⃗μ̄ is represented by the product of exponentials of elements in the block algebra 𝔤̄⊗F ≡ 𝔤̄⊗F(z^{[0]}). As seen in subsection 5.3 this has in particular the consequence that in all fibers over the moduli space M_{m} the induced maps on the space B ≡ B(z⃗) of chiral blocks have finite order and realize the group law of the finite abelian group Γ_{w}/Γ.

A particularly interesting situation is the one where the automorphism σ_{μ̄} is not inner, but still does not change the isomorphism class of a module H_{Λ}. More precisely, given an m-tuple Λ⃗ of integrable weights of the affine Lie algebra 𝔤, we associate to it the subgroup of Γ_{w} that leaves each of the irreducible highest weight modules H_{Λ_{s}} invariant up to isomorphism. This subgroup is precisely the stabilizer Γ_{Λ⃗} of Λ⃗ as defined in (6.7). For every m-tuple Λ⃗ the stabilizer Γ_{Λ⃗} definitely contains Γ as a subgroup. If it is larger than Γ, then it also contains elements which do not necessarily act as a multiple of the identity on the blocks. In this case we call the m-tuple Λ⃗ of 𝔤-weights a fixed point.

The results of subsection 5.3 tell us that each fiber of the vector bundle B of chiral blocks can be split into finitely many eigenspaces B^{ψ} under the action of the group Γ_{Λ⃗}, or rather, of the quotient by its subgroup Γ. Such an eigenspace is labelled by its eigenvalue ψ, which is a character of the group Γ_{Λ⃗}/Γ. Since the Knizhnik--Zamolodchikov connection is preserved under the map Θ⃗⋆ ; μ̄, it restricts to the various eigenspaces and they fit together into sub-vector bundles of B. In particular, the dimension of the eigenspaces B^{ψ} does not depend on the moduli. Hence as soon as Γ_{Λ⃗} is smaller than Γ, i.e. for fixed points, the bundle B of chiral blocks is reducible (as a vector bundle). To actually establish this fact, we have to show that the ranks of at least two subbundles B^{ψ} are non-zero, which in fact follows from the conjectured formula for the rank to be discussed below. We refer to the decomposition of the bundle B into the subbundles of eigenspaces under Θ⃗μ̄ as fixed point resolution. Our results imply that the Knizhnik--Zamolodchikov connection consistently restricts to these subbundles; thus even after fixed point resolution we are still given a Knizhnik--Zamolodchikov connection. The present situation should be compared with the situation in coset conformal field theories, where it is known [12] that the characters, i.e. the zero-point blocks on the torus, of fixed points decompose in a similar way under the action of an outer automorphism. This structural analogy should also explain the striking similarities in the modular matrices for extension modular invariants [8] and coset conformal field theories [12].

We pause for a side remark. One might be tempted to speculate that the converse is also true, i.e. that the bundle B splits into a direct sum of subbundles if and only if fixed points are involved. This, however, does not seem to be true, as the following example at higher genus shows. The bundle of zero-point blocks on the torus is given by the characters of the theory, and the representation of the mapping class group on this bundle is just given by the usual modular group representation on the characters. This bundle definitely does not involve fixed points. On the other hand it is known that the representation of the modular group is reducible if the theory contains non-trivial simple currents. (One well known example for this phenomenon is the fact that in superconformal field theories the character-valued indices of fields in the Ramond sector span a closed subspace under modular transformations.)

7.2 Trace formulæ

We have seen that the dimensions of the eigenspaces B^{ψ} do not depend on the moduli. In this subsection we present a conjecture for a general formula for these dimensions. Using Fourier transformation over the finite abelian group Γ_{w}/Γ, these dimensions are related to the traces of Θ⃗⋆ ; μ̄ on the fibers,

dim B^{ψ} = |Γ_{w}/Γ|^{−1} ∑_{ω⃗∈Γ_{w}/Γ} ψ^{∗}(ω⃗) tr_{B}Θ⃗⋆ ; μ̄ .  (7.1)

This identity is to be understood as follows. For every element ω⃗ in the coset Γ_{w}/Γ we choose a representative μ̄ ∈ Γ_{w}. But since the action of Θ⃗⋆ ; μ̄ is independent of the choice of representative, the expression is well-defined. In a first step, it should be regarded as an identity which holds in each fiber separately. Since the dimension dim B^{ψ} is independent of the choice of the fiber, by inverse Fourier transformation one concludes that the traces also do not depend on the point in moduli space over which the trace is taken.

It will be convenient not to write down directly a formula for the dimension of the eigenspace, but to present instead the trace tr_{B}Θ⃗⋆ ; μ̄ of the implementation on the space of chiral blocks. This trace depends on μ̄ ∈ Γ_{Λ⃗} only modulo Γ. Now recall that ω_{μ̄;s} denotes the diagram automorphism that is in the same class as σ_{μ̄;s}. We have also seen that the latter is in the same class as the ordinary single-shift automorphism σμ̄_{s}^{(0)}; accordingly, instead of ω_{μ̄;s} we may also use the notation ω_{μ̄_{s}} or, for brevity ω_{s}. The condition that ∑_{s=1}^{m} μ̄_{s} = 0 tells us that we have

∏_{s=1}^{m} ω_{s} = id .  (7.2)

(As diagram automorphisms are in one-to-one correspondence with classes of outer automorphisms, we identify the m-tuple ω⃗ of outer automorphisms with the corresponding element of the coset Γ_{w}/Γ, in agreement with the notation already chosen in (7.1).)

Now the results of [8] for the S-matrix of integer spin simple current modular invariants suggest, when combined with the ordinary Verlinde formula, that in the case m = 3 the traces

T_{Λ⃗;ω⃗} ≡ T_{Λ⃗;σ⃗_{μ̄}} := tr_{BΛ⃗} Θ⃗⋆ ; μ̄  (7.3)

should be given by

T_{Λ⃗;ω⃗} = ∑_{Λ′} \frac{S_{Λ_{1},Λ′}^{ω_{1}}S_{Λ_{2},Λ′}^{ω_{2}}S_{Λ_{3},Λ′}^{ω_{3}}}{S_{Ω,Λ′}} .  (7.4)

Let us explain the notation employed in this formula. The summation extends over all integrable 𝔤-weights at level k^{∨} which are fixed under each of the permutations ω_{s}^{⋆} for s = 1, 2, 3; Ω is the label for the vacuum primary field, while S_{Λ,Λ′} denotes the entries of the modular S-matrix of the WZW theory based on 𝔤, which is given by the Kac--Peterson formula [28]

S_{Λ,Λ′} = N ∑_{w̄∈\overline{W}} sign(w̄) exp [ − \frac{2πi}{k^{∨}+g^{∨}} (w̄(Λ̄ + ρ̄), Λ̄′ + ρ̄)] .  (7.5)

And finally, S_{Λ_{,}Λ′}^{ω} are the entries of the modular S-matrix for some other WZW theory. Namely, to each pair consisting of an affine Lie algebra 𝔤 and a diagram automorphism ω of 𝔤 one can associate another affine Lie algebra 𝔤^{ω}, the so-called orbit Lie algebra; S^{ω} is just given by the Kac--Peterson formula for 𝔤^{ω}. (For more details, in particular on how the fixed point weights Λ_{s} are to be interpreted as weights of 𝔤_{s}^{ω}, see [22,29]. For convenience, we have also listed in table 1 the orbit Lie algebras^{11} of untwisted affine Lie algebras with respect to all relevant diagram automorphisms.)

It is a remarkable empirical observation that, in all cases that have been checked numerically, the expression (7.4) gives an integral (though not necessarily non-negative) result.^{12} Notice Table 1: Orbit Lie algebras of untwisted affine Lie algebras [22].

11 B̃_{n}^{(2)} stands for the unique series of twisted affine Lie algebras whose characters furnish a module of the modular group and which have simple roots of three different lengths.

12 The relevant calculations have been performed with the program Kac which has been written by A.N.

Schellekens and is available at http://norma.nikhef.nl/˜ t58/kac.html.

Helpful discussions with Bert Schellekens are gratefully acknowledged.

\begin{matrix}𝔤 & ω & N & 𝔤^{ω} \\ A_{n−1}^{(1)} & ω_{n} & n & \{0\} \\ A_{n−1}^{(1)} & (ω_{n})^{n/N} & N < n & A_{(n/N)−1}^{(1)} \\ B_{n}^{(1)} & ω & 2 & B̃_{n−1}^{(2)} \\ C_{2}^{(1)} & ω & 2 & A_{1}^{(2)} \\ C_{2n}^{(1)} & ω & 2 & B̃_{n}^{(2)} \\ C_{2n+1}^{(1)} & ω & 2 & C_{n}^{(1)} \\ D_{n}^{(1)} & ω_{v} & 2 & C_{n−2}^{(1)} \\ D_{2n}^{(1)} & ω_{s} & 2 & B_{n}^{(1)} \\ D_{2n+1}^{(1)} & ω_{s} & 4 & C_{n−1}^{(1)} \\ E_{6}^{(1)} & ω & 3 & G_{2}^{(1)} \\ E_{7}^{(1)} & ω & 2 & F_{4}^{(1)} \end{matrix}

that this property is much stronger than the obvious requirement that the dimensions dim B^{ψ} of the eigenspaces must be integral. (Also, when the constraint (7.2) is not satisfied, then the expression on the right hand side of (7.4) is zero.)

We conjecture that the formula (7.4) indeed holds true and, moreover, that it directly generalizes to more complicated cases. To prepare the ground for this generalization, we write (7.4) in the equivalent form

T_{Λ⃗;ω⃗} = ∑_{Λ′} \frac{S_{Λ_{1},Λ′}^{ω_{1}}}{S_{Ω,Λ′}} · \frac{S_{Λ_{2},Λ′}^{ω_{2}}}{S_{Ω,Λ′}} · \frac{S_{Λ_{3},Λ′}^{ω_{3}}}{S_{Ω,Λ′}} · |S_{Ω,Λ′}|^{2} .  (7.6)

The generalization to all m ≥ 3 then simply consists in replacing the product ∏_{s=1}^{3} S_{Λ_{s},Λ′}^{ω_{s}}/S_{Ω,Λ′} by the analogous product ∏_{s=1}^{m} S_{Λ_{s},Λ′}^{ω_{s}}/S_{Ω,Λ′}.

7.3 Curves of higher genus

The last factor on the right hand side of the formula (7.6) should find its heuristic interpretation as a Weyl integration volume at genus zero (compare e.g. the derivation of the Verlinde formula in the framework of Chern--Simons theory [30]). Accordingly, for a surface of genus g we expect that the exponent gets replaced by the Euler number χ = 2 − 2g.

We have already seen in subsection 5.4 that with the appropriate modifications most of our constructions should generalize to curves of arbitrary genus. In contrast, it is not clear whether the results of section 4 can be generalized as well. We will see, however, that this issue is not really essential for our purposes. Since the group law of Γ_{w} is preserved, the only ambiguity in the implementation of an automorphism σ⃗_{μ̄} is a phase ψ(μ̄). As a consequence, every element of Γ is implemented on the space of chiral blocks by the multiple ψ(μ̄)·id of the identity, and the phase ψ(μ̄) must furnish a character of the free abelian group Γ. Moreover, it follows from the prescription (6.21) that the character depends smoothly on the modulus z⃗:

Θ⃗⋆ ; μ̄(z⃗) = ψ(μ̄; z⃗) id .  (7.7)

In this context we should keep in mind that from the point of view of conformal field theory it is more natural to define the chiral blocks as a projective space rather than as a vector space, i.e. a global phase does not possess any physical meaning. Indeed, at higher genus the Knizhnik--Zamolodchikov connection is not flat any longer, but only projectively flat. Accordingly all our considerations perfectly make sense for projective spaces of chiral blocks, and hence in particular we get a natural identification of the projective bundles obtained from the maps

Θ⃗⋆ ; μ̄ : BΛ⃗ → B_{ω^{⋆}Λ⃗} .  (7.8)

The eigenvalues are labelled by a moduli dependent character ψ(z⃗) of Γ_{Λ⃗}, but this character does not factorize necessarily any longer to a character of the quotient Γ_{Λ⃗}/Γ. But still, these characters must depend smoothly on the moduli, and also the corresponding eigenspaces depend smoothly on the moduli. In particular, the dimension of the eigenspaces B^{ψ} does not depend on the moduli, although the eigenvalue itself potentially does. Still, we prefer to work with the Fourier transform

T_{Λ⃗;ω⃗}^{g} := ∑_{ψ∈(Γ_{w}/Γ)^{⋆}} ψ(ω) dim B^{ψ}  (7.9)

of the dimension, which by abuse of language we will call the trace of Θ⃗⋆ ; μ̄ (actually, it can differ from the trace by a phase).

For a surface of any genus g the expression for the traces analogous to (7.4) is independent on the moduli, and we expect it to be

T_{Λ⃗;ω⃗}^{g} = ∑_{Λ′} |S_{Ω,Λ′}|^{2−2g} ∏_{s=1}^{m} \frac{S_{Λ_{s},Λ′}^{ω_{s}}}{S_{Ω,Λ′}} .  (7.10)

As non-trivial evidence for this formula, we show that it is consistent with factorization. Namely, suppose a curve C of genus g develops an ordinary double point and that its normal resolution consists of two curves C^{(1)}, C^{(2)} of genus g_{1} and g_{2}, respectively, with g_{1}+g_{2} = g. We assume that for i = 1, 2 the insertion points on C^{(i)} correspond to integrable 𝔤-weights Λ⃗^{(i)} and that we want to implement an m_{i}-tuple of automorphisms σμ̄^{(i)} satisfying ω^{⋆}^{(i)}Λ⃗^{(i)} = Λ⃗^{(i)}. According to (7.2) we require that ∏_{s=1}^{m_{1}} ωs^{(1)} · ∏_{s=m_{1}+1}^{m_{1}+m_{2}} ωs^{(2)} = id on the curve C, which implies that

ω_{∘} := ∏_{s=m_{1}+1}^{m_{1}+m_{2}} ωs^{(2)} = ( ∏_{s=1}^{m_{1}} ωs^{(1)})^{−1} .  (7.11)

We claim that the identity

T_{Λ⃗^{(1)},Λ⃗^{(2)};ω^{(1)},ω^{(2)}}^{g_{1}+g_{2}} = ∑_{Λ_{∘} fix} T_{Λ⃗^{(1)},Λ_{∘};ω^{(1)},ω_{∘}}^{g_{1}} · T_{Λ⃗^{(2)},Λ_{∘}^{+};ω^{(2)},ω_{∘}^{−1}}^{g_{2}}  (7.12)

holds, where the sum is over fixed points of ω_{∘}. Note that as a consequence of (7.11) the product of the automorphisms on each of the two components C^{(i)} of the normalization separately fulfils the constraint (7.2). An important consequence of (7.12) is that all traces for any genus will be integral if the traces (7.4) on the three-point blocks on a curve of genus zero are integral. We also expect the factorization rule (7.12) to play in important role for a rigorous proof of the consistency of the boundary conditions described in [15] on higher genus surfaces.

As for the proof of (7.12), we remark that the right hand side can be written as

T_{Λ⃗^{(1)},Λ′;ω^{(1)},ω_{∘}}^{g_{1}}· T_{Λ⃗^{(2)},Λ′^{+};ω^{(2)},ω_{∘}^{−1}}^{g_{2}}= ∑_{κ: ω_{∘}^{⋆}κ=κ}( ∑_{Λ′} |S_{Ω,Λ′}|^{2−2g_{1}} · \frac{S_{κ,Λ′}^{ω_{∘}}}{S_{Ω,Λ′}} ∏_{s=1}^{m_{1}} \frac{S_{Λ_{s},Λ′}^{ω_{s}}}{S_{Ω,Λ′}})

· ( ∑_{Λ′′} |S_{Ω,Λ′′}|^{2−2g_{2}} · \frac{S_{κ^{+},Λ′′}^{ω_{∘}^{−1}}}{S_{Ω,Λ′′}} ∏_{s=m_{1}+1}^{m_{1}+m_{2}} \frac{S_{Λ_{s},Λ′′}^{ω_{s}}}{S_{Ω,Λ′′}}) .

(7.13)

Using the fact [22] that S^{ω} = S^{ω^{−1}} as well as unitarity of this matrix (which follows from the Kac--Peterson formula (7.5)), the summation over κ can be carried out; yielding a factor of (S_{Ω,Λ′})^{−2}δ_{Λ′,Λ′′}. Then the summation over, say, Λ′′ is immediate and leads to

T_{Λ⃗^{(1)},Λ′;ω^{(1)},ω_{∘}}^{g_{1}} T_{Λ⃗^{(2)},Λ′^{+};ω^{(2)},ω_{∘}^{−1}}^{g_{2}} = ∑_{Λ′} |S_{Ω,Λ′}|^{2−2g_{1}−2g_{2}} ∏_{s=1}^{m_{1}+m_{2}} \frac{S_{Λ_{s},Λ′}^{ω_{s}}}{S_{Ω,Λ′}} = T_{Λ⃗^{(1)},Λ⃗^{(2)};ω^{(1)},ω^{(2)}}^{g_{1}+g_{2}} . (7.14)

This finishes the proof of (7.12). Unfortunately, a complete proof of (7.10) is not known at present and remains a challenge for future work. Note that such a proof would in particular include a proof of the ordinary Verlinde formula for WZW theories, namely for the special case when ω_{s} = id for all s.

Apart from the factorization property, there is another aspect of the formula (7.10) which may be regarded as evidence that the conjecture is indeed correct. Namely, the formula expresses the traces exclusively through matrices, namely S ≡ S^{id} and the various S^{ω}, which all share the standard properties (such as furnishing a unitary representation of the modular group and satisfying S_{Ω,Λ}^{ω} > 0) that are familiar from conformal field theory. It can therefore be expected that the eigenspaces are the correct building blocks for the spaces of chiral blocks of WZW theories based on non-simply connected groups, and that as a consequence of the properties of the modular matrices S^{ω} the Verlinde formula will hold automatically in these conformal field theories as well. Work on these issues is in progress.

A The Virasoro algebra

In this appendix we collect some information about multi-shift automorphisms of the Virasoro algebra and its semidirect sum with the affine Lie algebra 𝔤.

We first show that for an arbitrary automorphism σ of an untwisted affine Lie algebra 𝔤, the extension to the semi-direct sum with the Virasoro algebra is unique, if it exists. To see this, consider two maps σ_{i}: Vir → Vir ⊕ 𝔤, i = 1, 2, such that both (σ, σ_{1}) and (σ, σ_{2}) furnish an automorphism of the semi-direct sum Vir ⊕ 𝔤. Then we have

[σ_{1}(L_{n}), σ(x_{m})] = −m σ(x_{m+n}) = [σ_{2}(L_{n}), σ(x_{m})] ,  (A.1)

from which we learn that for every n the combination σ_{1}(L_{n}) − σ_{2}(L_{n}) commutes with all of 𝔤, so that σ_{1}(L_{n}) − σ_{2}(L_{n}) = ξ_{n}K + η_{n}C with ξ_{n}, η_{n} ∈ ℂ for n ∈ ℤ. But this in turn implies that [σ_{1}(L_{n}), σ_{1}(L_{m})] = [σ_{2}(L_{n}), σ_{2}(L_{m})], or more explicitly

(n − m) σ_{1}(L_{n+m}) + \frac{1}{24} (n^{3} − n) δ_{n+m,0} C = (n − m) σ_{2}(L_{n+m}) + \frac{1}{24} (n^{3} − n) δ_{n+m,0} C . (A.2)

This finally implies that ξ_{n} = 0 = η_{n} for all n, and hence σ_{1} = σ_{2} as claimed.

Next we check that the prescription (2.32) indeed provides us with an extension of the automorphism σ_{μ̄;s_{∘}} of 𝔤 as defined in (2.6). We first verify that the relation (2.31) is preserved when x̄ = H^{i}. We have

[σ_{μ̄;s_{∘}}(L_{n}), σ_{μ̄;s_{∘}}(H^{i}^{⊗}f)] = [L_{n} + ∑_{ℓ∈ℤ} ∑_{s}(μ̄_{s}, H_{n−ℓ})^{⊗}Res(t^{ℓ}φ_{s_{∘},s}), H^{i}^{⊗}f]

= −H^{i} ^{⊗} t^{n+1}df + K ∑_{ℓ∈ℤ} ∑_{s} μ̄_{s}^{i} Res(t^{ℓ}φ_{s_{∘},s}) Res(dt^{n−ℓ} f) . (A.3)

Integrating by parts within the second residue and using the identity

Res(fg) = ∑_{ℓ∈ℤ}Res(t^{ℓ−1}f) Res(t^{−ℓ}g) for f, g ∈ ℂ((t))  (A.4)

(which follows immediately by substituting f and g by their Laurent expansions), this reduces to

\begin{aligned}[σ_{μ̄;s_{∘}}(L_{n}), σ_{μ̄;s_{∘}}(H^{i}^{⊗}f)] & = −H^{i} ^{⊗} t^{n+1}df − K ∑_{s} μ̄_{s}^{i} Res(t^{n+1}φ_{s_{∘},s} df) \\ & = −σ_{μ̄;s_{∘}}(H^{i}^{⊗}t^{n+1}df) = σ_{μ̄;s_{∘}}([L_{n}, H^{i}^{⊗}f]) , \end{aligned}  (A.5)

which is the desired result. Similarly, with the help of the identities (2.11) and (2.23) we calculate

[σ_{μ̄;s_{∘}}(L_{n}), σ_{μ̄;s_{∘}}(Eᾱ^{⊗}f)] = [L_{n} + ∑_{ℓ∈ℤ} ∑_{s}(μ̄_{s}, H_{n−ℓ}) Res(t^{ℓ}φ_{s_{∘},s}), Eᾱ^{⊗}fΦᾱ]

= −Eᾱ ^{⊗} t^{n+1}d(fΦᾱ) + ∑_{ℓ∈ℤ} ∑_{s}(μ̄_{s}, ᾱ) Eᾱ ^{⊗} t^{n−ℓ}fΦᾱ Res(t^{ℓ}φ_{s_{∘},s})

= −Eᾱ ^{⊗} t^{n+1}d(fΦᾱ) + Eᾱ ^{⊗} fΦᾱ ∑_{s}(μ̄_{s}, ᾱ)φ_{s_{∘},s} t^{n+1}

= −Eᾱ ^{⊗} t^{n+1}d(fΦᾱ) + Eᾱ ^{⊗} t^{n+1}f dΦᾱ = −Eᾱ ^{⊗} t^{n+1} (df) Φᾱ

= −σ_{μ̄;s_{∘}}(Eᾱ^{⊗}t^{n+1}df) = σ_{μ̄;s_{∘}}([L_{n}, Eᾱ^{⊗}f]) .

(A.6)

Finally we check that we are indeed dealing with an automorphism of Vir_{s_{∘}}. We obtain

[σ_{μ̄;s_{∘}}(L_{n}), σ_{μ̄;s_{∘}}(L_{m})] = [L_{n} + ∑_{ℓ∈ℤ} ∑_{s}(μ̄_{s}, H_{n−ℓ})^{⊗}Res(t^{ℓ}φ_{s_{∘},s}),
L_{m} + ∑_{ℓ′∈ℤ} ∑_{s′}(μ̄_{s′}, H_{m−ℓ′})^{⊗}Res(t^{ℓ′}φ_{s_{∘},s′})]

= (n − m) L_{n+m} + \frac{1}{24} (n^{3} − n) δ_{n+m,0} C

+ ∑_{ℓ′∈ℤ} ∑_{s′}(μ̄_{s′}, H_{m−ℓ′+n})^{⊗} Res(t^{ℓ′}φ_{s_{∘},s′}) [−(m − ℓ′)]

− ∑_{ℓ∈ℤ} ∑_{s}(μ̄_{s}, H_{n−ℓ+m})^{⊗} Res(t^{ℓ}φ_{s_{∘},s}) [−(n − ℓ)]

+ K ∑_{ℓ,ℓ′∈ℤ}∑_{s,s′}(μ̄_{s},μ̄_{s′}) (n−ℓ)δ_{n−ℓ+m−ℓ′,0} Res(t^{ℓ}φ_{s_{∘},s}) Res(t^{ℓ′}φ_{s_{∘},s′}) .

(A.7)

The terms in the second and third line combine to (n − m) ∑_{ℓ∈ℤ} ∑_{s}(μ̄_{s}, H_{n+m−ℓ})^{⊗}Res(t^{ℓ}φ_{s_{∘},s}). The central term in the fourth line can be rewritten as

K ∑_{ℓ,ℓ′∈ℤ} ∑_{s,s′}(μ̄_{s}, μ̄_{s′}) (\frac{n−m}{2} + \frac{ℓ′−ℓ}{2}) δ_{n+m−ℓ−ℓ′,0} Res(t^{ℓ}φ_{s_{∘},s}) Res(t^{ℓ′}φ_{s_{∘},s′})

= \frac{n−m}{2} K ∑_{s,s′}(μ̄_{s}, μ̄_{s′}) ∑_{ℓ∈ℤ}Res(t^{ℓ}φ_{s_{∘},s}) Res(t^{n+m−ℓ}φ_{s_{∘},s′})

= \frac{n−m}{2} K ∑_{s,s′}(μ̄_{s}, μ̄_{s′}) ∑_{ℓ∈ℤ}Res(t^{n+m+1}φ_{s_{∘},s}φ_{s_{∘},s′}) ,

(A.8)

where in the first step we used the fact that except for the explicit factor (ℓ′ − ℓ)/2 the expression is symmetric in ℓ and ℓ′, and in the second step we employed again the identity (2.23). Collecting all terms, we see that indeed

[σ_{μ̄;s_{∘}}(L_{n}), σ_{μ̄;s_{∘}}(L_{m})] = σ_{μ̄;s_{∘}}([L_{n}, L_{m}])  (A.9)

as required.

B Inner multi-shift automorphisms

Here we collect some details about the derivation of some of the results stated in section 4. Let us consider the map ad_{X} for the element

X = Eᾱ^{⊗}f_{+} + E^{−ᾱ}^{⊗}f_{−}  (B.1)

of the block algebra 𝔤̄⊗F. We have

\begin{matrix}ad_{X}(H^{i}^{⊗}f) = −ᾱ^{i} (Eᾱ^{⊗}f_{+}f − E^{−ᾱ}^{⊗}f_{−}f) , \\ ad_{X}^{2}(H^{i}^{⊗}f) = 2ᾱ^{i} (ᾱ^{∨}, H)^{⊗} f_{+}f_{−}f , \\ ad_{X}^{3}(H^{i}^{⊗}f) = −4ᾱ^{i} (Eᾱ^{⊗}f 2_{+}f_{−}f − E^{−ᾱ}^{⊗}f_{+}f 2_{−}f) \end{matrix}  (B.2)

etc., and hence

exp(ad_{ξX})(H^{i}^{⊗}f) = H^{i}^{⊗}f − \frac{1}{2} ᾱ^{i} (Eᾱ^{⊗}f_{+} − E^{−ᾱ}^{⊗}f_{−}) sinh(2ξ\sqrt{f_{+}f_{−}}) (f_{+}f_{−})^{−1/2} f
+ \frac{1}{2} ᾱ^{i} (ᾱ^{∨}, H)^{⊗} [cosh(2ξ\sqrt{f_{+}f_{−}}) − 1] f .

(B.3)

In the special case where f_{−} = (f_{+})^{−1} (recall that this restricts f_{+} ∈ F(ℙ_{m}^{1}) to lie in the subalgebra F^{∗}(ℙ_{m}^{1})), this reduces to

exp(ad_{(iπ/2)X})(H^{i}^{⊗}f) = (w̄ᾱ(H))^{i} ^{⊗} f ,  (B.4)

where w̄ᾱ(H) = H − (ᾱ^{∨}, H)ᾱ is the image of H under the Weyl reflection w̄ᾱ. Note that this result does not depend on the function f_{+} at all; as a consequence, by making use of w̄2 ; ᾱ = id one immediately deduces the relation (4.4).

When applying ad_{X} to Eβ̄^{⊗}f ∈ 𝔤̄⊗F, we distinguish between several cases. First assume that β̄ = ±ᾱ. Then

exp(ad_{ξX})(E^{±ᾱ}^{⊗}f) = E^{±ᾱ}^{⊗}f ∓ \frac{1}{2} (ᾱ^{∨}, H)^{⊗} sinh(2ξ\sqrt{f_{+}f_{−}}) (f_{+}f_{−})^{−1/2} f_{∓}f

± \frac{1}{2} (Eᾱ^{⊗}f_{+} − E^{−ᾱ}^{⊗}f_{−}) [cosh(2ξ\sqrt{f_{+}f_{−}}) − 1] (f_{±})^{−1}f .

(B.5)

Similarly, for β̄ ≠ ±ᾱ we find the following. When neither β̄ + ᾱ nor β̄ − ᾱ is a 𝔤̄-root, we simply have ad_{X}(Eβ̄^{⊗}f) = 0. When either β̄ + ᾱ or β̄ − ᾱ (but not both) is a 𝔤̄-root, while β̄ ± 2ᾱ is not a 𝔤̄-root, then we obtain

exp(ad_{ξX})(Eβ̄^{⊗}f) = Eβ̄^{⊗} cosh(ξ\sqrt{ηf_{+}f_{−}}) f
+ e_{±ᾱ,β̄} E^{β̄±ᾱ}^{⊗} sinh(ξ\sqrt{ηf_{+}f_{−}}) (ηf_{+}f_{−})^{−1/2} f_{±}f ,

(B.6)

where η ≡ η_{±ᾱ}:= e_{±ᾱ,β̄}e_{∓ᾱ,β̄±ᾱ}, with e_{ᾱ,β̄} structure constants of the horizontal subalgebra 𝔤̄. Actually, from the fact that the ᾱ-string through β̄ has only two elements, it follows that e_{±ᾱ,β̄} and e_{∓ᾱ,β̄±ᾱ} can only take the values ±1, and that β̄ and β̄ ± ᾱ have the same length, so that the general identity (β̄ ± ᾱ, β̄ ± ᾱ)/(β̄, β̄) = e_{±ᾱ,β̄}/e_{∓ᾱ,β̄±ᾱ} tells us that η = 1. When the ᾱ-string through β̄ has more elements, then the calculations become still a bit more lengthy. We refrain from describing all different possibilities, because the calculations remain straightforward. As an illustration, let β̄ + ᾱ and β̄ + 2ᾱ, but neither β̄ − ᾱ nor β̄ + 3ᾱ be 𝔤̄-roots; then we have

\begin{matrix}ad_{X}(Eβ̄^{⊗}f) = e_{ᾱ,β̄} E^{ᾱ+β̄} ^{⊗} f_{+}f , \\ ad_{X}^{2}(Eβ̄^{⊗}f) = e_{ᾱ,β̄} (e_{ᾱ,ᾱ+β̄} E^{2ᾱ+β̄} ^{⊗} f_{+} + e_{−ᾱ,ᾱ+β̄} Eβ̄ ^{⊗} f_{−}) f_{+}f , \\ ad_{X}^{3}(Eβ̄^{⊗}f) = η′e_{ᾱ,β̄} E^{ᾱ+β̄} ^{⊗} f 2_{+}f_{−}f \end{matrix}  (B.7)

with η′:= e_{ᾱ,ᾱ+β̄}e_{−ᾱ,2ᾱ+β̄} + e_{−ᾱ,ᾱ+β̄}e_{ᾱ,β̄}, leading to

exp(ad_{ξX})(Eβ̄^{⊗}f) = Eβ̄^{⊗}f + e_{ᾱ,β̄} E^{ᾱ+β̄} ^{⊗} sinh(ξ\sqrt{η′f_{+}f_{−}}) (η′f_{+}f_{−})^{−1/2} f_{+}f
+ e_{ᾱ,β̄} (e_{ᾱ,ᾱ+β̄} E^{2ᾱ+β̄} ^{⊗} f_{+} + e_{−ᾱ,ᾱ+β̄} Eβ̄ ^{⊗} f_{−})
[cosh(ξ\sqrt{η′f_{+}f_{−}}) − 1] (η′f_{+}f_{−})^{−1} f_{+}f .

(B.8)

In the special case f_{+}f_{−} = 1 from (B.5) we get

exp(ad_{(iπ/2)X})(E^{±ᾱ}^{⊗}f) = E^{±ᾱ}^{⊗}f ∓ (Eᾱ^{⊗}f_{+} − E^{−ᾱ}^{⊗}f_{−}) (f_{±})^{−1}f = E^{∓ᾱ}^{⊗}f 2_{∓}f  (B.9)

for β̄ = ±ᾱ. When in addition Y ∈ 𝔤̄⊗F is of the same form as X, but with f_{±} replaced by g_{±} ∈ F^{∗}(ℙ_{m}^{1}), then it follows from (B.9) that

Ad_{iπ/2;X,Y} (E^{±ᾱ}^{⊗}f) = E^{±ᾱ}^{⊗}(f_{+})^{∓2}(g_{+})^{±2}f .  (B.10)

Analogously, the special case f_{+}f_{−} = 1 of (B.6) yields

exp(ad_{(iπ/2)X})(Eβ̄^{⊗}f) = i e_{±ᾱ,β̄} E^{β̄±ᾱ}^{⊗}f_{±}f .  (B.11)

Similarly, from (B.8) one obtains (using also the fact that β̄ and β̄ + 2ᾱ must be long roots while ᾱ and β̄ + ᾱ must be short roots, which implies the identities

\begin{matrix}e_{ᾱ,ᾱ+β̄}e_{−ᾱ,2ᾱ+β̄} = (e_{ᾱ,ᾱ+β̄})^{2} · \frac{(ᾱ+β̄,ᾱ+β̄)}{(2ᾱ+β̄,2ᾱ+β̄)} = 2^{2} · \frac{1}{2} = 2 , \\ e_{ᾱ,β̄}e_{−ᾱ,ᾱ+β̄} = (e_{ᾱ,β̄})^{2} · \frac{(β̄,β̄)}{(ᾱ+β̄,ᾱ+β̄)} = 1^{2} · 2 = 2 , \\ |e_{ᾱ,β̄}e_{ᾱ,ᾱ+β̄}| = 2 , \end{matrix}  (B.12)

so that in particular η′ = 4),

exp(ad_{(iπ/2)X})(Eβ̄^{⊗}f) = ± E^{2ᾱ+β̄} ^{⊗} f 2_{+}f .  (B.13)

Taking again also Y of the form described above, the relations (B.11) and (B.13) lead to

Ad_{iπ/2;X,Y} (Eβ̄^{⊗}f) = Eβ̄^{⊗}f_{±}g_{∓}f  and Ad_{iπ/2;X,Y} (Eβ̄^{⊗}f) = Eβ̄^{⊗}(f_{+}g_{−})^{2}f ,  (B.14)

respectively.

For the remaining cases, the calculations are completely parallel and lead to results analogous to (B.14), with f_{±}g_{∓} raised to the power −(ᾱ^{∨}, β̄). This finally leads to the formula (4.5) of the main text.

Next we comment on the analogous calculations for the case where the functions f are Laurent series (in the local coordinate ζ_{s} respectively the formal variable t). First we compute that, for f_{+}f_{−} = 1, the second formula in (B.2) changes to

ad_{X}^{2}(H^{i}^{⊗}f) = 2ᾱ^{i} (ᾱ^{∨}, H)^{⊗} f_{+}f_{−}f − 2ᾱ^{i}K \frac{2}{(ᾱ,ᾱ)} Res(f_{+}f df_{−}) ,  (B.15)

so that (B.3) acquires an additional contribution proportional to the central element,

−\frac{1}{2} (ᾱ^{∨})^{i}K Res(f_{+}fdf_{−}) (cosh(2ξ) − 1)) ,  (B.16)

which when specialized to ξ = iπ/2 becomes (ᾱ^{∨})^{i}K Res(f_{+}fdf_{−}). Performing the same calculation also with the analogous element Y in place of X, we then arrive at the formula (4.8).

In the calculation of exp(ad_{ξX})(Eβ̄^{⊗}f), a change occurs only for β̄ = ±ᾱ; in this case we get

ad_{X}(E^{±ᾱ}^{⊗}f) = ∓ (ᾱ^{∨}, H)^{⊗} f_{∓}f − K \frac{2}{(ᾱ,ᾱ)} Res(f_{∓}df) ,
ad_{X}^{2}(E^{±ᾱ}^{⊗}f) = ±2 (Eᾱ^{⊗}f_{+} − E^{−ᾱ}^{⊗}f_{−}) f_{∓}f ,
ad_{X}^{3}(E^{±ᾱ}^{⊗}f) = ∓4 (ᾱ^{∨}, H)^{⊗} f_{∓}f − 4K \frac{2}{(ᾱ,ᾱ)} Res(f_{∓}df) ,

(B.17)

etc. The residue terms add up to −(ᾱ, ᾱ)^{−1} K Res(f_{∓}df) sinh(2ξ), which vanishes for ξ = iπ/2, while the other terms reproduce (B.6).

Finally let us compute the action of ad_{X} on the Virasoro generators L_{n}. By (2.31) we find, for f_{+}f_{−} = 1,

ad_{X}(L_{n}) = Eᾱ ^{⊗} t^{n+1}df_{+} + E^{−ᾱ} ^{⊗} t^{n+1}df_{−} ,
ad_{X}^{2}(L_{n}) = −2 (ᾱ^{∨}, H^{i})^{⊗} f_{−}df_{+} t^{n+1} + 2 \frac{2}{(ᾱ,ᾱ)}K Res(t^{n+1}df_{+}df_{−}) ,
ad_{X}^{3}(L_{n}) = 4 (Eᾱ ^{⊗} t^{n+1}df_{+} + E^{−ᾱ} ^{⊗} t^{n+1}df_{−}) ,

(B.18)

and hence

exp(ad_{ξX})(L_{n}) = L_{n} + \frac{1}{2} sinh(2ξ) (Eᾱ^{⊗}t^{n+1}df_{+} + E^{−ᾱ}^{⊗}t^{n+1}df_{−})

− \frac{1}{2} [cosh(2ξ) − 1] ((ᾱ^{∨}, H)^{⊗} f_{−}df_{+} t^{n+1} − \frac{2}{(ᾱ,ᾱ)}K Res(t^{n+1}df_{+}df_{−})) . (B.19)

Specializing to ξ = iπ/2, we have

exp(ad_{(iπ/2)X})(L_{n}) = L_{n} + (ᾱ^{∨}, H)^{⊗} f_{−}df_{+} t^{n+1} − \frac{2}{(ᾱ,ᾱ)}K Res(t^{n+1}df_{+}df_{−}) .  (B.20)

C On the moduli dependence of inner automorphisms

Our first aim in this appendix is to check the formula (6.11), which asserts that the automorphism δ_{μ̄} acting as in (6.10) can be represented as δ_{μ̄} = exp(ad_{(μ̄_{u},H)⊗g_{s∘,u}}) for s_{∘} ≠ u and as δ_{μ̄} = exp(ad_{∑_{s≠u}(μ̄_{s},H)⊗g̃_{s}}) for s_{∘} = u, with g_{s_{∘},u} and g̃_{s} as defined in (6.12). First, definitely

exp(ad_{(μ̄_{u},H)⊗g_{s∘,u}})(K) = K = exp(ad_{(μ̄_{s},H)⊗g̃_{s}})(K) .  (C.1)

Second, using the relations dg_{s_{∘},u} = φ_{s_{∘},u} − φs_{∘},u^{[0]} and

[(μ̄_{u}, H)^{⊗} g_{s_{∘},u}, H^{i}^{⊗}f] = μ̄_{u}^{i} Res(dg_{s_{∘}} f) K  (C.2)

together with [K, ·] = 0, we have

exp(ad_{(μ̄_{u},H)⊗g_{s∘,u}})(H^{i}^{⊗}f) = H^{i}^{⊗}f + K μ̄_{u}^{i} Res((φ_{s_{∘},u} − φs_{∘},u^{[0]}) f) ,  (C.3)

and similarly

exp(ad_{(μ̄_{s},H)⊗g̃_{s}})(H^{i}^{⊗}f) = H^{i}^{⊗}f + K μ̄_{s}^{i} Res((φ_{u,s} − φu,s^{[0]}) f) .  (C.4)

And finally, by exponentiating the identity [(μ̄_{u}, H)^{⊗}g_{s_{∘},u}, Eβ̄^{⊗}f] = (μ̄_{u}, β̄)Eβ̄^{⊗}g_{s_{∘},u}f we obtain

exp(ad_{(μ̄_{u},H)⊗g_{s∘,u}})(Eβ̄^{⊗}f) = Eβ̄ ^{⊗} exp((μ̄_{u}, β̄) g_{s_{∘},u}) f = Eβ̄ ^{⊗} f ·(\frac{t+z_{s_{0}}−z_{u}}{t+z_{s_{0}}−z_{u}^{[0]}})^{(μ̄_{u},β̄)} , (C.5)

and analogously

exp(ad_{(μ̄_{s},H)⊗g̃_{s}})(Eβ̄^{⊗}f) = Eβ̄ ^{⊗} f ·(\frac{t+z_{u}−z_{s}}{t+z_{u}^{[0]}−z_{s}})^{(μ̄_{s},β̄)} .  (C.6)

Putting these results together we see that δ_{μ̄} is indeed given by (6.11).

Next we derive the formula (6.15) for the commutator [∇_{u}, Θ_{μ̄;s_{∘}}]. Recall that ∇_{u} = ∂/∂z_{u} + L−1^{(u)}. Thus we first study the effect of differentiating with respect to z_{u}. Employing the identities

\frac{∂}{∂z_{u}} g_{s_{∘}} = −φ_{s_{∘},u} for s_{∘} ≠ u and  \frac{∂}{∂z_{u}} g̃_{s}|_{z_{u}=z_{u}^{[0]}} = φ_{u,s} for s ≠ u ,  (C.7)

we derive

\frac{∂}{∂z_{u}} Θ_{μ̄;s_{∘}}(z⃗)|_{z_{u}=z_{u}^{[0]}} = ( − (μ̄_{u}, H) φs_{∘},u^{[0]} + K \frac{∂}{∂z_{u}} h̃_{s_{∘}}|_{z_{u}=z_{u}^{[0]}}) Θ_{μ̄;s_{∘}}(z^{[0]}) for s_{∘} ≠ u ,
\frac{∂}{∂z_{u}} Θ_{μ̄;s_{∘}}(z⃗)|_{z_{u}=z_{u}^{[0]}} = ( ∑_{{s=1 \atop s≠u}}^{m}(μ̄_{s}, H) φu,s^{[0]} + K \frac{∂}{∂z_{u}} h̃_{u}|_{z_{u}=z_{u}^{[0]}}) Θ_{μ̄;s_{∘}}(z^{[0]}) for s_{∘} = u . (C.8)

Concerning the commutation with L−1^{(u)}, we use Θ_{μ̄;s}(z⃗)L−1^{(s)} = σ_{μ̄;s}(L−1^{(s)}) Θ_{μ̄;s}(z⃗) which follows by (2.34). Also, according to (2.32) we have explicitly

σ_{μ̄;u}(L−1^{(u)}) = L−1^{(u)} + ∑_{ℓ∈ℤ} ∑_{s=1}^{m}(μ̄_{s},H_{−1−ℓ}) Res(t^{ℓ}φ_{u,s}) + \frac{1}{2} K ∑_{s,s′=1}^{m}(μ̄_{s},μ̄_{s′}) Res(φ_{u,s}φ_{u,s′}) . (C.9)

The two terms of the right hand side can be rewritten with the help of

∑_{ℓ∈ℤ}H_{−1−ℓ}^{i} Res(t^{ℓ}φ_{u,s}) ≡ ∑_{ℓ∈ℤ}H^{i} ^{⊗} t^{−1−ℓ} Res(t^{ℓ}φ_{u,s}) = ∑_{ℓ∈ℤ}H^{i} ^{⊗} (φ_{u,s})_{−1−ℓ} t^{−1−ℓ} = H^{i} ^{⊗} φ_{u,s}

(C.10)

and

∑_{s,s′=1}^{m}(μ̄_{s},μ̄_{s′}) Res(φ_{u,s}φ_{u,s′}) = 2 ∑_{{s=1 \atop s≠u}}^{m} \frac{(μ̄_{s}, μ̄_{u})}{z_{u} − z_{s}}  (C.11)

so as to obtain

[Θ_{μ̄;u}, L−1^{(u)}] = (σ_{μ̄;u}(L−1^{(u)}) − L−1^{(u)}) Θ_{μ̄;u} = ( ∑_{s=1}^{m}(μ̄_{s},H)^{⊗} φ_{u,s} + K ∑_{{s=1 \atop s≠u}}^{m} \frac{(μ̄_{s},μ̄_{u})}{z_{u} − z_{s}}) Θ_{μ̄;u} . (C.12)

Combining this with (C.8), we finally get

[∇_{u}, Θ_{μ̄;s_{∘}}(z⃗)] = −(μ̄_{u}, H)^{⊗}φs_{∘},u^{[0]} Θ_{μ̄;s_{∘}}(z^{[0]}) + K \frac{∂}{∂z_{u}} h̃_{s_{∘}}|_{z_{u}=z_{u}^{[0]}} Θ_{μ̄;s_{∘}}(z^{[0]}) for s_{∘} ≠ u , (C.13)

while for s_{∘} = u several terms cancel, leading to

[∇_{u}, Θ_{μ̄;u}(z⃗)]|_{z_{u}=z_{u}^{[0]}} = −(μ̄_{u}, H)^{⊗}t^{−1} Θ_{μ̄;u}(z^{[0]}) + K (\frac{∂}{∂z_{u}} h̃_{u}|_{z_{u}=z_{u}^{[0]}} − ∑_{{s=1 \atop s≠u}}^{m} \frac{(μ̄_{s},μ̄_{u})}{z_{u} − z_{s}}) Θ_{μ̄;u}(z^{[0]}) .

(C.14)

This is indeed equivalent to the formula (6.15).

D On the implementation of inner automorphisms

In this appendix we present the calculations which show that the specific implementation of inner multi-shift automorphisms that was described in (5.7) satisfies the relation (6.41) and explain why this implies that the implementation can be consistently chosen at all points of the moduli space M_{m}.

We consider first the case of a single coroot ᾱs^{∨} for some fixed s, i.e. μ̄_{s} = ᾱs^{∨} while μ̄_{s} = 0 for s′ ≠ s. We write

X_{s_{∘},s} ≡ X_{s_{∘},s;α_{s}}(z⃗) := Eᾱ_{s} ^{⊗} φ_{s_{∘},s} + E^{−ᾱ_{s}} ^{⊗} φ_{s_{∘},s}^{−1}  (D.15)

and

Xs_{∘},s^{[0]} := X_{s_{∘},s;α_{s}}(z^{[0]}) = Eᾱ_{s} ^{⊗} φs_{∘},s^{[0]} + E^{−ᾱ_{s}} ^{⊗} (φs_{∘},s^{[0]})^{−1} .  (D.16)

We claim that X_{s_{∘},s} and Xs_{∘},s^{[0]} are related by (6.40), which in terms of the present notations reads

exp(\frac{1}{2} iπ X_{s_{∘},s}) = U_{s_{∘},s}(z⃗) exp(\frac{1}{2} iπ Xs_{∘},s^{[0]}) U_{s_{∘},s}^{−1}(z⃗) ,  (D.17)

where

U_{s_{∘},s}(z⃗) ≡ U_{s_{∘},s;ᾱ_{s}^{∨}}(z⃗) := exp ( − \frac{1}{2} Hᾱ_{s} ^{⊗} g_{s_{∘},s}(z⃗))  (D.18)

with g_{s_{∘},s} as defined in (6.22) and Hᾱ_{s} ≡ (ᾱs^{∨}, H). To prove (D.17), we show that both sides satisfy the same differential equation; the relation then holds because owing to U_{s_{∘},s}(z^{[0]}) = 1 both sides are identical at z⃗ = z^{[0]}. To obtain the derivative of the left hand side of (D.17) we first compute

\begin{aligned}\frac{∂}{∂z_{u}} X_{s_{∘},s} & = (δ_{u,s} − δ_{u,s_{∘}}) (Eᾱ_{s}^{⊗}φ_{s_{∘},s}^{2} + E^{−ᾱ_{s}}^{⊗}id) \\ & = −\frac{1}{2} (δ_{u,s} − δ_{u,s_{∘}}) ad_{X_{s∘,s}}(Hᾱ_{s} ^{⊗} φ_{s_{∘},s}) ≡ \frac{1}{2} (δ_{u,s} − δ_{u,s_{∘}}) ad_{Hᾱs⊗φ_{s∘,s}}(X_{s_{∘},s}) , \end{aligned} (D.19)

where in the transition to the second line we used the formula (B.2). It follows immediately that

\frac{∂}{∂z_{u}} exp (\frac{iπ}{2} X_{s_{∘},s}) = \frac{1}{2} (δ_{u,s} − δ_{u,s_{∘}}) (Hᾱ_{s}^{⊗}φ_{s_{∘},s} exp (\frac{iπ}{2} X_{s_{∘},s}) − exp (\frac{iπ}{2} X_{s_{∘},s}) Hᾱ_{s}^{⊗}φ_{s_{∘},s}) .

(D.20)

As for the right hand side of (D.17), we have

\begin{aligned}\frac{∂U_{s∘,s}(z⃗)}{∂z_{u}} & = −\frac{1}{2} U_{s_{∘},s}(z⃗) · Hᾱ_{s} ^{⊗} \frac{∂}{∂z_{u}} g_{s_{∘},s} \\ & = −\frac{1}{2} U_{s_{∘},s}(z⃗) · Hᾱ_{s} ^{⊗} φ_{s_{∘},s} (δ_{u,s_{∘}} − δ_{u,s}) ≡ \frac{1}{2} (δ_{u,s} − δ_{u,s_{∘}}) Hᾱ_{s} ^{⊗} φ_{s_{∘},s} · U_{s_{∘},s}(z⃗) \end{aligned}

(D.21)

and

\frac{∂}{∂z_{u}} U_{s_{∘},s}^{−1}(z⃗) = −U_{s_{∘},s}^{−1}(z⃗) \frac{∂U_{s∘,s}(z⃗)}{∂z_{u}} U_{s_{∘},s}^{−1}(z⃗) = −\frac{1}{2} (δ_{u,s} − δ_{u,s_{∘}}) U_{s_{∘},s}^{−1}(z⃗) · Hᾱ_{s} ^{⊗} φ_{s_{∘},s}; (D.22)

therefore the product U_{s_{∘},s}(z⃗) exp(\frac{1}{2} iπ Xs_{∘},s^{[0]}) U_{s_{∘},s}^{−1}(z⃗) indeed satisfies the same differential equation as exp(\frac{1}{2} iπ X_{s_{∘},s}) in (D.20).

Next we note that according to the result (B.4) and (B.16), we also have

exp(ad_{(iπ/2)X_{s∘,s}})(Hβ̄^{⊗}f) = w̄ᾱ(Hβ̄)^{⊗} f − (ᾱ^{∨}, β̄^{∨}) K Res(φ_{s_{∘},s}^{−1}dφ_{s_{∘},s} f) .  (D.23)

Thus in particular

exp(ad_{(iπ/2)X_{s∘,s}})(Hᾱ_{s}^{⊗}f) = −Hᾱ_{s} ^{⊗} f − (ᾱs^{∨}, ᾱs^{∨}) K Res(φ_{s_{∘},s}^{−1}dφ_{s_{∘},s} f) ,
exp(ad_{−(iπ/2)Y_{s}})(Hᾱ_{s}^{⊗}f) = −Hᾱ_{s} ^{⊗} f ,

(D.24)

where we introduced

Y_{s} := (Eᾱ_{s} + E^{−ᾱ_{s}})^{⊗} id .  (D.25)

In other words, exp(−\frac{1}{2} iπ Y_{s}) Hᾱ_{s}^{⊗}f = −Hᾱ_{s}^{⊗}f exp(−\frac{1}{2} iπ Y_{s}), which implies in particular that

exp(−\frac{1}{2} iπ Y_{s}) U_{s_{∘},s}^{−1}(z⃗) = U_{s_{∘},s}(z⃗) exp(−\frac{1}{2} iπ Y_{s}) ,  (D.26)

while

exp (\frac{iπ}{2} X_{s_{∘},s}) U_{s_{∘},s}(z⃗) = U_{s_{∘},s}^{−1}(z⃗) exp (\frac{iπ}{2} X_{s_{∘},s})·exp (−\frac{2}{(ᾱ_{s},ᾱ_{s})} K Res(φ_{s_{∘},s}^{−1}dφ_{s_{∘},s} g_{s_{∘},s})) , (D.27)

which by dφ_{s_{∘},s} = −φ_{s_{∘},s}^{2} and

Res(φ_{s_{∘},s′}g_{s_{∘},s}) = \{ \begin{matrix}δ_{s_{∘},s′} ln \frac{z_{s∘}−z_{s}}{z_{s∘}^{[0]}−z_{s}^{[0]}} & for s ≠ s_{∘} , \\ 0 & for s = s_{∘} \end{matrix}  (D.28)

reduces to

exp(\frac{1}{2} iπ X_{s_{∘},s}) U_{s_{∘},s}(z⃗) = U_{s_{∘},s}^{−1}(z⃗) exp(\frac{1}{2} iπ X_{s_{∘},s}) .  (D.29)

Now according to our prescription (5.7), at each individual point z⃗ in the moduli space the inner automorphism Ad_{iπ/2;X_{s∘,s},Y_{s}} of the block algebra 𝔤̄⊗F is implemented by

Θ̃_{ᾱ_{s}^{∨};s_{∘}}(z⃗) = exp(−\frac{1}{2} iπ Y_{s}) exp(\frac{1}{2} iπ X_{s_{∘},s}) .  (D.30)

Combining the results (D.17), (D.26) and (D.29), we learn that this twisted intertwiner satisfies

Θ̃_{ᾱ_{s}^{∨};s_{∘}}(z^{[0]}) = exp(−\frac{1}{2} iπ Y_{s}) exp(\frac{1}{2} iπ Xs_{∘},s^{[0]})
= exp(−\frac{1}{2} iπ Y_{s}) U_{s_{∘},s}^{−1}(z⃗) exp(\frac{1}{2} iπ X_{s_{∘},s}) U_{s_{∘},s}(z⃗)
= U_{s_{∘},s}^{2}(z⃗) exp(−\frac{1}{2} iπ Y_{s}) exp(\frac{1}{2} iπ X_{s_{∘},s}) = U_{s_{∘},s}^{2}(z⃗) Θ̃_{ᾱ_{s}^{∨};s_{∘}}(z⃗) .

(D.31)

We now return to the general case where the components μ̄ are arbitrary elements of the coroot lattice. We note that because of

Res(dg_{s_{∘},s} g_{s_{∘},s′}) = 0  (D.32)

for all values of s, s′ any two operators of the form (D.18) commute (when taken at the same point z⃗ in the moduli space M_{m} of course). Moreover, according to (4.8), up to central terms U_{s_{∘},s}(z⃗) also commutes with the twisted intertwiner Θ̃_{β̄_{s′}^{∨};s_{∘}}(z⃗) for s′ ≠ s and arbitrary coroots β̄^{∨} ≡ β̄s′^{∨}; more precisely, using the identity (D.28) we find

Θ̃_{β̄_{s′}^{∨};s_{∘}}(z⃗) U_{s_{∘},s}(z⃗) = U_{s_{∘},s}(z⃗) Θ̃_{β̄_{s′}^{∨};s_{∘}}(z⃗) · exp[ − \frac{1}{2} (β̄s′^{∨}, (ᾱs^{∨})) K Res(φ_{s_{∘},s′} g_{s_{∘},s})]

= U_{s_{∘},s}(z⃗) Θ̃_{β̄_{s′}^{∨};s_{∘}}(z⃗) · exp[ − \frac{1}{2} (β̄s′^{∨}, (ᾱs^{∨})) δ_{s_{∘},s′} K ln \frac{z_{s∘}−z_{s}}{z_{s∘}^{[0]}−z_{s}^{[0]}}] .

(D.33)

The final expression will involve the square of the exponential with the central element, because it is U_{s_{∘},s}^{2}(z⃗) that we commute to the left. Moreover, taking into account the chosen ordering of the factors in Θ̃_{μ̄;s_{∘}}, such a term will appear if and only if s >s_{∘}. Thus defining Θ̃_{μ̄;s_{∘}} according to the prescription (5.7), i.e. as

Θ̃_{μ̄;s_{∘}}(z⃗) := ∏_{s}^{→}( ∏_{i_{s}}^{→} exp(−\frac{1}{2} iπ Y_{α_{is};s_{∘}}) exp(\frac{1}{2} iπ X_{α_{is};s,s_{∘}}))  (D.34)

when μ̄_{s} = ∑_{i_{s}=1} ᾱi_{s}^{∨} for s = 1, 2, ... , m−1, the equation (D.31) generalizes to

Θ̃_{μ̄;s_{∘}}(^{z^{[0]}}) = U_{s_{∘};μ̄}^{2}(z⃗) Θ̃_{μ̄;s_{∘}}(z⃗) · exp[ − ∑_{{s=1 \atop s>s∘}}^{m−1}(μ̄_{s_{∘}}, μ̄_{s}) K ln \frac{z_{s∘}−z_{s}}{z_{s∘}^{[0]}−z_{s}^{[0]}}]  (D.35)

with

U_{s_{∘};μ̄}(z⃗) := exp ( − ∑_{s=1}^{m−1} \frac{1}{2} (μ̄_{s}, H)^{⊗} g_{s_{∘},s}(z⃗)) .  (D.36)

Using the invariance of the exponent under exchange of s with s_{∘}, (D.35) may also be rewritten as

Θ̃_{μ̄;s_{∘}}(^{z^{[0]}}) = U_{s_{∘};μ̄}^{2}(z⃗) Θ̃_{μ̄;s_{∘}}(z⃗) · exp[ − \frac{1}{2} ∑_{{s=1 \atop s≠s∘}}^{m−1}(μ̄_{s_{∘}}, μ̄_{s}) K ln \frac{z_{s∘}−z_{s}}{z_{s∘}^{[0]}−z_{s}^{[0]}}] .  (D.37)

We are now finally in a position to make contact to the relation (6.21) that corresponds to the compatibility with the Knizhnik--Zamolodchikov connection. First, comparison of our results with the definition (6.34)^{13} (see also (6.22) and (6.26)), tells us that we may rewrite (D.37) in the form of (6.41), i.e.

Θ̃_{μ̄;s_{∘}}(z⃗) = A_{μ̄;s_{∘}}(z⃗; z^{[0]}) Θ̃_{μ̄;s_{∘}}(z^{[0]}) .  (D.38)

Moreover, we already know that the implementation is consistent at the specific point z^{[0]}, i.e. we have Θ̃_{μ̄;s_{∘}}(z^{[0]}) = Θ_{μ̄;s_{∘}}(z^{[0]}). Together with the relation (6.21) between Θ_{μ̄;s_{∘}}(z⃗) and Θ_{μ̄;s_{∘}}(z^{[0]}) the result (D.38) therefore implies that in fact

Θ̃_{μ̄;s_{∘}}(z⃗) = A_{μ̄;s_{∘}}(z⃗; z^{[0]}) Θ_{μ̄;s_{∘}}(z^{[0]}) = Θ_{μ̄;s_{∘}}(z⃗)  (D.39)

at every point z⃗ in the moduli space M and hence that, as claimed, the implementation is consistent on all of M.

13 The summation over s in (6.34) now extends only up to m−1, owing to our decision to keep one puncture at z_{m} = ∞ when dealing with inner multi-shift automorphisms.

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