Fuzzy Complex Grassmannian Spaces and their Star Products

arXiv:hep-th/0111020 v2 7 Jan 2003 DIAS-STP-01-16 hep-th/0111020 October 2002 Fuzzy Complex Grassmannian Spaces and their Star Products Brian P. Dolan 1,a and Oliver Jahn 2,b aDepartment of Mathematical Physics NUI Maynooth, Maynooth, Ireland bDublin Institute for Advanced Studies 10 Burlington Road, Dublin 4, Ireland Abstract We derive an explicit expression for an associative star product on noncommutative versions of complex Grassmannian spaces, in particular for the case of complex 2-planes. Our expression is in terms of a finite sum of derivatives. This generalises previous results for complex projective spaces and gives a discrete approximation for the Grassmannians in terms of a non-commutative algebra, represented by matrix multiplication in a finitedimensional matrix algebra. The matrices are restricted to have a dimension which is precisely determined by the harmonic expansion of functions on the commutative Grassmannian, truncated at a finite level. In the limit of infinitedimensional matrices we recover the commutative algebra of functions on the complex Grassmannians. [email protected] 2[email protected] 1 Introduction There has recently been much interest in non-commutative geometry, [1, 2], both as a novel direction in string theory [3] and as a new tool in quantum field theory [4]. In the latter approach the theory is formulated on a “fuzzy” space, a series of discrete approximations to a continuous space-time manifold M. The space of fields on each approximation is finite-dimensional, so the fuzzy space serves as a regulator, similar to a lattice. In contrast to the latter, the truncation enjoys all symmetries of the approximated continuous manifold. This has several useful implications including the absence of a doubling problem for chiral fermions [5, 6]. Topologically non-trivial field configurations can be included and the chiral anomaly emerges naturally in this approach [7, 8]. In more detail, a fuzzy space is given by a series of finite-dimensional algebras ALthat approximate the commutative algebra of functions on Min the following way: each ALis identified with a subset of functions on Mand the product in AL induces a (non-commutative) product of functions in this subset, the so-called star product; in the limit L→ ∞, the subset should exhaust the set of all functions and the star product go over to the (commutative) pointwise product of functions. In the present case, the algebras ALare full matrix algebras. The fuzzy spaces are thus matrix geometries, which go over to the usual continuous manifold as the size of the matrix is taken to infinity. Non-commutative star products are known to exist for every Poisson manifold, in particular for symplectic manifolds [9]. Star products realised by finite-dimensional matrix algebras have been constructed in [10] for all homogeneous K¨ahler manifolds provided a certain quantisation condition on the metric is satisfied. These algebras can also be obtained using generalised coherent states [11] or the method of orbits for irreducible representations of Lie groups [12, 13]. The relation between these approaches has been discussed in [14] and that to deformation quantisation in [15, 16]. For the relation to Fourier transformation on group space see [17]. In terms of functions on the manifold, the above formulations provide an expression of the star product as an integral over the analytical continuation of the functions. This is not very convenient for explicit calculations in non-commutative field theory. An explicit, local formula in terms of a finite number of derivatives is so far only known for complex projective spaces [18] including the case of the fuzzy 2-sphere [19] already treated in [20]. In this paper, we derive an analogous formula for fuzzy complex Grassmannians, that is finite matrix geometry approximations to the usual Grassmannians, GN k∼ =U(N)/[U(k)×U(N−k)], which are homogeneous spaces isomorphic to the space of complex k-planes in CN. The matrix geometries consist of matrices acting on the irreducible representation of SU(N) which is given by the L-fold Young product of the representation on antisymmetric k-tensors. As Lis increased the continuous Grassmannian is recovered. The special case k= 1 requires the L-fold symmetric product of the fundamental representation of SU(N), as in [18]. The star product for the case k= 2 is constructed explicitly, using globally well-defined but over-complete co-ordinates on the Grassmannian. The result is expressed as a finite sum over multiple derivatives of the functions, which are decomposed into 1 irreducible representations of the stability group S[U(k)×U(N−k)] acting on the tangent space. It is shown that the star product reduces to the usual commutative product as L→ ∞. The particular case G4 2may be of some interest in field theory as it is a symplectic manifold which has S4as a Lagrangian sub-manifold. The corresponding matrix geometries can be viewed as non-commutative versions of T∗S4, the co-tangent bundle to S4[21]. A star product on the complex Grassmannians as an infinite sum over derivatives is known [22, 23]. However, this formula cannot be restricted to finite-dimensional sub-algebras and therefore cannot serve as a star product on a fuzzy approximation to the manifold. The layout of the paper is as follows: in section 2 we describe the complex Grassmannians, GN k, in terms of projection operators acting in CN, we introduce a set of global co-ordinates which are over-complete and satisfy a set of quadratic constraints which ensures that they indeed describe GN k; in section 3 we analyse the algebra of functions in terms of representations of SU(N); in section 4 we describe the finite matrix geometries that define the fuzzy Grassmannians GN k,F; section 5 gives the construction of the star product for the special case k= 2, GN 2,F, while section 6 presents a conjecture on the possible form for k > 2 and some observations on the construction; section 7 gives a summary and conclusions and some technical details are relegated to the appendices. 2 Complex Grassmannian spaces The complex Grassmannian space GN k∼ =U(N)/[U(k)×U(N−k)] = SU(N)/S[U(k)× U(N−k)] can be represented as the space of all Hermitean rank-kprojectors Pacting on CN. This is easily seen since any such projector can be diagonalised by an element of U(N) and there is still a residual adjoint action of U(k)×U(N−k) which leaves it invariant. It will be convenient to describe GN kusing a redundant set of globally well-defined co-ordinates, ξA,A= 1,...,N2−1. First we introduce orthonormal Hermitean generators tAof the Lie algebra of SU(N) satisfying tAtB=1 NδAB +1 √2(dABC+ifABC)tC,(1) where fABCare the structure constants of SU(N) and dABCthe components of the usual symmetric traceless tensor. The tAare normalised so that tr(tAtB) = δAB. This allows us to parameterise Pin terms of N2−1 real parameters ξA, P=k N+ξAtA.(2) The condition that Pis as projector translates into ξAξA=k(N−k) Nand 1 √2dABCξAξB=N−2k NξC,(3) and ξAnow parameterise GN kwhen these conditions are imposed. This construction embeds the Grassmannian in the space of traceless Hermitian matrices, which we identify with RN2−1parameterised by the unrestricted ξA. 2 As discussed in detail in reference [18] for the case of CPN−1≡GN 1, the complex structure and metric are encoded in a Hermitean projector KAB≡trPtA(1 −P)tB=1 2PAB+iJAB(4) where JAB=√2fABC ξCis the complex structure on GN kand PAB=PBA=−(J2)AB (indices are raised and lowered with the flat metric δA Bof RN2−1). The matrix KAB projects derivatives with respect to ξAonto the holomorphic tangent space of the Grassmannian when acting on the right and onto the anti-holomorphic tangent space when acting on the left; so ∇A≡KAB∂/∂ξBis a holomorphic derivative and ¯ ∇B≡KAB∂/∂ξAan anti-holomorphic one. To see that KABis a projector we use the completeness relation for the generators, (tA)i j(tA)k l=δilδkj−1 Nδijδkl,(5) to show that KABtB=PtA(1 −P) and tBKBA= (1 −P)tAP.(6) (These relations will prove very useful in the ensuing analysis.) Examining the real and imaginary parts of KABseparately reveals that P=−J2and PJ =JP =J. This means that Pitself is a projector onto the tangent space of GN k, with rank 2k(N−k). An important observation for the following analysis is that the differential operators KAB∂/∂ξBcommute with the constraints (3), as they must do, since K projects onto the tangent space. This can also be proven using (6). It is this fact that allows us to use the global co-ordinates, rather than local co-ordinates, in the final differential expression for the star product. Covariant derivatives can be constructed by projecting derivatives with respect to the flat coordinates ξAto the tangent space. Multiple covariant holomorphic derivatives are thus defined as ∇A1···∇Anf(ξ)≡KA1 B1···KAn Bn∂B1KB2 C2∂C2···KBn Cn∂Cnf(ξ)(7) where ∂A=∂/∂ξA. In our case there is a simplification because KABKCD(∂BKDE) = 0 (8) which follows from the definition (4) of Kand the completeness relation (5). It implies ∇A1···∇Anf(ξ) = KA1 B1···KAn Bn∂B1···∂Bnf(ξ).(9) 3 Harmonic analysis In order to obtain a finite-dimensional truncation of the space of functions on the Grassmannian, which is compatible with the symmetries, we decompose the space of functions into irreducible representations of the isometry group G=SU(N). We think of GN kas the space G/H of (right) cosets in Gwith respect to H= S[U(k)×U(N−k)], the subset of matrices in U(k)×U(N−k) with unit determinant. 3 Functions on G/H can thus be considered as functions on Gthat are invariant under the left action of H. They transform under Gaccording to the right action. The full space of functions on Gis spanned by the matrix elements DJ MM′of all irreducible unitary representations Jof G. Decomposing into irreducible representations of H, we may write the first component index as M= (n, j, m) where j labels the irreducible representations of H,mthe corresponding components and n distinguishes copies of equivalent representations. The left action of His then given by DJ (n,j,m)M′(h−1g) = X m′′ DJ (n,j,m) (n,j,m′′)(h−1)DJ (n,j,m′′ )M′(g).(10) The H-invariant matrix elements are those for which the first index corresponds to the trivial representation of H, labelled for instance by j=m= 0. The space of functions on G/H is thus spanned by the matrix elements DJ (n,0,0) M′. Under the right action of G, DJ (n,0,0) M′(gg′) = X M′′ DJ (n,0,0) M′′ (g)DJ M′′ M′(g′),(11) so, for fixed Jand n, the DJ (n,0,0) M′span the vector space of the representation J of G. The space of functions on GN kthus contains all irreducible representations of SU(N) that contain the trivial representation upon restriction to S[U(k)×U(N−k)]. The multiplicities are given by the multiplicity of the trivial representation in the restriction. We will describe representations of SU(N) by Young diagrams. These will be denoted by their symbols J= [j1, j2,...,jN−1] where jidenotes the number of columns of height iof the diagram.3The fundamental representation, for instance, has J= [1,0,...] and the adjoint one J= [1,0,...,0,1]. Note that the complex conjugate representation is given by J∗= [jN−1,...,j1]. In appendix C, we show that a representation Jof SU(N) contains the trivial representation of S[U(k)× U(N−k)] if and only if it appears in the direct product ML≡[0k−1, L, 0N−k−1]⊗[0N−k−1, L, 0k−1] (12) for L≥n/N where n=Pijiis the number of boxes in the diagram Jand 0k−1 stands for k−1 zero entries. The multiplicity in the restriction is the same as that in the direct product. The MLsatisfy M1⊂M2⊂ ···, so they provide a hierarchy of truncations of the space of functions on the Grassmannian. The representation [0k−1, L, 0N−k−1] is the Young product of Lanti-symmetric k-tensors, for instance for k= 2, L= 5. As an example, for N= 6 and k= 2, the first truncation is M1= [0,1,0,0,0] ⊗[0,0,0,1,0] = [0,0,0,0,0] ⊕[1,0,0,0,1] ⊕[0,1,0,1,0] , ⊗= 1 ⊕ ⊕ (13) 3Note that this symbol is different from the highest weight vector sometimes used to describe a diagram, the entries of the latter being the number of boxes in each row. 4 the second is M2= [0,2,0,0,0] ⊗[0,0,0,2,0] = [0,0,0,0,0] ⊕[1,0,0,0,1] ⊕[0,1,0,1,0] ⊕[2,0,0,0,2] ⊕[1,1,0,1,1] ⊕[0,2,0,2,0] . ⊗= 1 ⊕ ⊕ ⊕ ⊕ ⊕ (14) Although not needed in the following, we would like to state for illustrational purposes that the full harmonic analysis on GN k(the “union” of all ML) is given by the representations J=([m1,...,mk,0,...,0, mk,...,m1] if 2k < N , [m1,...,mk−1,2mk, mk−1,...,m1] if 2k=N , (15) with mi= 0,1,..., each representation occurring once. The case 2k > N can be obtained by replacing kby N−k, since GN k∼ =GN N−k. The representations (15) correspond to Young diagrams which can be obtained by putting a diagram with at most krows next to its conjugate (which has at least N−k≥kboxes in any column), as can be verified for the examples given above. This result is also derived in appendix C. 4 Matrix geometry In the previous section, we have obtained a series of truncations of the space of functions on the Grassmannian. We will now see that these carry a natural product. The representation content MLof a given truncation can in fact be realised as an algebra of matrices in a representation Jof SU(N): since such matrices transform under SU(N) by conjugation, they form the representation space of J⊗J∗; so ML, as introduced in eq. (12), is equivalent to the space of matrices in the representation [0k−1, L, 0N−k−1]. Since the matrix product respects the action of SU(N), the algebra MLhas the same symmetries as GN k. In order to obtain the corresponding product of (truncated) functions, we shall now construct an injective map from MLto the space of functions on the Grassmannian which also respects the group action (an equivariant map). This map automatically provides the notions of differentiation and integration needed for the construction of actions: one just has to map the corresponding notions for functions back to matrices. Equivariance guarantees that they are compatible with the truncation. The map will also provide a non-commutative product for functions in the image of ML, the star product. If the star product tends to the point-wise product in the limit L→ ∞, we have succeeded in constructing a fuzzy GN k. Since we will restrict ourselves to k= 2 in the following sections, we present the map only for this case. The generalisation to other values of kshould be obvious. 5 For GN 2the basic building block will be the anti-symmetric representation , corresponding to L= 1. The first non-trivial truncation of functions therefore requires using [N(N−1)/2]×[N(N−1)/2] matrices. A function on GN 2is associated with such a matrix ˆ Fby restricting the tensor product of the fundamental projector (2) to the anti-symmetric representation , ρ≡(P ⊗P)a,(16) and constructing F1(ξ) = tr[ρ(ξ)ˆ F].(17) Since Phas rank 2, ρhas rank 1: let the plane onto which Pprojects be spanned by the vectors ~v and ~w;ρthen projects onto the 1-dimensional subspace of the representation space of spanned by the anti-symmetric product of ~v and ~w (an explicit proof is given in appendix B). A general truncation requires taking the L-fold (Young) product [0, L, 0,...] = ··· ··· of , which has dimension nN L=(N+L−1)!(N+L−2)! (N−1)!L!(N−2)!(L+1)! , and using nN L×nN Lmatrices. Equations (13) and (14), for instance, show the decomposition of 15 ×15 and 105 ×105 matrices as harmonics for G6 2. In the following we shall drop trailing zeros in symbols of representations, so the above representations will be denoted by [0, L]. The Young product can be obtained as a component of the symmetric tensor product, so a projector can be constructed by restricting the L-fold tensor product of ρto the representation [0, L], ρL= ( Ltimes z}| { ρ⊗···⊗ρ)[0,L],(18) (of course ρ1=ρ) and a function can be associated with any nN L×nN Lmatrix ˆ Fby FL(ξ) = tr[ρL(ξ)ˆ F].(19) Appendix B contains a proof that the map (19) is injective. The matrix geometries introduced here coincide with those obtained from complex line bundles [10] or generalised coherent states [11], see [14, 15]. FLis usually called the covariant symbol of the operator ˆ Fin these formulations, and injectivity is well known and follows from an analyticity argument. The relation with coherent states will be discussed in some more detail in appendix B. 5 Star product on GN 2 Multiplication of truncated functions on the Grassmannian can now be defined using matrix multiplication. The star product of two functions, FL= tr(ρLˆ F) and GL= tr(ρLˆ G), is obtained from the matrix product through the map (FL⋆ GL)(ξ) = trρL(ξ)ˆ Fˆ G.(20) By construction this is an associative product and it keeps within the class of functions truncated at level L. Our aim is to find an explicit expression for this star 6 product, purely in terms of FLand GLand their derivatives, thus eliminating the explicit reference to matrices. By orthonormality of the matrix elements in the representation [0, L], Rdµ(g)D[0,L] M1M2(g−1)D[0,L] M3M4(g) = (1/nN L)δM1M4δM2M3,ˆ Fcan be expanded as ˆ F=Zdµ(g)˜ F(g)D[0,L](g) (21) with ˜ F(g)≡nN LtrD[0,L](g−1)ˆ F.(22) Inserting this into (19), we obtain FL(ξ) = Zdµ(g)ωL(ξ, g)˜ F(g) (23) with ωL(ξ, g)≡trρL(ξ)D[0,L](g)(24) and the star product can be expressed as (FL⋆ GL)(ξ) = Zdµ(g)Zdµ(g′)ωL(ξ, gg′)˜ F(g)˜ G(g′).(25) We seek an expression for the star product in terms of derivatives acting on FL(ξ) and GL(ξ). By eqs. (25) and (23), this can be achieved by deriving an expression for ωL(ξ, gg′) in terms of derivatives of ωL(ξ, g) and ωL(ξ, g′) with respect to ξ. The latter is greatly facilitated by the observation that ωLcan be expressed in terms of ω1, ωL(ξ, g) = [ω1(ξ, g)]L,(26) because, by eq. (18), ρLfactorises into rank-1 projectors ρand D[0,L]acts as a direct product as well. The reason behind this relation is that the representation [0, L] when projected to S[U(2) ×U(N−2)] by ρLfactorises as [0,1]Lsince all other irreducible components of the product involve tensors that are anti-symmetric in 3 or more indices and therefore vanish in SU(2). As a first step, we have to find an expression for ω1(ξ, gg′). This is a straightforward but somewhat lengthy exercise. It is deferred to appendix A and yields ω1(ξ, gg′) = ω1(ξ, g)1 + ←− ∂AKAB−→ ∂B+1 4←− ∂A←− ∂BKACKBD−→ ∂C−→ ∂Dω1(ξ, g′) (27) where ∂A=∂/∂ξAand Kis the projector onto the holomorphic tangent space introduced in eq. (4). Substituting (27) in (25) and interchanging differentiation with respect to ξwith integration over gand g′, we now have the star product at level one, (F1⋆ G1)(ξ) = F1(ξ)1 + ←− ∂AKAB−→ ∂B+1 4←− ∂A←− ∂BKACKBD−→ ∂C−→ ∂DG1(ξ).(28) 7 For higher Lwe have to consider ωL= (ω1)L. Equation (27) implies ωL(ξ, gg′) = X n+m≤L L! n!m! (L−n−m)! (ωω′)L−n−m(∂Aω)KAB(∂Bω′)n ×1 4(∂C∂Dω)KCEKDF (∂E∂Fω′)m (29) where we have used the abbreviations ω≡ω1(ξ, g) and ω′≡ω1(ξ, g′). The righthand side of this equation has to be expressed in terms of multiple derivatives acting on ωL(ξ, g) and ωL(ξ, g′). It contains several different terms with a given number of derivatives. This means that we have to distinguish components of multiple derivatives of ωL. To this end, we decompose multiple holomorphic derivatives ∇A1···∇AnωLas defined in eq. (9) with respect to irreducible representations of the stability group Hwhich acts on the tangent space. It will be sufficient to consider the subgroup H0=SU(2) ×SU(N−2) of H. Representations of H0will be denoted by (J, J′) where Jis a representation of the first factor and J′one of the second. To find the representation content of a single holomorphic derivative ∇A=KAB∂B, note that the fundamental representation [1] of SU(N) decomposes as [1]H0= ([1],[0]) ⊕([0],[1]) (30) into the fundamental representations of SU(2) and SU(N−2) upon restriction to H0. The two components can be obtained by projection with Pand 1 −P. Now use eqs. (4) and (5) to write ∇Ain terms of (anti-)fundamental indices, (tA)ij∇Af(ξ) = (1 −P)tBPi j∂Bf(ξ).(31) The matrix tB∂Bftransforms like the traceless component of [1]×[1]∗under SU(N). Since the index iis projected by 1 − P to ([0],[1]) while jis projected by P to ([1],[0])∗, we find that the holomorphic derivative transforms like ([0],[1]) ⊗ ([1],[0])∗= ([1]∗,[1]). Note that tracelessness is guaranteed by the projections in eq. (31). By the same reasoning, an anti-holomorphic derivative ¯ ∇A=KBA∂B transforms like ([1],[1]∗). A multiple holomorphic derivative transforms like the symmetric tensor product of ncopies of the representation ([1]∗,[1]). In order to obtain an explicit expression for the decomposition of this product, it turns out to be useful to first decompose the tensor product of ncopies of the fundamental representation of SU(N). This can be done by considering the action of the symmetric group Sn, whose elements permute the factors in the tensor product [28]. The latter can then be decomposed into irreducible representations of SU(N)×Snwith the help of character projection operators. They provide the following decomposition of unity, 1 = X |J|=n PJwhere PJ≡dJ n!X π∈Sn χJ(π)π . (32) Here, the sum is over all Young diagrams with nboxes, χJis the character of the symmetric group in the representation Jand dJthe dimension of that representation. 8 Since ρ1= (P ⊗P)ais a rank-1 projector, the simple product can be written as ω(ξ, g)ω(ξ, g′) = tr(P ⊗P)a(g⊗g)a(P ⊗P)a(g′⊗g′)a.(64) Using 1 = (P ⊗ P)a+ (P ⊗ (1 − P))a+ ((1 − P)⊗ P)a+ ((1 − P)⊗(1 − P))a, eqs. (62), (63) and (64) can be combined to ω(ξ, gg′) = ω(ξ, g)1 + ←− ∂AKAB−→ ∂B+1 4←− ∂A←− ∂BKACKBD−→ ∂C−→ ∂Dω(ξ, g′),(65) which is the desired expression. B Coherent states We show that the projector ρLas given in eq. (18) has rank 1, and we present a simple argument for why the map from matrices to functions is injective. To this end, we require a more explicit representation for ρL. The vector space of the irreducible representation of SU(N) with symbol J= [0, L] can be realised as a sub-space with certain symmetry properties of the space of 2L-index tensors. We construct it as the image of a Young symmetriser. We first assign tensor indices to the boxes in the Young diagram of the representation by putting the numbers 1,2,...,2Lin ascending order into one column after the other, for instance 1 3 5 7 2468 for [0,4]. The Young symmetriser is now defined as Y[0,L]=2L L+ 1ALSL, AL= L Y i=1 1 2(1 −τi),SL=1 L!X π1∈R1 π1 1 L!X π2∈R2 π2 (66) where τiinterchanges the two boxes of the ith column of the diagram and Ridenotes the set of permutations that permute the boxes of row i. So SLsymmetrises the rows of the diagrams, while ALanti-symmetrises the columns. Both are symmetric projectors. The Young symmetriser is a projector, Y2 [0,L]=Y[0,L], but not symmetric. Operators in the vector space of [0, L] can be unambiguously described as operators ˆ Fon 2L-tensors that satisfy ˆ FY[0,L]=Y[0,L]ˆ F=ˆ F . (67) Now we can prove that ρLhas rank 1 by expressing the rank-2 projector Pin terms of an orthonormal basis |ϕi,|ψiof the complex plane onto which it projects, P=|ϕihϕ|+|ψihψ|. The level-1 projector ρwas defined in (16) as the projection of the tensor product of Pwith itself to the anti-symmetric representation [0,1]. Since Y[0,1] reduces to a single anti-symmetrisation, ρ≡(P ⊗P)a= (P ⊗P)Y[0,1] =|ϕψihϕψ|(68) 15 where |ϕψi ≡ 1 √2|ϕi|ψi−|ψi|ϕi.(69) For the projector at level L, we obtain ρL= (ρ⊗···⊗ρ)Y[0,L]=|ϕψiLhϕψ|LY[0,L](70) The state |ϕψicompletely characterises the plane that corresponds to a point in GN 2. It is therefore natural that it occurs as a fundamental object in the construction. The states |ϕψiLcoincide, up to a conventional phase, with the generalised coherent states discussed in [11]. Since ˆ FY[0,L]=Y[0,L]ˆ F, FL(ξ) = hϕψ|Lˆ F|ϕψiL.(71) So FLis the covariant symbol, as defined in [11], of the operator ˆ F. This expression can be used to show that the map is injective. We have to show that ˆ Fcan be reconstructed from FL. Since |ϕψiL= 2L/2AL(|ϕi|ψi)L, FL(ξ) = 2Lhϕ|hψ|LALˆ FAL|ϕi|ψiL. Due to the anti-symmetrisation between |ϕiand |ψithis function can be homogeneously extended to general (non-orthonormal) |ϕiand |ψi. We choose |ϕi= PN n=1 an|niand |ψi=PN n=1 bn|niwith canonical basis vectors |ni. By differentiating with respect to a,band their complex conjugates, all matrix elements of SLALˆ FALSLcan be obtained. Using the symmetry (67), we obtain SLˆ Fand thus also 2L L+1ALSLˆ F=ˆ F. C Restrictions and direct products We shall derive the relation between the restriction of representations of G=SU(N) to H=S[U(k)×U(N−k)] and the direct product of certain representations used in section 3. In this appendix, we will allow for columns of height N,J= [j1, j2,...,jN], in diagrams describing representations of SU(N). These do not lead to new representations, since representations differing only by jNare unitarily equivalent, but this generalisation will make formulas much simpler. We embed Hinto SU(N) as ei(N−k)ϕU′0 0 e−ikϕU′′(72) where U′∈SU(k) and U′′ ∈SU(N−k) and eiϕ∈U(1). This shows that H= [SU(k)×SU(N−k)×U(1)]/Znwhere nis the least common multiple of kand N−k. Representations of Hcan thus be considered as representations of SU(k)× SU(N−k)×U(1) that represent Zntrivially. This fixes the charge qof the U(1) factor eiqϕ of the representation modulo n. We will denote these representations as (J′, J′′)qwhere J′and J′′ are symbols of SU(k) and SU(N−k) representations, respectively, and qis the charge of the U(1) representation. 16 The restriction of an SU(N) representation to Hcan be written as JH=M J′,J′′ mJ J′,J′′ (J′, J′′)(N−k)|J′|−k|J′′|.(73) Here, |J|=Pijiis the number of boxes in the diagram Jand we assume that the diagrams have been chosen such that the total number of boxes in J′and J′′ is the same as in J, |J|=|J′|+|J′′|.(74) Note that the U(1) representation is determined by the SU(k)×SU(N−k) representation, so the multiplicities mJ J′,J′′ are the same as for the restriction from SU(N) to the latter. It is known ([24, 25], also see [26, 27]) that these can be obtained from the decomposition of the direct product of the SU(N) representations with diagrams J′and J′′, J′⊗J′′ =M J mJ J′,J′′ Jin SU(N). (75) Here, again, the restriction (74) on the number of boxes applies. Note that in eq. (75) J′and J′′ are interpreted as SU(N) representations while they are interpreted as SU(k) respectively SU(N−k) representations in eq. (73). We are interested in the case where the trivial representation of Happears on the right-hand side of (73). This means that J′and J′′ only have columns of height kand N−k, respectively, J′= [0k−1, L′,0N−k−1] and J′′ = [0N−k−1, L′′,0k−1] where 0k−1stands for k−1 zero entries, etc. In addition, the U(1) charge has to vanish, (N−k)|J′|=k|J′′|. Since |J′|=L′kand |J′′|=L′′(N−k), this implies L′=L′′ ≡L, so that J′and J′′ are complex conjugate representations of SU(N). We conclude that a representation of SU(N) contains the trivial representation of S[U(k)×U(N−k)] if and only if it appears in the decomposition of the direct product ML≡[0k−1, L, 0N−k−1]⊗[0N−k−1, L, 0k−1] (76) for some L. The multiplicities are given by the multiplicities mJ [0k−1,L,0N−k−1],[0N−k−1,L,0k−1] in the product. Note that the multiplicity does not depend on the number jNof columns of height Nin the diagram Jchosen for a given representation. This means that a representation appears in MLif and only if NL ≥ |J|where Jis the minimal diagram (jN= 0) of the representation. Therefore ML⊂ML′if L < L′. Decomposition of the direct product For illustrational purposes, we will explicitly perform the decomposition of the direct product (76) into irreducible representations. We can assume k≤N−ksince GN k∼ =GN N−k. The decomposition is achieved by Young diagram techniques. Recall the rules for decomposing the direct product of two irreducible representations of SU(N) [28]: 1. Label each box in the second diagram by its row number. 17 nN−k+1 boxes → nN−k+2 boxes → nN−k+kboxes → 1·· ·· ·· ·· ·· ·· ·· ·· ·· 1 2·· ·· ·· ·· ·· ·· 2 ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· k·· k 1·· 1 2 ·· 2·· ·· k·· k 2·· 2·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· k·· k ←n11’s ←n22’s ←nkk’s Figure 1: Decomposition of the tensor product [0N−k−1, L, 0k−1]⊗[0k−1, L, 0N−k−1]. 2. Attach all boxes with 1’s to the first diagram, then all boxes with 2’s and so on, such that (a) at all stages the intermediate diagram corresponds to an irreducible representation of SU(N), i.e. all columns start in the first row and are connected and the height of the columns monotonically decreases from left to right, (b) no column contains any number more than once, and (c) when counted from the right, the n-th idoes not appear before the n-th i−1. Applying these rules to our case, we have to attach boxes with Lcopies of each of the numbers 1,...,kto a rectangle of height N−kand width L. This is indicated in figure 1. We have already anticipated some facts about the resulting distribution of boxes, which we will explain now. The number of 1’s in the first row has been denoted by n1. The remaining L−n11’s have to be in the N−k+ 1-st row. Denoting the number of 2’s in the second row by n2, there can be at most n1−n2 2’s in row N−k+ 1, because of rule 2c. There must thus be at least L−n12’s in row N−k+ 2. However, owing to rule 2b, the number of 2’s in that row is at most L−n1. Therefore, there have to be exactly n1−n22’s in row N−k+ 1 and L−n1 2’s in row N−k+ 2. By the same reasoning, we find that the number of i’s in row N−k+lfor l > 0 is equal to the number of i−l+ 1’s in the row N−k+ 1 which is in turn equal to ni−l−ni−l+1 if i≥land 0 if i < l (we have put n0≡L). Adding up, we find for the number of boxes in row N−k+l, nN−k+l= k X i=l (ni−l−ni−l+1) = L−nk−l+1 for l= 1,...,k. (77) Graphically, this means that the subdiagram of columns L+ 1, L + 2,...combines, after a rotation by π, with the remainder of the diagram to a rectangle of width Land height N. In terms of symbols J= [j1,...,jN], where jiis the number of columns of height i, this implies J=([m1,...,mk,0,...,0, mk,...,m1] if 2k < N , [m1,...,mk−1,2mk, mk−1,...,m1] if 2k=N(78) 18 with mi=ni−ni+1 where nk+1 ≡0. All non-negative values of misatisfying Pk i=1 mi=n1≤Loccur. Furthermore, each diagram can be obtained in only one way, since it is determined by the numbers ni(i= 1,...,k). So all multiplicities equal 1. D Symmetric group In this appendix, we shall provide a proof of the factorisation property (38). On the way, we will recall some facts about representations of the symmetric group and the associated projectors PJused in the text. These projectors can be considered as elements of the group algebra R[Sj]≡span Sn={A=Pπ∈SnAππ|Aπ∈R}, the set of formal linear combinations of group elements. The vector space R[Sj] carries a representation of the algebra R[Sj] whose action is given by left multiplication. This representation restricts to a representation of the subgroup Snof R[Sj]. It is usually called the regular representation. Sometimes it is convenient to identify R[Sj] with the algebra F(Sn) of functions on the group Snby setting A(π)≡Aπ. The action of a group element πis then given by (πA)(σ) = A(π−1σ). F(Sn) can in fact be considered as the dual vector space of R[Sj] since each function on the group can be linearly and uniquely extended to a function on R[Sj]. The identification of R[Sj] with its dual space can be obtained from the inner product hA, Bi ≡ X π∈Sn AπBπ=1 n!tr(ATB) (79) by putting A(B) = hA, Bi. The trace in eq. (79) is over the regular representation and we have set AT=PπAππ−1. Of particular importance are the central elements of R[Sj], that are invariant under conjugation with any group element π∈Sn,A=πAπ−1. They correspond to class functions, i.e. functions that depend only on the conjugacy class of their argument. An orthogonal basis in the subspace of class functions is given by the characters χJassociated with the irreducible representations Jof Sn, hχJ, χJ′i=n!δJJ′.(80) So every class function can be expanded as A=X J AJχJwith AJ=1 n!hχJ, Ai.(81) To each A∈R[Sj], one can associate a central element Aby averaging with respect to conjugation, A≡1 n!X π∈Sn πAπ−1=1 n!X JhχJ, AiχJ(82) where we have used (81) and the invariance of χJunder conjugation. The regular representation is in general reducible. It contains each irreducible representation Jwith a multiplicity that is given by the dimension dJof the representation. The component containing all copies of an irreducible representation J 19 can be obtained as the image of the symmetric projection operator introduced in eq. (32), in the dual picture, PJ=dJ n!χJ.(83) The decomposition of unity 1 = X J PJ(84) provides a decomposition of R[Sj] into orthogonal subspaces [29]. Now we will show how the averaged tensor product of two projectors PJ1and PJ2onto irreducible representations of Sn1and Sn2can be expressed in terms of irreducible projectors. PJ1⊗PJ2can be extended to R[Sj] where n=n1+n2. By (82), we have PJ1⊗PJ2=X J aJχJ(85) with aJ=1 n!hχJ, PJ1⊗PJ2i.(86) The restriction of χJto (π, σ)∈Sn1×Sn2decomposes into irreducible characters as χJ(π, σ) = X J1,J2 cJ J1J2χJ1(π)χJ2(σ) (87) where cJ J1J2∈Zare multiplicities or Clebsch-Gordan coefficients. With (80) we get aJ=1 n!X J′ 1,J′ 2 cJ J′ 1J′ 2hχJ′ 1, PJ1ihχJ′ 2, PJ2i=dJ1dJ2 n!cJ J1J2(88) and therefore PJ1⊗PJ2=X J dJ1dJ2 n!cJ J1J2χJ=X J dJ1dJ2 dJ cJ J1J2PJ.(89) By iteration, this result can be generalised to multiple products, 1 dJ1···dJm PJ1⊗···⊗PJm=X J cJ J1...Jm 1 dJ PJ.(90) Note that symmetrisation with respect to Snimplies symmetrisation with respect to Sn′⊂Sn, A⊗B⊗C=A⊗B⊗C . (91) Now we can prove eq. (38). The right-hand side of this equation can be written as a single trace like in eq. (36) but with PJreplaced by P[l]⊗P⊗m [0,1]. Owing to the symmetric tensors Sand Tall factors in the trace except P[l]⊗P⊗m [0,1] are symmetric under conjugation, so P[l]⊗P⊗m [0,1] can be replaced by its symmetrised version P[l]⊗P⊗m [0,1]. Now we can insert eq. (90). The only term on the right-hand side that does not vanish when projected by P⊗nto a representation of SU(2) is J= [l, m, 0,...] with multiplicity 1. The dimensions of the representations [l] and [0,1] (of the symmetric group) are 1, while the dimension of [l, m] appearing in the denominator of (90) just cancels that on the right-hand side of (38), so we obtain the left-hand side. 20 E Projection of multiple derivatives We compute the product of multiple (anti-)holomorphic derivatives of ωLand ω′ L projected to the representation ([l, m]∗,[l, m]) of the stability group, X(L) l,m ≡(∂A1···∂Al+2mωL)KA1...Al+2m,B1...Bl+2m [l,m](∂B1···∂Bl+2mω′L).(92) By eq. (9) and since K[l,m]contains the projector K, the derivatives in this equation are really covariant derivatives, holomorphic ones acting on ω′and anti-holomorphic ones on ω. Equation (38) implies that K[l,m]can be replaced by d[l,m]K[l]⊗K⊗m [0,1], X(L) l,m =d[l,m](∂l+2mωL)K[l]⊗K⊗m [0,1](∂l+2mω′L) (93) where we have introduced an index-free notation. Using the second equality of eq. 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