ORBIFOLDS OF SYMPLECTIC FERMION ALGEBRAS THOMAS CREUTZIG AND ANDREW R. LINSHAW ABSTRACT. We present a systematic study of the orbifolds of the rank n symplectic fermion algebra A(n), which has full automorphism group Sp(2n). First, we show that A(n) Sp(2n) and A(n) GL(n) are W-algebras of type W(2, 4,..., 2n) and W(2, 3,..., 2n + ), respectively. Using these results, we find minimal strong finite generating sets for A(mn) Sp(2n) and A(mn) GL(n) for all m, n. We compute the characters of the irreducible representations of A(mn) Sp(2n) SO(m) and A(mn) GL(n) GL(m) appearing inside A(mn), and we express these characters using partial theta functions. Finally, we give a complete solution to the Hilbert problem for A(n); we show that for any reductive group G of automorphisms, A(n) G is strongly finitely generated.. INTRODUCTION Vertex algebras are a fundamental class of algebraic structures that arose out of conformal field theory and have applications in a diverse range of subjects. Given a vertex algebra V and a group G of automorphisms of V, the invariant subalgebra V G is called an orbifold of V. Many interesting vertex algebras can be constructed either as orbifolds or as extensions of orbifolds. A spectacular example is the Moonshine vertex algebra V, whose full automorphism group is the Monster, and whose graded character is j(τ) 744 where j(τ) is the modular invariant j-function [FLM]. In physics, rational vertex algebras, of which V is an example, correspond to rational two-dimensional conformal field theories. A rational vertex algebra V has only finitely many irreducible, admissible modules, and any admissible V-module is completely reducible. On the other hand, many interesting vertex algebras admit modules which are reducible but indecomposable. Such vertex algebras are called logarithmic as the corresponding conformal field theories often have logarithmic singularities in their correlation functions. For many years after Zhu s thesis appeared in 990 [Z], it was believed that rationality and the C 2 -cofiniteness conditions were equivalent. However, this was disproven by the construction of the W p,q -triplet algebras [KI, FGST], which are logarithmic and C 2 -cofinite [AM, TW]. One of the first logarithmic conformal field theories studied in the physics literature was the symplectic fermion theory [KII]. The symplectic fermion algebra A(n) of rank n has odd generators e i, f i for i =,..., n satisfying operator product expansions e i (z)f j (w) δ i,j (z w) 2. It is an important example of a free field algebra, and the simplest triplet algebra W 2, coincides with the orbifold A() Z/2Z. It is also known that A(n) Z/2Z is logarithmic and Key words and phrases. free field algebra, symplectic fermions, invariant theory; reductive group action; orbifold; strong finite generation; W-algebra; character formula; theta function.
C 2 -cofinite [A], but little is known about the structure and representation theory of more general orbifolds of A(n). Our goal in this paper is to carry out a systematic study of the orbifolds A(n) G, where G is any reductive group of automorphisms of A(n). The full automorphism group of A(n) is the symplectic group Sp(2n), and in order to describe A(n) G for a general G, a detailed understanding of the structure and representation theory of A(n) Sp(2n) is necessary. As we shall see, all irreducible, admissible A(n) Sp(2n) -modules are highest-weight modules, and A(n) G is completely reducible as an A(n) Sp(2n) -module. Our study of A(n) Sp(2n) is based on classical invariant theory, and follows the approach developed in [LI, LII, LIII, LIV, LV]. First, A(n) admits an Sp(2n)-invariant filtration such that gr(a(n)) = k 0 U k as supercommutative rings. Here each U k is a copy of the standard Sp(2n)-representation C 2n. As a vector space, A(n) Sp(2n) is isomorphic to R = ( k 0 U k) Sp(2n), and we have isomorphisms of graded commutative rings gr(a(n) Sp(2n) ) = gr(a(n)) Sp(2n) = R. In this sense, we regard A(n) Sp(2n) as a deformation of the classical invariant ring R. By an odd analogue of Weyl s first and second fundamental theorems of invariant theory for the standard representation of Sp(2n), R is generated by quadratics {q a,b 0 a b}, and the ideal of relations among the q a,b s is generated by elements {p I I = (i 0,..., i 2n+ ), 0 i 0 i 2n+ } of degree n +, which are analogues of Pfaffians. We obtain a corresponding strong generating set {ω a,b } for A(n) Sp(2n), as well as generators {P I } for the ideal of relations among the ω a,b s, which correspond to the classical relations with suitable quantum corrections. In fact, there is a more economical strong generating set {j 2k = ω 0,2k k 0}, and the sets {ω a,b 0 a b} and { i j 2k i, k 0} are related by a linear change of variables. The relation of minimal weight among the generators occurs at weight 2n + 2, and corresponds to I = (0, 0,..., 0). A key technical result in this paper is the analysis of the quantum corrections of the above classical relations. For each p I, there is a certain correction term R I appearing in P I which we call the remainder. We show that R I satisfies a recursive formula which implies that the relation of minimal weight has the form (.) j 2n = Q(j 0, j 2,..., j 2n 2 ). Here Q is a normally ordered polynomial in j 0, j 2,... j 2n 2 and their derivatives. We call (.) a decoupling relation, and by applying the operator j 2 repeatedly, we can construct higher decoupling relations j 2m = Q m (j 0, j 2,..., j 2n 2 ) for all m > n. This shows that {j 0, j 2,..., j 2n 2 } is a minimal strong generating set for A(n) Sp(2n), and in particular A(n) Sp(2n) is of type W(2, 4,..., 2n); see Theorem 3.. Using this result, we give a minimal strong generating set for A(mn) Sp(2n) for all m, n. Similarly, using the invariant theory of GL(n) we show that A(n) GL(n) is of type 2
W(2, 3,..., 2n + ), and we give a minimal strong generating set for A(mn) GL(n) for all m, n. Character decompositions. By a general theorem of Kac and Radul [KR], A(mn) decomposes as a sum of irreducible A(mn) G -modules for every reductive group G Sp(2mn) (see also [DLM]). Recently, the characters of some orbifolds of the βγ-system of rank n have been computed [BCR]. The task was to find Fourier coefficients of certain negative index meromorphic Jacobi forms, and one way to solve the decomposition problem used the denominator identity of affine Lie superalgebras. We will instead use the denominator identity of the finite-dimensional Lie superalgebra spo(2n m) to find the character decomposition when G = Sp(2n) SO(m), and the denominator identity of gl(n m) to find the character decomposition for G = GL(n) GL(m). In all these cases we obtain explicit character formulas, and they are expressed using partial theta functions. The character of A(mn) is q mn 2 ( + q j ) 2mn = j= with the Dedekind eta function η(q) = q 24 ( q n ). n= ( ) η (q 2 2mn ) η (q) In order to perform the character decomposition, we need to refine the grading. For this let g 0 = gl(m) gl(n) or g 0 = sp(2m) so(n) be the even subalgebra of g = gl(m n), respectively g = spo(2m n). Then the set of odd roots of g is contained in the weight lattice of g 0, and the A(mn)-algebra character, graded by both the weight lattice of g 0 and by conformal dimension, is ch[a(mn)] = q mn 2 The character can be rewritten as ch[a(mn)] = e ρ q mn e α 2 2 ( + e α ) α + n= α n= ( + e α q n ). ( + e α q n ) ( + e α q n ) = e ρ α + where ρ is the odd Weyl vector of g, + is the set of its positive odd roots and ϑ(z; q) = n Z z n+ 2 q 2(n+ 2) 2 ϑ (e α ; q) ( + e α ) η(q), is a standard Jacobi theta function. Let P + be the set of dominant weights of the even subalgebra g 0. Then our result, Theorem 5.4, states that the graded character of A(mn) decomposes as ch[a(mn)] = ch Λ B Λ. Λ L P + Here ch Λ is the irreducible highest-weight representation of g 0 of highest-weight Λ. The branching function is B Λ = W η(q) + W ɛ(w) ω W ((n α),(m β )) I ω(λ+ρ0 ) ρ 0 q 3 2(n α+ 2) 2 P mβ (q)
and P n (q) := q 2(n+ 2) 2 m=0 ( ) m q 2(m 2 2m(n+ 2)) is a partial theta function. Further, ρ 0 is the Weyl vector of g 0, L is the root lattice of g, W is its Weyl group and W the subgroup of W corresponding to the larger subalgebra of g 0 and finally I λ is a subset of Z +. More details on these objects are found in Section 5. Summing over all representations corresponding to either so(m) or gl(m), one then obtains the character decomposition for G = Sp(2n), and respectively for G = GL(n). In particular, the character formula for A(n) Sp(2n) implies that it is freely generated; there are no nontrivial normally ordered relations among the generators j 0, j 2,..., j 2n 2. This yields an explicit classification of the irreducible, admissible A(n) Sp(2n) -modules. One of the important properties of rational vertex algebras is that characters of modules are the components of a vector-valued modular form for the modular group SL(2, Z) [Z]. In the non-rational case, the relation to modularity is unclear. However, there are examples of affine vertex superalgebra whose characters are built out of mock modular forms [KWI]. For the singlet vertex algebra W(2, 2p ) for p 2, the module characters are, up to a prefactor of η(q), partial theta functions [F], and W(2, 3) coincides with A() GL(). The characters are then not modular, but their modular group action is still known [Zw, CM]. They carry an infinite-dimensional modular group representation that is compatible with the expectations of vertex algebras and conformal field theory in the cases of a family of vertex superalgebras [AC] and the singlet vertex algebras [CM]. The Hilbert problem for A(n). A vertex algebra A is called strongly finitely generated if there exists a finite set of generators such that the set of iterated Wick products of the generators and their derivatives spans A. Recall Hilbert s theorem that if a reductive group G acts on a finite-dimensional complex vector space V, the ring O(V ) G of invariant polynomial functions is finitely generated [HI, HII]. This theorem was very influential in the development of commutative algebra and algebraic geometry. In fact, Hilbert s basis theorem, Nullstellensatz, and syzygy theorem were all introduced in connection with this problem. The analogous question for vertex algebras is the following. Problem.. Given a simple, strongly finitely generated vertex algebra A and a reductive group G of automorphisms of A, is A G strongly finitely generated? This problem was solved affirmatively when V is the Heisenberg vertex algebra H(n) [LIII, LIV], and when V is the βγ-system S(n) or free fermion algebra F(n) [LV]. The approach is the same in all these cases and is based on the following observations. () The Zhu algebra of V Aut(V) is abelian, which implies that its irreducible, admissible modules are all highest-weight modules. (2) For any reductive G, V G has a strong generating set that lies in the direct sum of finitely many irreducible V Aut(V) -modules. (3) The strong finite generation of V Aut(V) implies that these modules all have a certain finiteness property. In the case V = A(n), these conditions are satisfied, so the same approach yields a solution to the Hilbert problem for A(n); see Theorem 6.2. The method is essentially constructive and it reveals how A(n) G arises as an extension of A(n) Sp(2n) by irreducible modules. 4
A free field algebra is any vertex algebra of the form H(n) F(m) S(r) A(s), m, n, r, s 0, where V(0) is declared to be C for V = H, S, F, A. Many interesting vertex algebras have free field realizations, that is, embeddings into such algebras. In addition, many vertex algebras can be regarded as deformations of free field algebras. For example, if g = g 0 g is a Lie superalgebra with a nondegenerate, supersymmetric bilinear form, the corresponding affine vertex superalgebra is a deformation of H(n) A(m), where n = dim(g 0 ) and 2m = dim(g ). By combining Theorem 6.2 with the results of [LIII, LIV, LV], one can establish the strong finite generation of a broad class of orbifolds of free field algebras, as well as their deformations. This has applications to the longstanding problem of describing coset vertex algebras, which will be developed in a separate paper. 2. VERTEX ALGEBRAS In this section, we define vertex algebras, which have been discussed from various points of view in the literature (see for example [B, FLM, K, FBZ]). We will follow the formalism developed in [LZ] and partly in [LiI]. Let V = V 0 V be a super vector space over C, and let z, w be formal variables. By QO(V ), we mean the space of all linear maps V V ((z)) = { n Z v(n)z n v(n) V, v(n) = 0 for n >> 0}. Each element a QO(V ) can be uniquely represented as a power series a = a(z) = n Z a(n)z n End(V )[[z, z ]]. We refer to a(n) as the n th Fourier mode of a(z). Each a QO(V ) is assumed to be of the form a = a 0 + a where a i : V j V i+j ((z)) for i, j Z/2Z, and we write a i = i. There is a set of nonassociative bilinear operations n on QO(V ), indexed by n Z, which we call the n th circle products. For homogeneous a, b QO(V ), they are defined by a(w) n b(w) = Res z a(z)b(w) ι z > w (z w) n ( ) a b Res z b(w)a(z) ι w > z (z w) n. Here ι z > w f(z, w) C[[z, z, w, w ]] denotes the power series expansion of a rational function f in the region z > w. We usually omit the symbol ι z > w and just write (z w) to mean the expansion in the region z > w, and write (w z) to mean the expansion in w > z. For a, b QO(V ), we have the following identity of formal power series, known as the operator product expansion (OPE) formula. (2.) a(z)b(w) = n 0 a(w) n b(w) (z w) n + : a(z)b(w) :. Here : a(z)b(w) : = a(z) b(w) + ( ) a b b(w)a(z) +, where a(z) = n<0 a(n)z n and a(z) + = n 0 a(n)z n. Often, (2.) is written in the form a(z)b(w) n 0 a(w) n b(w) (z w) n, where means equal modulo the term : a(z)b(w) :, which is regular at z = w. 5
Note that : a(w)b(w) : is a well-defined element of QO(V ). It is called the Wick product of a and b, and it coincides with a b. The other negative circle products are given by n! a(z) n b(z) = : ( n a(z))b(z) :, = d dz. For a (z),..., a k (z) QO(V ), the k-fold iterated Wick product is defined to be (2.2) : a (z)a 2 (z) a k (z) : = : a (z)b(z) :, b(z) = : a 2 (z) a k (z) :. We often omit the formal variable z when no confusion can arise. The set QO(V ) is a nonassociative algebra with the operations n, which satisfy n a = δ n, a for all n, and a n = δ n, a for n. A subspace A QO(V ) containing which is closed under the circle products will be called a quantum operator algebra (QOA). A subset S = {a i i I} of A is said to generate A if every a A can be written as a linear combination of nonassociative words in the letters a i, n, for i I and n Z. We say that S strongly generates A if every a A can be written as a linear combination of words in the letters a i, n for n < 0. Equivalently, A is spanned by {: k a i km a im : i,..., i m I, k,..., k m 0}. We say that S freely generates A if there are no nontrivial normally ordered polynomial relations among the generators and their derivatives. We say that a, b QO(V ) quantum commute if (z w) N [a(z), b(w)] = 0 for some N 0. Here [, ] denotes the super bracket. This condition implies that a n b = 0 for n N, so (2.) becomes a finite sum. Finally, a commutative QOA is a QOA whose elements pairwise quantum commute, and this notion is well known to be equivalent to the notion of a vertex algebra in the sense of [FLM]. Our main example in this paper is the symplectic fermion algebra A(n) of rank n, which is freely generated by odd elements {e i, f i i =,..., n} satisfying (2.3) e i (z)f j (w) δ i,j (z w) 2, f j (z)e i (w) δ i,j (z w) 2, e i (z)e j (w) 0, f i (z)f j (w) 0. We give A(n) the conformal structure (2.4) L A = n : e i f i : i= of central charge 2n, under which e i, f i are primary of weight one. The full automorphism group of A(n) is the symplectic group Sp(2n). It acts linearly on the generators, which span a copy of the standard Sp(2n)-module C 2n. Category R. Let R be the category of vertex algebras A equipped with a Z 0 -filtration (2.5) A (0) A () A (2), A = k 0 A (k) such that A (0) = C, and for all a A (k), b A (l), we have { A(k+l) n < 0 (2.6) a n b A (k+l ) n 0. Elements a(z) A (d) \ A (d ) are said to have degree d. Filtrations on vertex algebras satisfying (2.6) were introduced in [LiII], and are known as good increasing filtrations. Setting A ( ) = {0}, the associated graded object gr(a) = 6
k 0 A (k)/a (k ) is a Z 0 -graded associative, (super)commutative algebra with a unit under a product induced by the Wick product on A. For each r we have the projection (2.7) φ r : A (r) A (r) /A (r ) gr(a). Moreover, gr(a) has a derivation of degree zero (induced by the operator on A), and for each a A (d) and n 0, the operator a n on A induces a derivation of degree d k on gr(a), which we denote by a(n). Here k = sup{j A (r) n A (s) A (r+s j) r, s, n 0}, as in [LL]. Finally, these derivations give gr(a) the structure of a vertex Poisson algebra. The assignment A gr(a) is a functor from R to the category of Z 0 -graded (super)commutative rings with a differential of degree 0, which we call -rings. A -ring is just an abelian vertex algebra, that is, a vertex algebra V in which [a(z), b(w)] = 0 for all a, b V. A -ring A is said to be generated by a subset {a i i I} if { k a i i I, k 0} generates A as a ring. The key feature of R is the following reconstruction property [LL]. Lemma 2.. Let A be a vertex algebra in R and let {a i i I} be a set of generators for gr(a) as a -ring, where a i is homogeneous of degree d i. If a i (z) A (di ) are vertex operators such that φ di (a i (z)) = a i, then A is strongly generated as a vertex algebra by {a i (z) i I}. As shown in [LI], there is a similar reconstruction property for kernels of surjective morphisms in R. Let f : A B be a morphism in R with kernel J, such that f maps A (k) onto B (k) for all k 0. The kernel J of the induced map gr(f) : gr(a) gr(b) is a homogeneous -ideal (i.e., J J). A set {a i i I} such that a i is homogeneous of degree d i is said to generate J as a -ideal if { k a i i I, k 0} generates J as an ideal. Lemma 2.2. Let {a i i I} be a generating set for J as a -ideal, where a i is homogeneous of degree d i. Then there exist vertex operators a i (z) A (di ) with φ di (a i (z)) = a i, such that {a i (z) i I} generates J as a vertex algebra ideal. We now define a good increasing filtration on the symplectic fermion algebra A(n). First, A(n) has a basis consisting of the normally ordered monomials (2.8) : I e In e n J f Jn f n :. In this notation, I k = (i k,..., i k r k ) and J k = (j k,..., j k s k ) are lists of integers satisfying 0 i k < < i k r k and 0 j k < < j k s k, and We have a Z 0 -grading I k e k = : ik ek ik r k e k :, J k f k = : jk f k jk s k f k :. (2.9) A(n) = d 0 A(n) (d), where A(n) (d) is spanned by monomials of the form (2.8) of total degree d = n k= r k + s k. Finally, we define the filtration A(n) (d) = d i=0 A(n)(i). This filtration satisfies (2.6), and we have an isomorphism of Sp(2n)-modules (2.0) A(n) = gr(a(n)), 7
and an isomorphism of graded supercommutative rings (2.) gr(a(n)) = k 0 U k. Here U k is the copy of the standard Sp(2n)-module C 2n with basis {e i k, f k i }. In this notation, e i k and f k i are the images of k e i (z) and k f i (z) in gr(a(n)). The -ring structure on k 0 U k is defined by e i k = ei k+ and f i k = f i k+. For any reductive group G Sp(2n), this filtration is G-invariant and is inherited by A(n) G. We obtain a linear isomorphism A(n) G = gr(a(n) G ) and isomorphisms of -rings (2.2) gr(a(n) G ) ( ) = gr(a(n)) G G. = U k Finally, the weight grading on A(n) is inherited by gr(a(n)) and (2.2) preserves weight as well as degree, where wt(e i k ) = wt(f i k ) = k +. k 0 3. THE STRUCTURE OF A(n) Sp(2n) Recall Weyl s first and second fundamental theorems of invariant theory for the standard representation of Sp(2n) (Theorems 6..A and 6..B of [W]). Theorem 3.. For k 0, let U k be the copy of the standard Sp(2n)-module C 2n with symplectic basis {x i,k, y i,k i =,..., n}. Then (Sym k 0 U k) Sp(2n) is generated by the quadratics (3.) q a,b = 2 n ( ) xi,a y i,b x i,b y i,a, 0 a < b. i= For a > b, define q a,b = q b,a, and let {Q a,b a, b 0} be commuting indeterminates satisfying Q a,b = Q b,a and no other algebraic relations. The kernel I n of the homomorphism (3.2) C[Q a,b ] (Sym k 0 U k ) Sp(2n), Q a,b q a,b, is generated by the degree n + Pfaffians p I, which are indexed by lists I = (i 0,..., i 2n+ ) of integers satisfying (3.3) 0 i 0 < < i 2n+. For n = and I = (i 0, i, i 2, i 3 ), we have and for n > they are defined inductively by p I = q i0,i q i2,i 3 q i0,i 2 q i,i 3 + q i0,i 3 q i,i 2, 2n+ (3.4) p I = ( ) r+ q i0,i r p Ir, where I r = (i,..., î r,..., i 2n+ ) is obtained from I by omitting i 0 and i r. r= There is an analogue of this theorem when the symmetric algebra Sym k 0 U k is replaced by the exterior algebra k 0 U k. It is a special case of Sergeev s first and second 8
fundamental theorems of invariant theory for Osp(m, 2n) (Theorem.3 of [SI] and Theorem 4.5 of [SII]). The generators of ( k 0 U k) Sp(2n) are (3.5) q a,b = n ( ) xi,a y i,b + x i,b y i,a, 0 a b. 2 i= For a > b, define q a,b = q b,a, and let {Q a,b a, b 0} be commuting indeterminates satisfying Q a,b = Q b,a and no other algebraic relations. The kernel I n of the homomorphism (3.6) C[Q a,b ] ( k 0 U k ) Sp(2n), Q a,b q a,b, is generated by elements p I of degree n + which are indexed by lists I = (i 0,..., i 2n+ ) satisfying (3.7) 0 i 0 i 2n+. For n = and I = (i 0, i, i 2, i 3 ), we have (3.8) p I = q i0,i q i2,i 3 + q i0,i 2 q i,i 3 + q i0,i 3 q i,i 2, and for n > they are defined inductively by (3.9) p I = 2n+ r= q i0,i r p Ir, where I r = (i,..., î r,..., i 2n+ ) is obtained from I by omitting i 0 and i r. The generators q a,b of R correspond to vertex operators (3.0) ω a,b = n ( : a e i b f i : + : b e i a f i : ), 0 a b, 2 i= of A(n) Sp(2n), satisfying φ 2 (ω a,b ) = q a,b. By Lemma 2., {ω a,b 0 a b} strongly generates A(n) Sp(2n). In fact, there is a more economical strong generating set. For each m 0, let A m denote the vector space spanned by {ω a,b a + b = m}, which has weight m + 2. We have dim(a 2m ) = m + = dim(a 2m+ ) for m 0, so (3.) dim ( A 2m / (A 2m ) ) =, dim ( A 2m+ / (A 2m ) ) = 0. For m 0, define (3.2) j 2m = ω 0,2m, which is clearly not a total derivative. We have (3.3) A 2m = (A 2m ) j 2m = 2 (A 2m 2 ) j 2m, where j 2m is the linear span of j 2m. Similarly, (3.4) A 2m+ = 2 (A 2m ) j 2m = 3 (A 2m 2 ) j 2m. Moreover, { 2i j 2m 2i 0 i m} and { 2i+ j 2m 2i 0 i m} are bases of A 2m and A 2m+, respectively, so each ω a,b A 2m and ω c,d A 2m+ can be expressed uniquely as m m (3.5) ω a,b = λ i 2i j 2m 2i, ω c,d = µ i 2i+ j 2m 2i i=0 for constants λ i, µ i. Hence {j 2m m 0} is also a strong generating set for A(n) Sp(2n). 9 i=0
Theorem 3.2. A(n) Sp(2n) is generated by j 0 and j 2 as a vertex algebra. Proof. It suffices to show that each j 2k can be generated by these elements. This follows from the calculation j 2 j 2k = (2k + 4)j 2k+2 + 2 ω, where ω is a linear combination of 2i j 2k 2i for i = 0,..., k. Consider the category of all vertex algebras with generators {J 2m m 0}, which satisfy the same OPE relations as the generators {j 2m m 0} of A(n) Sp(2n). Since the vector space with basis {} { l j 2m l, m 0} is closed under n for all n 0, it forms a Lie conformal algebra. By Theorem 7.2 of [BK], this category possesses a universal object M n, which is freely generated by {J 2m m 0}. Then A(n) Sp(2n) is a quotient of M n by an ideal I n, and since A(n) Sp(2n) is a simple vertex algebra, I n is a maximal ideal. Let π n : M n A(n) Sp(2n), J 2m j 2m denote the quotient map. Using (3.5), which holds in A(n) Sp(2n) for all n, we can define an alternative strong generating set {Ω a,b 0 a b} for M n by the same formula: for a + b = 2m and c + d = 2m +, m m Ω a,b = λ i 2i J 2m 2i, Ω c,d = µ i 2i+ J 2m 2i. i=0 Clearly π n (Ω a,b ) = ω a,b. We will use the same notation A m to denote the linear span of {Ω a,b a+b = m}, when no confusion can arise. Finally, M n has a good increasing filtration in which (M n ) (2k) is spanned by iterated Wick products of the generators J 2m and their derivatives, of length at most k, and (M n ) (2k+) = (M n ) (2k). Equipped with this filtration, M n lies in the category R, and π n is a morphism in R. The structure of the ideal I n. Under the identifications gr(m n ) = C[Q a,b ], gr(a(n) Sp(2n) ) = ( k 0 U k ) Sp(2n) = C[qa,b ]/I n, i=0 gr(π n ) is just the quotient map (3.2). Lemma 3.3. For each I = (i 0, i,..., i 2n+ ), there exists a unique element (3.6) P I (M n ) (2n+2) I n of weight 2n + 2 + 2n+ a=0 i a, satisfying (3.7) φ 2n+2 (P I ) = p I. These elements generate I n as a vertex algebra ideal. Proof. Clearly π n maps each filtered piece (M n ) (k) onto (A(n) Sp(2n) ) (k), so the hypotheses of Lemma 2.2 are satisfied. Since I n = Ker(gr(π n )) is generated by {p I }, we can apply Lemma 2.2 to find P I (M n ) (2n+2) I n satisfying φ 2n+2 (P I ) = p I, such that {P I } generates I n. If P I also satisfies (3.7), we would have P I P I (M n) (2n) I n. Since there are no relations in A(n) Sp(2n) of degree less than 2n + 2, we have P I P I = 0. 0
Let P I denote the vector space with basis {P I } where I satisfies (3.3). We have P I = (M n ) (2n+2) I n, and clearly P I is a module over the Lie algebra P generated by {J 2m (k) m, k 0}, since P preserves both the filtration on M n and the ideal I n. It will be convenient to work instead with the generating set {Ω a,b (a + b + w) 0 a b, a + b + w 0} for P. Note that Ω a,b (a + b + w) is homogeneous of weight w. The action of P by derivations of degree zero on gr(m n ) coming from the vertex Poisson algebra structure is independent of n, and is specified by the action of P on the generators Ω c,d. We compute (3.8) Ω a,b (a + b + w)(ω c,d ) = λ a,b,w,c (Ω c+w,d ) + λ a,b,w,d (Ω c,d+w ), where (3.9) λ a,b,w,c = {( ) a+ (a+c+)! + (c b+w)! ( )b+ (b+c+)! c b + w 0 (c a+w)!. 0 c b + w < 0 The action of P on P I is by weighted derivation in the following sense. Given I = (i 0,..., i 2n+ ) and p = Ω a,b (a + b + w) P, we have (3.20) p(p I ) = 2n+ r=0 λ r P I r, where I r = (i 0,..., i r, i r + w, i r+,..., i 2n+ ). If i r + w = i s for some s =,..., 2n + we have λ r = 0, and otherwise λ r = λ a,b,w,ir. For each n, there is a distinguished element P 0 P I, defined by P 0 = P I, I = (0, 0,..., 0). It is the unique element of I n of minimal weight 2n + 2, and is a singular vector in M n. Theorem 3.4. P 0 generates I n as a vertex algebra ideal. We need a preliminary lemma in order to prove this statement. For simplicity of notation, we take n =, but our lemma holds for any n. In this case, A is generated by e, f. Recall from (2.) that e j and f j denote the images of j e and j f in gr(a), respectively. Let W gr(a) be the vector space with basis {e j, f j j 0}, and for each m 0, let W m be the subspace with basis {e j, f j 0 j m}. Let φ : W W be a linear map of weight w, such that (3.2) φ(e j ) = c j e j+w, φ(f j ) = c j f j+w for constants c j C. For example, the restriction of j 2k (2k + w) to W is such a map for 2k + w 0. Lemma 3.5. Fix w and m 0, and let φ be a linear map satisfying (3.2). Then the restriction φ Wm can be expressed uniquely as a linear combination of the operators j 2k (2k + w) Wm 0 2k + w 2m +.
Proof. Suppose first that w is odd, and let k j = j + (w ), for j = 0,..., m. We need 2 to show that φ Wm can be expressed uniquely as a linear combination of the operators j 2k j (2j) Wm for j = 0,..., m. We calculate (3.22) j 2k j (2j)(e i ) = λ 0,2kj,w,i(e i+w ), j 2k j (2j)(f i ) = λ 0,2kj,w,i(f i+w ), where λ 0,2kj,w,i is given by (3.9). Let M w be the (m + ) (m + ) matrix with entries Mi,j w = λ 0,2kj,w,i, for i, j = 0,..., m. Let c be the column vector in C m+ with transpose (c 0,..., c m ). Given an arbitrary linear combination ψ = t 0 j 2k 0 (0) + t j 2k (2) + + t m j 2km (2m) of the operators j 2k j (2j) for 0 j m, let t be the column vector with transpose (t 0,..., t m ). Note that φ Wm = ψ Wm precisely when M w t = c, so in order to prove the claim, it suffices to show that M w is invertible. But this is clear since each 2 2 minor [ ] M w i,j Mi,j+ w M w i+,j M w i+,j+ has positive determinant. Finally, if w is even, the same argument shows that for k j = j + w, φ can be expressed uniquely as a linear combination of the operators 2 j2k j (2j + ) for j = 0,..., m. Since (3.22) holds for all n with e j and f j replaced with e i j and f i j, it follows that Lemma 3.5 holds for any n. More precisely, let W gr(a(n)) be the vector space with basis {e i j, f i j i =,..., n, j 0}, and let W m W be the subspace with basis {e i j, f i j i =,..., n, 0 j m}. Let φ : W W be a linear map of weight w taking (3.23) e i j c j e i j+w, f i j c j f i j+w, i =,..., n, where c j is independent of i. Then φ Wm can be expressed uniquely as a linear combination of j 2k (2k + w) Wm for 0 2k + w 2m +. Proof of Theorem 3.4. Since I n is generated by P I as a vertex algebra ideal, it suffices to show that P I is generated by P 0 as a module over P. Let I n denote the ideal in M n generated by P 0, and let P I [m] denote the homogeneous subspace of P I of weight m. This space is trivial for m < 2n + 2, and is spanned by P 0 for m = 2n + 2. We shall prove that P I [m] I n by induction on m. Fix m > 2n + 2, and assume that P I [m ] I n. Fix I = (i 0, i,..., i 2n+ ) such that P I has weight m = 2n + 2 + 2n+ k=0 i k. Since m > 2n + 2, there is some k for which i k > 0. Let k be the first element where this happens, and let I = (0,..., 0, i k, i k+,..., i 2n+ ). Clearly P I I n. By Lemma 3.5, we can find p P such that p(e i r) = c r e i r+, p(f i r) = c r f i r+ where c r = for r = i k and c r = 0 for all other r i 2n+. It is immediate from the weighted derivation property (3.20) that p(p I ) = P I. Therefore P I I n. 2
Normal ordering and quantum corrections. Given a homogeneous p gr(m n ) = C[Q a,b ] of degree k in the Q a,b, a normal ordering of p will be a choice of normally ordered polynomial P (M n ) (2k), obtained by replacing Q a,b by Ω a,b, and by replacing ordinary products with iterated Wick products. Of course P is not unique, but for any choice we have φ 2k (P ) = p. For the rest of this section, P 2k, E 2k, F 2k, etc., will denote elements of (M n ) (2k) which are homogeneous, normally ordered polynomials of degree k in the Ω a,b. Let P 2n+2 I (M n ) (2n+2) be some normal ordering of p I, so that φ 2n+2 (P 2n+2 I ) = p I. Then π n (P 2n+2 I ) (A(n) Sp(2n) ) (2n), and φ 2n (π n (P 2n+2 I )) gr(a(n) Sp(2n) ) can be expressed uniquely as a polynomial of degree n in the variables q a,b. Choose some normal ordering of the corresponding polynomial in the variables Ω a,b, and call this element P 2n I. Then P 2n+2 I φ 2n+2 (P 2n+2 I + P 2n I satisfies + PI 2n ) = p I, π n (P 2n+2 I + PI 2n ) (A(n) Sp(2n) ) (2n 2). Continuing this process, we arrive at an element n+ k= P I 2k φ 2n+2 ( n+ ) = p I. By Lemma 3.6, must have k= P I 2k n+ (3.24) P I = PI 2k. k= in the kernel of π n, such that The term P 2 I lies in the space A m spanned by {Ω a,b a + b = m}, for m = 2n + 2n+ a=0 i a. By (3.3), for all even integers m we have a projection pr m : A m J m. For all I = (i 0, i,..., i 2n+ ) such that m = 2n + 2n+ a=0 i a is even, define the remainder (3.25) R I = pr m (P 2 I ). Lemma 3.6. Fix P I I n with I = (i 0, i,..., i 2n+ ) and m = 2n + 2n+ a=0 i a even. Suppose that P I = n+ k= P I 2k and P I = n+ 2k k= P I are two different decompositions of P I of the form (3.24). Then PI 2 P I 2 2 (A m 2 ). In particular, R I is independent of the choice of decomposition of P I. Proof. The argument is the same as the proof of Lemma 4.7 of [LI]. Lemma 3.7. Let R 0 denote the remainder of the element P 0. The condition R 0 0 is equivalent to the existence of a decoupling relation in A(n) Sp(2n) of the form (3.26) j 2n = Q(j 0, j 2,..., j 2n 2 ), where Q is a normally ordered polynomial in j 0, j 2,..., j 2n 2 and their derivatives. Proof. Let P 0 = n+ k= P 0 2k be a decomposition of P 0 of the form (3.24). If R 0 0, we have P0 2 = λj 2n + 2 ω for some λ 0 and ω A 2n 2. Since P 0 has weight 2n + 2 and P0 2k has degree k in the J 2m and their derivatives, P0 2k must depend only on J 0, J 2,..., J 2n 2 and their derivatives, for 2 k n. It follows that P λ 0 has the form (3.27) J 2n Q(J 0, J 2,..., J 2n 2 ). Applying π n : M n A(n) Sp(2n) yields the desired relation, since π n (P 0 ) = 0. The converse holds because P 0 is the unique element of I n of weight 2n + 2, up to scalar multiples. 3
Lemma 3.8. Suppose that R 0 0. Then for all m n, there exists a decoupling relation (3.28) j 2m = Q m (j 0, j 2,..., j 2n 2 ). Here Q m is a normally ordered polynomial in j 0, j 2,..., j 2n 2, and their derivatives. Proof. It suffices to find elements J 2m Q m (J 0, J 2,..., J 2n 2 ) I n, so we assume inductively that Q l exists for n l < m. Choose a decomposition Q m = d k= Q 2k m, where Q 2k m is a normally ordered polynomial of degree k in J 0, J 2,..., J 2n 2 and their derivatives. In particular, n Q 2 m = c i 2m 2i 2 J 2i, for constants c 0,..., c n. We apply the operator J 2 P, which satisfies i=0 J 2 J 2m = (2m + 4)J 2m+2 + m λ k 2k J 2m 2k, for constants λ k. Since Q l exists for n l < m, whenever m k n, we can use the element ) (J 2k 2m 2k Q m k (J 0, J 2,..., J 2n 2 ) I n to express 2k J 2m 2k as a normally ordered polynomial in J 0, J 2,..., J 2n 2 and their derivatives, modulo I n. ( Moreover, J 2 d ) k= Q2k m can be expressed in the form d k= E2k, where each E 2k is a normally ordered polynomial in J 0, J 2,..., J 2n and their derivatives. If J 2n or its derivatives appear in E 2k, we can use (3.27) to eliminate J 2n and any of its derivatives, modulo I n. Hence J 2 ( d k= k= ) Q 2k m can be expressed modulo I n in the form d k= F 2k, where d d, and F 2k is a normally ordered polynomial in J 0, J 2,..., J 2n 2 and their derivatives. It follows that ( ) 2m + 4 J 2 J 2m Q m (J 0, J 2,..., J 2n 2 ) can be expressed as an element of I n of the desired form. 4
A recursive formula for R I. In this section we find a recursive formula for R I for any I = (i 0, i,..., i 2n+ ) such that wt(p I ) = 2n + 2 + 2n+ a=0 i a is even. It will be clear from our formula that R 0 0. We introduce the notation (3.29) R I = R n (I)J m, m = 2n + so that R n (I) denotes the coefficient of J m in pr m (P 2 I ). For n = and I = (i 0, i, i 2, i 3 ) the following formula is easy to obtain using the fact that pr m (Ω a,b ) = ( ) m J m for m = a + b. 2n+ a=0 i a, ( ( ) i 0 +i 2 + ( ) i 0+i 3 + ( ) i +i 2 + ( ) i +i 3 (3.30) R (I) = 8 2 + i 0 + i + ( )i 0+i + ( ) i 0+i 3 + ( ) i +i 2 + ( ) i 2+i 3 2 + i 0 + i 2 + ( )i 0+i + ( ) i 0+i 2 + ( ) i +i 3 + ( ) i 2+i 3 2 + i + i 2 + ( )i 0+i + ( ) i 0+i 2 + ( ) i +i 3 + ( ) i 2+i 3 2 + i 0 + i 3 + ( )i 0+i + ( ) i 0+i 3 + ( ) i +i 2 + ( ) i 2+i 3 2 + i + i 3 + ( )i 0+i 2 + ( ) i 0+i 3 + ( ) i +i 2 + ( ) i +i 3 2 + i 2 + i 3 ). Assume that R n (J) has been defined for all J. Recall first that A(n) is a graded algebra with Z 0 grading (2.9), which specifies a linear isomorphism A(n) = k 0 U k, U k = C 2n. Since A(n) Sp(2n) is a graded subalgebra of A(n), we obtain an isomorphism of graded vector spaces (3.3) i n : A(n) Sp(2n) ( k 0 U k ) Sp(2n). Let p ( k 0 U k) Sp(2n) be homogeneous of degree 2d, and let f = (i n ) (p) (A(n) Sp(2n) ) (2d) be the homogeneous corresponding element. Let F (M n ) (2d) be an element satisfying π n (F ) = f, where π n : M n A(n) Sp(2n) is the projection. We can write F = d k= F 2k, where F 2k is a normally ordered polynomial of degree k in the Ω a,b. Next, consider the rank n + symplectic fermion algebra A(n + ), and let q a,b ( k 0 Ũ k ) Sp(2n+2) = gr(a(n + )) Sp(2n+2) = gr(a(n + ) Sp(2n+2) ) 5
be the generator given by (3.), where Ũk = C 2n+2. Let p be the polynomial of degree 2d obtained from p by replacing each q a,b with q a,b, and let f = (i n+ ) ( p) (A(n + )) Sp(2n+2) ) (2d) 2k be the corresponding homogeneous element. Finally, let F M n+ be the element obtained from F 2k by replacing each Ω a,b with the corresponding generator Ω a,b M n+, and let F = d F i= 2k. Lemma 3.9. Fix n, and let P I be an element of I n given by Lemma 3.6. There exists a decomposition P I = n+ k= P I 2k of the form (3.24) such that the corresponding vertex operator P I = n+ k= P 2k I M n+ has the property that π n+ ( P I ) lies in the homogeneous subspace (A(n + ) Sp(2n+2) ) (2n+2). Proof. The argument is the same as the proof of Corollary 4.4 of [LI], and is omitted. Recall that p I has an expansion p I = 2n+ r= q i 0,i r p Ir, where I r = (i,..., î r,..., i 2n+ ) is obtained from I by omitting i 0 and i r. Let P Ir M n be the element corresponding to p Ir. By Lemma 3.9, there exists a decomposition n P Ir = such that the corresponding element P Ir lies in (A(n) Sp(2n) ) (2n). We have (3.32) 2n+ r= : Ω i0,i r PIr : = i= = n i= 2n+ r= P 2i I r P 2i I r M n has the property that π n ( P Ir ) n i= : Ω i0,i r P 2i I r :. The right hand side of (3.32) consists of normally ordered monomials of degree at least 2 in the generators Ω a,b, and hence contributes nothing to R n (I). Since π n ( P Ir ) is homogeneous of degree 2n, π n (: Ω i0,i r PIr :) consists of a piece of degree 2n + 2 and a piece of degree 2n coming from all double contractions of Ω i0,i r with terms in P Ir, which lower the degree by two. The component of ( 2n+ ) π n : Ω i0,i r PIr : A(n) Sp(2n) r= in degree 2n + 2 must cancel since this sum corresponds to p I, which is a relation among the variables q a,b. The component of : Ω i0,i r PIr : in degree 2n is (3.33) S r = ( ( ) P i 0+ Ir,a 2 i a 0 + i a + 2 + ) P Ir,a ( )ir+ i a r + i a + 2 In this notation, for a {i 0,..., i 2n+ }\{i 0, i r }, I r,a is obtained from I r = (i,..., î r,..., i 2n+ ) by replacing i a with i a + i 0 + i r + 2. It follows that ( 2n+ ) ( 2n+ (3.34) π n : Ω i0,i r PIr : = π n S r ). r= 6 r=
Combining (3.32) and (3.34), we can regard 2n+ r= n i= : Ω i0,i r P 2i I r as a decomposition of P I of the form P I = n+ k= P I 2k where the leading term P 2n+2 I = 2n+ 2n r=0 : Ω i0,i r P I r :. Therefore R n (I) is the negative of the sum of the terms R n (J) corresponding to each P J appearing in 2n+ r=0 S r, so we obtain the following result. Theorem 3.0. R n (I) satisfies the recursive formula (3.35) R n (I) = 2n+ ( ( ) i R 0+ n (I r,a ) 2 i a 0 + i a + 2 + ( )ir+ a r= : n r=0 S r ) R n (I r,a ). i r + i a + 2 Now suppose that all the entries i 0,..., i 2n+ appearing in I are all even. Clearly each I r,a appearing in (3.35) consists only of even entries as well. In the case n = and I = (i 0, i, i 2, i 3 ), (3.30) reduces to R (I) = ( ) + + + + +, 2 2 + i 0 + i 2 + i 0 + i 2 2 + i + i 2 2 + i 0 + i 3 2 + i + i 3 2 + i 2 + i 3 so in particular R (I) 0. By induction on n, it is immediate from (3.35) that R n (I) 0 whenever I has even entries. Theorem 3.. For all n, A(n) Sp(2n) has a minimal strong generating set {j 0, j 2,..., j 2n 2 }, and is therefore a W-algebra of type W(2, 4,..., 2n). Proof. For I = (0, 0,..., 0), we have R n (I) 0, so R 0 = R n (I)J 2n 0. The claim then follows from Lemma 3.8. Using Theorem 3., it is now straightforward to describe A(mn) Sp(2n) for all m, n. Denote the generators of A(mn) by e i,j, f i,j for i =,..., n and j =,..., m, which satisfy e i,j (z)f k,l (w) δ i,k δ j,l (z w) 2. Under the action of Sp(2n), {e i,j, f i,j i =,..., n} spans a copy of the standard Sp(2n)- module C 2n for each j =,..., m. Define ω j,k a,b = n ( : a e i,j b f i,k : + : b e i,k a f i,j : ). 2 i= By Theorem 3., {ω j,k a,b j, k m, a, b 0} strongly generates A(mn)Sp(2n). In fact, {ω j,j 0,2k k 0, j =,..., m} {ω j,k 0,l j < k m, l 0} also strongly generates A(mn) Sp(2n). This is clear because the corresponding elements of gr(a(mn) Sp(2n) ) generate gr(a(mn) Sp(2n) ) as a -ring. Theorem 3.2. A(mn) Sp(2n) has a minimal strong generating set (3.36) {ω j,j 0,2r j m, 0 r n } {ω j,k 0,s j < k m, 0 s 2n }. 7
Proof. By Theorem 3., we have decoupling relations (3.37) ω j,j 0,2r = Q r (ω j,j 0,0,..., ω j,j 0,2n 2), for all r n and j =,..., m. We now construct decoupling relations expressing each ω j,k 0,a as a normally polynomial in the generators (3.36) and their derivatives, for j < k and a 2n. We need the following calculations. ω j,k 0,0 ω j,j 0,2r = (r + )ω j,k 0,2r + ω, ω j,k 0,0 ω j,j 0,2r+ = ω j,k 0,2r+ + ν. Here ω is a linear combination of 2r t ω j,k 0,t for t = 0,..., 2r, and ν is a linear combination of 2r t+ ω j,k 0,t for t = 0,..., 2r. Therefore applying the operator ω0,0 j,k to (3.37) yields a relation ω j,k 0,2r = Q r (ω0,0, j,j ω0,2, j,j... ω0,2n 2, j,j ω0,0, j,k ω0,, j,k..., ω0,2n ), j,k for all r n. Similarly, applying ω j,k 0,0 to the derivative of (3.37) yields a relation ω j,k 0,2r+ = Q r (ω j,j 0,0, ω j,j 0,2,... ω j,j 0,2n 2, ω j,k 0,0, ω j,k 0,,..., ω j,k 0,2n ), for all r n. This shows that (3.36) is a strong generating set for A(mn) Sp(2n). It is minimal because there are no normally ordered relations of weight less than 2n + 2. 4. THE STRUCTURE OF A(n) GL(n) The subgroup GL(n) Sp(2n) acts on A(n) such that the generators {e i } and {f i } of A(n) span copies of the standard GL(n)-modules C n and (C ) n, respectively. In this section, we use a similar approach to find a minimal strong generating set for A(n) GL(n). First, we have isomorphisms gr(a(n) GL(n) ) ( = gr(a(n)) GL(n) = R := (V j Vj ) ) GL(n), where V j = C n and Vj = (C n ) as GL(n)-modules. By an odd analogue of Weyl s first and second fundamental theorems of invariant theory for the standard representation of GL(n), R is generated by the quadratics p a,b = n i= xi ayb i where {xi a} is a basis for V a and {yb i } is the dual basis for Vb. The ideal of relations is generated by elements d I,J of degree n +, which are indexed by lists I = (i 0, i,..., i n ) and J = (j 0, j,..., j n ) of integers satisfying 0 i 0 i n and 0 j 0 j n ), which are analogous to determinants but without the signs. (This is a special case of Theorems 2. and 2.2 of [SII]). For n =, j 0 d I,J = p i0,j 0 p i,j + p i,j 0 p i0,j, and for n >, d I,J is defined inductively by n d I,J = p ir,j0 d, Ir,J r=0 where I r = (i 0,..., î r..., i n ) is obtained from I by omitting i r, and J = (j,..., j n ) is obtained from J by omitting j 0. The generators p a,b correspond to strong generators n γ a,b = : a e i b f i :, a, b 0, i= 8
for A(n) GL(n), where wt(γ a,b ) = a + b + 2. Let A m be spanned by {γ a,b a + b = m}. Then dim(a m ) = m + and dim(a m / A m ) =, so A m = A m h m, h m = γ 0,m. Since { a h m a a = 0,..., m} and {γ a,m a a = 0,..., m} are both bases for A m, {h k k 0} strongly generates A(n) GL(n). Lemma 4.. A(n) GL(n) is generated as a vertex algebra by h 0 and h. Proof. This is immediate from the following calculation. h h k = (k + 3)h k+ + 2 h k, k. There exists a vertex algebra N n which is freely generated by {H k k 0} with same OPE relations as {h k k 0}, such that A(n) GL(n) is a quotient of N n by an ideal J n under a map π n : N n A(n) GL(n), H k h k. We have an alternative strong generating set {Γ a,b a, b 0} for N n satisfying π n (Γ a,b ) = γ a,b. There is a good increasing filtration on N n where (N n ) (2k) is spanned by iterated Wick products of H m and their derivatives of length at most k, and (N n ) (2k+) = (N n ) (2k), and π n preserves filtrations. Lemma 4.2. For each I and J as above, there exists a unique element D I,J (N n ) (2n+2) J n of weight 2n + 2 + n+ a=0 (i a + j a ), satisfying φ 2n+2 (D I,J ) = d I,J. These elements generate J n as a vertex algebra ideal. Each D I,J can be written in the form n+ (4.) D I,J = k= D 2k I,J, where DI,J 2k is a normally ordered polynomial of degree k in the generators Γ a,b. The term DI,J 2 lies in the space A m for m = 2n + n a=0 (i a + j a ) = m. We have the projection pr m : A m H m, and we define the remainder R I,J = pr m (D 2 I,J). It is independent of the choice of decomposition (4.). The element of J n of minimal weight corresponds to I = (0,..., 0) = J, and has weight 2n + 2. We denote this element by D 0 and we denote its remainder by R 0. The condition R 0 0 is equivalent to the existence of a decoupling relation (4.2) h 2n = P (h 0, h,..., h 2n ), where P is a normally ordered polynomial in h 0, h,..., h 2n and their derivatives. From this relation it is easy to construct decoupling relations h m = P m (h 0, h,..., h 2n ) for all m > 2n, so the condition R 0 0 implies that {h 0, h,..., h 2n } is a minimal strong generating set for A(n) GL(n). 9
To prove this, we need to analyze the quantum corrections of D I,J. Write n R I,J = R n (I, J)H m, m = 2n + (i a + j a ), so that R n (I, J) denotes the coefficient of H m in pr m (DI,J 2 ). For n = and I = (i 0, i ), J = (j 0, j ) we have ( ) (4.3) R (I, J) = ( ) +j 0+j + + +. 2 + i 0 + j 0 2 + i + j 0 2 + i 0 + j 2 + i + j Using the same method as the previous section, one can show that R n (I, J) satisfies the following recursive formula. (4.4) R n (I, J) = n r=0 ( ( ( ) j 0 k a=0 ) R n (I r,k, J ) + ( ) ir 2 + i k + j 0 l ( Rn (I r, J l ) 2 + j l + i r In this notation, I r = (i 0,..., î r,..., i n ) is obtained from I by omitting i r. For k = 0,..., n and k r, I r,k is obtained from I r by replacing the entry i k with i k + i r + j 0 + 2. Similarly, J = (j,..., j n ) is obtained from J by omitting j 0, and for l =,..., n, J l is obtained from J by replacing j l with j l + i r + j 0 + 2. Suppose that all entries of I and J are even. Then for each R n (K, L) appearing in (4.4), all entries of K and L are even. It is immediate from (4.3) that R (I, J) 0, and by induction on n, it follows from (4.4) that R n (I, J) is nonzero whenever I and J consist of even numbers. Specializing to the case I = (0,..., 0) = J, we see that R 0 0, as desired. Finally, this implies Theorem 4.3. A(n) GL(n) has a minimal strong generating set {h 0, h,..., h 2n }, and is therefore of type W(2, 3,..., 2n + ). )). An immediate consequence is Theorem 4.4. For all m, A(mn) GL(n) has a minimal strong generating set n γ j,k 0,l = : e i,j l f i,k :, j k m, 0 l 2n. i= Proof. The argument is similar to the proof of Theorem 3.2, and is omitted. 5. CHARACTER DECOMPOSITIONS A general result of Kac and Radul (see Section of [KR]) states that for an associative algebra A and a Lie algebra g, every (g, A)-module V with the properties that it is an irreducible A-module and a direct sum of a countable number of finite-dimensional irreducible g-modules decomposes as V = ( ) E V E, E where the sum is over all irreducible g-modules E and V E is an irreducible A g -module. Kac and Radul then remark that the same statement is true if we replace g by a group G (see also [DLM] for similar results). The symplectic fermion algebra A(mn) is simple and 20
graded by conformal weight. Each graded subspace is finite-dimensional and G-invariant for any reductive G Sp(2mn). Hence the assumptions of [KR] apply, so that A(mn) = E ( E A(mn) E ), where the sum is over all finite-dimensional irreducible representations of G, and A(mn) E is an irreducible representation of A(mn) G. The purpose of this section is to find the characters of the A(mn) E for G = Sp(2n) SO(n) and also for G = GL(m) GL(n). We will use the denominator identities of finite-dimensional classical Lie superalgebras to solve this problem. We need some notation and results on Lie superalgebras. For this, we use the book [CW], and the article [KWII]. Define the lattice L m,n := δ Z δ m Z ɛ Z ɛ n Z of signature (m, n), where the bilinear product is defined by (δ i, δ j ) = δ i,j, (ɛ a, ɛ b ) = δ a,b, (δ i, ɛ b ) = 0, for all i, j m and a, b n. We restrict to the case m > n. The root systems of various Lie superalgebras can be constructed as subsets of L m,n. We are interested in three examples. Example 5.. Let g = gl (m n). Then the root system is the disjoint union of even and odd roots, = 0, where 0 = {δ i δ j, ɛ a ɛ b i, j m, a, b n } = {± (δ i ɛ a ) i m, a n }. A Lie superalgebra usually allows for inequivalent choices of positive roots and positive simple roots. We are interested in a choice of simple positive roots that has as many odd isotropic roots as possible. This is Π = {( ) a (ɛ a δ a ), ɛ b ɛ b+, δ i δ i+ a, b n, b odd, i m, i even } so that positive roots are and + = + 0 + + 0 = {δ i δ j, ɛ a ɛ b i < j m, a < b n } + = {ɛ a δ i, δ j ɛ b a n, a i m, a even, j m, j b n, j odd } (5.) S = {( ) a (ɛ a δ a ) a m } Π is a maximal isotropic subset of simple positive odd roots. For every positive root, define the Weyl reflection (x, α) r α (x) := x 2 (α, α) α. The Weyl group W is then the group generated by the r α for α in + 0. Finally, let W be the subgroup of the Weyl group generated by the roots of gl(m) if m > n, and otherwise the subgroup generated by the roots of gl(n). 2
Example 5.2. Let g = spo (2m 2n + ). Then the odd and even roots of the root system = 0 are 0 = {±δ i ± δ j, ±ɛ a ± ɛ b, ±2δ p, ±ɛ q }, = {±δ p ± ɛ q, ±δ p }, where i < j m, a < b n, p m, q n. A choice of simple positive roots that has as many odd isotropic roots as possible is Π = {( ) a (ɛ a δ a ), ɛ b ɛ b+, δ i δ i+, δ m a, b m, b odd, i n, i even} so that positive roots + = + 0 + split into even and odd roots as + 0 = {δ i ± δ j, ɛ a ± ɛ b, 2δ p, ɛ q } + = {ɛ a ± δ i, δ j ± ɛ b, δ p a n, a i m, a even, j m, j b n, j odd } where i < j m, a < b n, p m, q n unless otherwise indicated. Then (5.) is a maximal isotropic subset of simple positive odd roots. The Weyl group W is again the Weyl group of the even subalgebra. If m > n, W is defined to be the Weyl group of sp(2m), and otherwise it is the Weyl group of so(2n + ). Example 5.3. Let g = spo (2m 2n). Then the root system is the disjoint union of 0 = {±δ i ± δ j, ±ɛ a ± ɛ b, ±2δ p }, = {±δ p ± ɛ q }, where i < j m, a < b n, p m, q n. In this case, a choice of simple positive roots that has as many odd isotropic roots as possible is Π = {( ) a (ɛ a δ a ), ɛ b ɛ b+, δ i δ i+ a, b m, b odd, i n, i even} so that positive roots are the disjoint union of + 0 = {δ i ± δ j, ɛ a ± ɛ b, 2δ p } + = {ɛ a ± δ i, δ j ± ɛ b a n, a i m, a even, j m, j b n, j odd } where i < j m, a < b n, p m, q n unless otherwise indicated. Then again (5.) is a maximal isotropic subset of simple positive odd roots. The Weyl group W is as before the Weyl group of the even subalgebra. For m > n, W is the Weyl group of sp(2m), and otherwise it is the Weyl group of so(2n). The Weyl vector of a Lie superalgebra is defined to be ρ := ρ 0 ρ with even and odd Weyl vectors ρ 0 = α, ρ = α. 2 2 α + 0 We denote the root lattice of g by L. Note that this lattice is spanned by the odd roots in our examples. In order to state the main result of this section, we need to define for every λ in L the set { ( ) I λ := (n α ) α + \S, (m β) β S Z + } n α α + m β β = λ as well as the partial theta function P n (q) := q 2(n+ 2) 2 m=0 α + α + \S β S ( ) m q 2(m 2 2m(n+ 2)). 22