Reciprocity Laws in the Verlinde Formulae for the Classical Groups
Reciprocity Laws in the Verlinde Formulae for the Classical Groups
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经典群 Verlinde 公式中的互易律
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发表时间：
1996
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通讯作者：
S. M. J. Wilson
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作者：
W. Oxbury;S. M. J. Wilson
对每个单连通经典李群计算 Verlinde 公式，结果表明，所得公式在秩和能级交换方面遵循一定的互易律。曲线上 G 丛模空间上的 θ 线丛截面空间之间的一些对应对偶性被推测但尚未证明。简介 本文的目的是证明经典简单复李群的 Verlinde 公式表现出奇怪的互反律，类似于已经在酉群 [B1],[DT],[Z] 中观察到的互反律。对于每个这样的群 G 和非负整数 l, g ∈ Z，一个自然数 Nl(G) = Nl(G, g) 相关联，称为 Verlinde 数。这在下面第 2 节中定义；有关其在融合环形式主义中的推导，请参阅[B2]。那么刚才提到的基本互反律是： (0.1) Nl(SLn) ng = Nn(SLl) lg 。给定属 g 的平滑射影复曲线 C 和任何还原复代数群 G，存在 C 上代数主 G 丛的模空间 M(G)，其连通分量是由 G 的基本群索引的正态不可约射影簇。与任何有限维表示 G → GL(V ) 相关联，可以在 M(G) 上构造自然行列束 θ(V )。此外，当 G 是单连通时，有一个与表示相关联的非负整数 mV（参见 [KNR]），使得 Verlinde 数 NmV (G, g) 恰好是截面空间的维数： (0.2) dimH(M(G),θ(V )) = NmV (G, g)。特别是，Donagi 和 Tu 表明，作为这一点和 (0.1) 的结果，有 (0.3) dimH(M(GLn, 0),L) = dimH(M(SLl),L)，其中 L = θ(V )，V 是每一侧的标准表示，其中 M(GLn, 0) 表示模空间在 0 度的分量；他们推测各自的向量空间是正则对偶的（参见 [DT] 的完整内容，编辑于 1995 年 3 月 6 日收到，修订后的形式于 1995 年 5 月 21 日。1991 Mathematics subject Classification. Primary 14D20, 14H15。c ©1996 American Mathematical Society 2689 许可或版权限制可能适用于再分发；参见http://www.ams.org/journal-terms-of-use 2690 W. M. OXBURY 和 S. M. J. WILSON 故事）。在本文中，我们发现了更多此类维数的重合，尽管我们不会继续寻找向量空间同构的后续问题。第一种情况（实际上也是最简单的）是辛群的情况：对于所有 l，n ≥ 1，(0.4) Nl(Spn) = Nn(Spl)。其直接结果是：(0.5) 定理。设G = Spn为n≥1阶的复辛群，V = C为其标准表示；令 L = θ(V ) ∈ Pic ( M(Spn) ) 。那么对于任何 l ≥ 1，dimH(M(Spn),L) = dimH(M(Spl),L)。注意到在 l = 1 的情况下，模空间 M(Sp1) =M(SL2) = MC(2,O) 是具有平凡行列式的半稳定秩 2 向量丛，然后 L 是 PicMC(2,O) 的通常充足生成器，我们有： (0.6) 推论。对于任何 n ≥ 1，dimH(MC(2,O),L) = dimH(M(Spn),L)。人们可能会认为这表明了众所周知的对偶性 H0(J(C),θ⊗n) ∼= H(M(SLn),L) 的四元离子类似物。 （事实上，雅可比 J(C) 可以被视为圆群中 π1(C) 的表示空间，而 MC(2,O) 则是其在单位四元数的 3 球体中的表示空间。）另一方面，自旋群的互易定律则更加微妙。为了解释这些，我们需要注意 Verlinde 数 Nl(G) 被定义为与 G 关联的仿射李代数的 l 级可积表示的有限集 Pl 上的某个和。现在，对于每个经典群，我们考虑该仿射李代数的扩展 Dynkin 图的对称性有限群 Γ，以及它对每个 l ∈ N 对 Pl 的自然作用。事实证明，对于每个 G，我们获得将 Nl(G) 替换为修改后的 Verlinde 数 Ñl(G) 后的互易性，定义为轨道空间 Pl/Γ 上的和（参见下面的第 3 节）。特别是，对于辛情况， Γ 是微不足道的，并且具有直线互易关系 (0.4)；而对于酉情况 G = SLn，Γ 是循环群 Zn，正是这一点解释了 (0.1) 中出现的 n（或 l）的幂。对于奇数和偶数自旋群 Γ 分别为 Z2 和 Z2 × Z2，由此产生的 Ñl(Spinm) 互易定律如定理 (4.13) 所示。然而不幸的是，如何从几何上解释这一点还很不清楚，除了一种情况：l,m 都是奇数。为此，需要考虑二分量模空间 N (m) = M(Spinm) ∪M(Spinm)，其中 M(Spinm) 表示 C 上具有奇数次固定旋量范数的 Clifford 丛的模变。等效地，N (m) 是二元模空间 M(SOm) 的 étale 覆盖。对于 m = 3，N (3) 是 MC(2,O) 和 MC(2,O(x)) 的并集，对于 x ∈ C。现在考虑线束 θ(C)→N (m)，其中 C 是 Clifford 群的标准正交表示。我们的结果似乎暗示着以下互反律：如果存在，对于每个奇数 m ∈N，线束 L → N (m) 使得 L = θ(C)，那么它满足： (0.7) dimH(N (m),L) = dimH(N (l),L) 对于 l,m 都是奇数。许可或版权限制可能适用于再分发；请参阅 http://www.ams.org/journal-terms-of-use 经典群的 VERLINDE 公式 2691 这在本文的最后部分进行了解释；然而，其有效性取决于 M(Spinm) 上未经证明的 Verlinde 公式（猜想 (5.2)），该公式概括了奇数阶向量丛的“扭曲”Verlinde 公式。致谢。作者要感谢 Patrick Dorey 进行了一些有用的对话，并提供了 Dynkin 图。 1. 一些三角恒等式 在本节中，我们首先证明一系列令人惊讶的三角恒等式，这些三角恒等式支撑了稍后描述的互易定律。首先，令 p 为正整数，并令 f(r) = fp(r) = 4 sin (rπ/p) = (1− zep)(1− ze−r p )，其中 zep = e。给定一个有限集 U = {u1, . 。 。 , ur} 的有理数，我们将考虑以下乘积（其中空乘积被视为 1）： Πp(U) = ∏ 1≤i<j≤r ( f(ui − uj)f(ui + uj) ) ; Φp(U) = Πp(U)Np(U) 其中 Np(U) = r ∏
The Verlinde formula is computed for each of the simply-connected classical Lie groups, and it is shown that the resulting formula obeys certain reciprocity laws with respect to the exchange of the rank and the level. Some corresponding dualities between spaces of sections of theta line bundles over moduli spaces of G-bundles on curves are conjectured but not proved. Introduction The purpose of this article is to show that the Verlinde formulae for the classical simple complex Lie groups exhibit curious reciprocity laws analogous to those already observed for the unitary groups [B1],[DT],[Z]. To each such group G and nonnegative integers l, g ∈ Z one associates a natural number Nl(G) = Nl(G, g), called the Verlinde number. This is defined in §2 below; for its derivation in the formalism of fusion rings see [B2]. Then the basic reciprocity law just alluded to is: (0.1) Nl(SLn) ng = Nn(SLl) lg . Given a smooth projective complex curve C of genus g, and any reductive complex algebraic group G, there exists a moduli space M(G) for algebraic principal G-bundles over C, whose connected components are normal irreducible projective varieties indexed by the fundamental group of G. Associated to any finite dimensional representation G → GL(V ), one can construct a natural determinant line bundle Θ(V ) over M(G). Moreover, when G is simply-connected there is a nonnegative integer mV associated to the representation (see [KNR]) such that the Verlinde number NmV (G, g) is precisely the dimension of the space of sections: (0.2) dimH(M(G),Θ(V )) = NmV (G, g). In particular, Donagi and Tu showed that as a consequence of this and of (0.1) one has (0.3) dimH(M(GLn, 0),L) = dimH(M(SLl),L), where L = Θ(V ), V being the standard representation on each side, and where M(GLn, 0) denotes the component of moduli space at degree 0; and they conjectured that the respective vector spaces are canonically dual (see [DT] for the full Received by the editors March 6, 1995 and, in revised form, May 21, 1995. 1991 Mathematics Subject Classification. Primary 14D20, 14H15. c ©1996 American Mathematical Society 2689 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2690 W. M. OXBURY AND S. M. J. WILSON story). In the present paper we find some more coincidences of dimension of this kind, though we shall not pursue the subsequent question of finding isomorphisms of the vector spaces. The first case—and indeed the simplest—is that of the symplectic groups: (0.4) Nl(Spn) = Nn(Spl) for all l, n ≥ 1. The immediate consequence of this is then: (0.5) Theorem. Let G = Spn be the complex symplectic group of rank n ≥ 1, and V = C be its standard representation; and let L = Θ(V ) ∈ Pic ( M(Spn) ) . Then for any l ≥ 1, dimH(M(Spn),L) = dimH(M(Spl),L). Noting that in the case l = 1 the moduli space M(Sp1) =M(SL2) = MC(2,O) is that of semistable rank 2 vector bundles with trivial determinant, and that then L is the usual ample generator of PicMC(2,O), we have: (0.6) Corollary. For any n ≥ 1, dimH(MC(2,O),L) = dimH(M(Spn),L). One may perhaps view this as indicating a quaternionic analogue of the well-known duality H0(J(C),Θ⊗n) ∼= H(M(SLn),L). (Indeed, while the Jacobian J(C) can be viewed as the space of representations of π1(C) in the circle group, MC(2,O) is that of its representations in the 3-sphere of unit quaternions.) The reciprocity laws for the spin groups, on the other hand, are rather more subtle. To explain these, we need to note that the Verlinde number Nl(G) is defined as a certain sum over the finite set Pl of integrable representations of level l of the affine Lie algebra associated to G. We now consider, for each of the classical groups, a finite group Γ of symmetries of the extended Dynkin diagram of this affine Lie algebra, and its natural action on Pl for each l ∈ N. It turns out that for each G one obtains reciprocity after replacing Nl(G) by a modified Verlinde number Ñl(G) defined as a sum over the orbit space Pl/Γ (see §3 below). In particular, for the symplectic case Γ is trivial and one has the straight reciprocity relation (0.4); while for the unitary case G = SLn, Γ is the cyclic group Zn, and it is precisely this that accounts for the power of n (resp. l) arising in (0.1). For the odd and even spin groups Γ is Z2 and Z2 × Z2 respectively, and the resulting reciprocity law for Ñl(Spinm) is stated as theorem (4.13). Unfortunately, however, it is far from clear how to interpret this geometrically, except in one case: that of l,m both odd. For this one needs to consider the two-component moduli space N (m) = M(Spinm) ∪M(Spinm), where M(Spinm) denotes the moduli variety of Clifford bundles on C with fixed spinor norm of odd degree. Equivalently N (m) is an étale cover of the two-component moduli space M(SOm). For m = 3, N (3) is the union of MC(2,O) and MC(2,O(x)), for x ∈ C. Now consider the line bundle Θ(C)→N (m) where C is the standard orthogonal representation of the Clifford group. Our results appear to imply the following reciprocity law: if there exists, for each odd m ∈N, a line bundle L → N (m) such that L = Θ(C), then it satisfies: (0.7) dimH(N (m),L) = dimH(N (l),L) for l,m both odd. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use VERLINDE FORMULAE FOR CLASSICAL GROUPS 2691 This is explained in the final section of the paper; its validity, however, depends on an unproved Verlinde formula on M(Spinm) (Conjecture (5.2)), which generalises the ‘twisted’ Verlinde formula for rank 2 vector bundles of odd degree. Acknowledgements. The authors would like to thank Patrick Dorey for some useful conversations, as well as contributing the Dynkin diagrams. 1. Some trigonometric identities We begin by proving, in this section, a family of rather surprising trigonometric identities which underpin the reciprocity laws to be described later on. To begin, let p be a positive integer and let f(r) = fp(r) = 4 sin (rπ/p) = (1− ζ p)(1− ζ−r p ), where ζp = e. Given a finite set U = {u1, . . . , ur} of rational numbers, we shall consider the following products (where an empty product is deemed to be 1): Πp(U) = ∏ 1≤i<j≤r ( f(ui − uj)f(ui + uj) ) ; Φp(U) = Πp(U)Np(U) where Np(U) = r ∏