Modular invariant representations of infinite-dimensional Lie algebras and superalgebras.
Modular invariant representations of infinite-dimensional Lie algebras and superalgebras.
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DOI：
10.1073/pnas.85.14.4956
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发表时间：
1988-07
影响因子：
11.1
通讯作者：
V. Kac;M. Wakimoto
中科院分区：
文献类型：
--
作者：
V. Kac;M. Wakimoto
在本文中，我们推出了一个程序来描述和分类无限维李代数和超代数的模不变表示。本文证明了Kac-Moody代数[unk]的一类最高权表示L（lambda）的特征公式，推广了Weyl-Kac特征公式[Kac，V.G.（1974）Funct. Anal. Appl.8，68-70]。在仿射[unk]的情况下，这个类包括任意有理水平m = t/u的模不变表示，其中t [unk] Z和u [unk] N是互质的，并且m + g >/= g/u（g是对偶Coxeter数）。我们用θ函数描述了这些表示的特征，并计算了它们的渐近性，推广了Kac和Peterson [Kac，V.G. & Peterson，D. H.（1984）Adv.Math.53，125-264]以及Kac和Wakimoto [Kac，V.G. & Wakimoto，M.（1988）Adv.Math.70，156-234]对于u = 1（可积）的情况。我们详细地计算了[unk] = A（1）（（1））的情况，特别是对它的所有模不变表示进行了分类。此外，我们证明了Virasoro代数Vir的模不变表示正是Belavin等人的“极小级数”[Belavin，A.一、Polyakov，A. M. & Zamolodchikov，A. B。（1984）Nucl. Phys. B 241，333-380]中使用Feigin和Fuchs [Feigin，B. L. & Fuchs，D. B。（1984）Lect. Notes Math.1060，230-245]。我们证明了张紧A（1）（（1））的基本表示和模不变表示产生Vir的所有模不变表示，推广了戈达德等人的结果[戈达德P.，肯特，A. & Olive，D.（1986）Commun. Math.Phys.103，105-119]和Kac和Wakimoto [Kac，V.G. & Wakimoto，M.（1986）Lect. Notes Phys.261，345-371]在单一情况下。我们还研究了一般的分支函数。这些结果推广到Kac [Kac，V.G.（1978）Adv.Math.30，85-136]和N = 1超Virasoro代数。我们详细计算了超代数B（0，1）（（1））的情形，特别表明，限制到它的偶数部分又产生Vir的所有模不变表示。这些结果给出了关于正能量表示的渐近行为和模不变表示的分类的一般性结论。
In this paper, we launch a program to describe and classify modular invariant representations of infinite-dimensional Lie algebras and superalgebras. We prove a character formula for a large class of highest weight representations L(lambda) of a Kac-Moody algebra [unk] with a symmetrizable Cartan matrix, generalizing the Weyl-Kac character formula [Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70]. In the case of an affine [unk], this class includes modular invariant representations of arbitrary rational level m = t/u, where t [unk] Z and u [unk] N are relatively prime and m + g >/= g/u (g is the dual Coxeter number). We write the characters of these representations in terms of theta functions and calculate their asymptotics, generalizing the results of Kac and Peterson [Kac, V. G. & Peterson, D. H. (1984) Adv. Math. 53, 125-264] and of Kac and Wakimoto [Kac, V. G. & Wakimoto, M. (1988) Adv. Math. 70, 156-234] for the u = 1 (integrable) case. We work out in detail the case [unk] = A(1) ((1)), in particular classifying all its modular invariant representations. Furthermore, we show that the modular invariant representations of the Virasoro algebra Vir are precisely the "minimal series" of Belavin et al. [Belavin, A. A., Polyakov, A. M. & Zamolodchikov, A. B. (1984) Nucl. Phys. B 241, 333-380] using the character formulas of Feigin and Fuchs [Feigin, B. L. & Fuchs, D. B. (1984) Lect. Notes Math. 1060, 230-245]. We show that tensoring the basic representation and modular invariant representations of A(1) ((1)) produces all modular invariant representations of Vir generalizing the results of Goddard et al. [Goddard P., Kent, A. & Olive, D. (1986) Commun. Math. Phys. 103, 105-119] and of Kac and Wakimoto [Kac, V. G. & Wakimoto, M. (1986) Lect. Notes Phys. 261, 345-371] in the unitary case. We study the general branching functions as well. All these results are generalized to the Kac-Moody superalgebras introduced by Kac [Kac, V. G. (1978) Adv. Math. 30, 85-136] and to N = 1 super Virasoro algebras. We work out in detail the case of the superalgebra B(0, 1)((1)), showing, in particular, that restricting to its even part produces again all modular invariant representations of Vir. These results lead to general conjectures about asymptotic behavior of positive energy representations and classification of modular invariant representations.