Duality of subregular W-algebras and principal W-superalgebras
Duality of subregular W-algebras and principal W-superalgebras
复制标题
次正则 W-代数和主 W-超代数的对偶性
DOI：
10.1016/j.aim.2021.107685
复制
发表时间：
2021
影响因子：
1.7
通讯作者：
Nakatsuka Shigenori
中科院分区：
文献类型：
--
作者：
Creutzig Thomas;Genra Naoki;Nakatsuka Shigenori
摘要证明了A、B型次正则w -代数与sl （1| n）、osp （2| 2 n）型主w -超代数之间的Feigin-Frenkel型对偶性。A类情况证明了Feigin和Semikhatov的一个猜想。设(g1, g2)=(sl n+ 1, sl (1| n+ 1))或（所以2n + 1， osp (2|2n)）设r为g1的密度。让k是一个复杂的数量和ℓ定义为r (k + h 1∨)(ℓ+ h 2∨)= 1 h我∨双重Coxeter数字的g。我们的第一个主要结果是,海森堡叠合组ck (g 1)和Cℓ(2 g)这些W-algebras这些双水平同构,即C k C (g 1)≃ℓ(2 g)通用k。我们决定通用水平而且建立类似结果的简单的叠合组上有W-algebras。我们的第二个结果是一个新的Kazama-Suzuki型协集构造：我们证明了k层的次正则w代数乘以格顶点超代数vz的对角线Heisenberg协集是对偶层上的主w超代数。相反地，在水平l上主w -超代数乘以晶格顶点超代数V - 1z的对角海森堡余集是对偶水平k上的次正则w -代数。这再次证明了普适w -代数以及单商。我们证明了Kazama-Suzuki型构造的一个结果是，如果对偶水平上的简单次正则w -代数也是如此，则在水平r上的简单主w -超代数及其Heisenberg协集是有理数和/或c2有限的。给出了许多新的c2 -有限性和合理性的结果。
Abstract We prove Feigin-Frenkel type dualities between subregular W-algebras of type A, B and principal W-superalgebras of type sl (1| n), osp (2| 2 n). The type A case proves a conjecture of Feigin and Semikhatov. Let (g 1, g 2)=(sl n+ 1, sl (1| n+ 1)) or (so 2 n+ 1, osp (2| 2 n)) and let r be the lacity of g 1. Let k be a complex number and ℓ defined by r (k+ h 1∨)(ℓ+ h 2∨)= 1 with h i∨ the dual Coxeter numbers of the g i. Our first main result is that the Heisenberg cosets C k (g 1) and C ℓ (g 2) of these W-algebras at these dual levels are isomorphic, ie C k (g 1)≃ C ℓ (g 2) for generic k. We determine the generic levels and furthermore establish analogous results for the cosets of the simple quotients of the W-algebras. Our second result is a novel Kazama-Suzuki type coset construction: We show that a diagonal Heisenberg coset of the subregular W-algebra at level k times the lattice vertex superalgebra V Z is the principal W-superalgebra at the dual level ℓ. Conversely a diagonal Heisenberg coset of the principal W-superalgebra at level ℓ times the lattice vertex superalgebra V− 1 Z is the subregular W-algebra at the dual level k. Again this is proven for the universal W-algebras as well as for the simple quotients. We show that a consequence of the Kazama-Suzuki type construction is that the simple principal W-superalgebra and its Heisenberg coset at level ℓ are rational and/or C 2-cofinite if the same is true for the simple subregular W-algebra at dual level ℓ. This gives many new C 2-cofiniteness and rationality results.