论文标题
次常规W-代数和主要W-Superalgebras的双重性
Duality of subregular W-algebras and principal W-superalgebras
论文作者
论文摘要
我们证明了A，B型的次规则W-代数之间的Feigin-Frenkel类型二元性，$ \ Mathfrak型的类型W-Superalgebras {sl}（1 | n），\ Mathfrak {osp}（osp}（osp}（2 | 2n）$。 A型情况证明了Feigin和Semikhatov的猜想。令$（\ Mathfrak {g} _1，\ Mathfrak {g} _2）=（\ Mathfrak {sl} _ {n+1}，\ Mathfrak {sl}（1 | n+1）（1 | n+1）） $ r $是$ \ mathfrak {g} _1 $的间隙。令k为一个复杂的数字，$ \ ell $由$ r（k+h^\ vee_1）定义（\ ell+h^\ vee_2）= 1 $，用$ h^\ vee_i $ $ h^\ vee_i $ $ \ mathfrak g_i $的dual coxeter数字。我们的第一个主要结果是，这些双重级别的Heisenberg Cosets $ \ Mathcal C^K（\ Mathfrak G_1）$和$ \ Mathcal c^\ ell（\ Mathfrak G_2）$在这些双重级别上是等级的，即同在c^\ ell（\ Mathfrak g_2）$用于通用k。我们确定了通用水平，并为W-代数的简单商的宇宙库建立了类似的结果。 我们的第二个结果是一种小说的Kazama-Suzuki型coset构造：我们表明，在$ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ v _ {\ mathbb z} $上是主要w-superalgebra的dual $ $ $ $ \ el e \ e el e \ el e Ell $。相反，主要的W-Superalgebra的对角线Heisenberg coset在级别$ \ ell $ times lattice顶点superalgebra $ v _ {\ sqrt {-1} \ mathbb z} $是二级k的下规则w- algebra。同样，这对于通用的W-Algebras以及简单的商证明了这一点。我们表明，Kazama-Suzuki型结构的结果是，如果简单的dual Level $ \ ell $ \ ell $ \ el $ \ ell $ \ ell $ \ ell $的简单主要W-Algebra对于简单的亚规格w- algebra而言，则简单的主要W-Superalgebra及其Heisenberg coset是合理的和/或C_2-Cofinite。这给出了许多新的C_2 cofinity和合理性结果。
We prove Feigin-Frenkel type dualities between subregular W-algebras of type A, B and principal W-superalgebras of type $\mathfrak{sl}(1|n), \mathfrak{osp}(2|2n)$. The type A case proves a conjecture of Feigin and Semikhatov. Let $(\mathfrak{g}_1,\mathfrak{g}_2) = (\mathfrak{sl}_{n+1},\mathfrak{sl}(1|n+1))$ or $(\mathfrak{so}_{2n+1}, \mathfrak{osp}(2|2n))$ and let $r$ be the lacity of $\mathfrak{g}_1$. Let k be a complex number and $\ell$ defined by $r(k+h^\vee_1)(\ell+h^\vee_2)=1$ with $h^\vee_i$ the dual Coxeter numbers of the $\mathfrak g_i$. Our first main result is that the Heisenberg cosets $\mathcal C^k(\mathfrak g_1)$ and $\mathcal C^\ell(\mathfrak g_2)$ of these W-algebras at these dual levels are isomorphic, i.e. $\mathcal C^k(\mathfrak g_1) \simeq \mathcal C^\ell(\mathfrak g_2)$ 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 $V_{\mathbb Z}$ is the principal W-superalgebra at the dual level $\ell$. Conversely a diagonal Heisenberg coset of the principal W-superalgebra at level $\ell$ times the lattice vertex superalgebra $V_{\sqrt{-1}\mathbb Z}$ 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 $\ell$ are rational and/or C_2-cofinite if the same is true for the simple subregular W-algebra at dual level $\ell$. This gives many new C_2-cofiniteness and rationality results.
