论文标题
$ \ MATHCAL {W} $ - 代数的试验
Trialities of $\mathcal{W}$-algebras
论文作者
论文摘要
We prove the conjecture of Gaiotto and Rapčák that the $Y$-algebras $Y_{L,M,N}[ψ]$ with one of the parameters $L,M,N$ zero, are simple one-parameter quotients of the universal two-parameter $\mathcal{W}_{1+\infty}$-algebra, and satisfy a symmetry known as试验。这些$ y $ - 代数被定义为某些非主要$ \ MATHCAL {W} $ - 代数 - 代数和$ \ Mathcal {W} $ - Superalgebras的costex subsalgebras，而通过其亲戚subsalgebras，这是三个此类代数之间的同性恋。我们结果的特殊情况提供了有关$ \ Mathcal {w} $ - 类型$ a $代数的文献中许多定理和开放猜想的新的和统一的证据。这包括（1）feigin-frenkel二元性，（2）由于Arakawa和US引起的主要$ \ MATHCAL {W} $ - 代数的coset实现，（3）Feigin和Semikhatov和Semikhatov的猜想的猜测性试验性的次数$ \ MATHCAL $ \ MATHCAL $ \ MATHCAL {顶点超级甲虫，（4）由于Arakawa和van Ekeren而导致的次数$ \ MATHCAL {W} $ - 代数的合理性，（5）识别均为$ \ Mathcal {W} $ algebras的Heisenberg cosets与原理$ \ Mathercal conteration contrenge contrenge contrenge contrenge conterution contrenge conterution-Al and and and a imentation-al and contrenge and conterutiate 25年前。最后，我们在显式截断曲线上证明了Procházka和Rapčák的猜想，将简单的$ y $ -Algebras视为$ \ Mathcal {w} _ {1+ \ infty} $ - 商人，以及其最小的强生成类型。
We prove the conjecture of Gaiotto and Rapčák that the $Y$-algebras $Y_{L,M,N}[ψ]$ with one of the parameters $L,M,N$ zero, are simple one-parameter quotients of the universal two-parameter $\mathcal{W}_{1+\infty}$-algebra, and satisfy a symmetry known as triality. These $Y$-algebras are defined as the cosets of certain non-principal $\mathcal{W}$-algebras and $\mathcal{W}$-superalgebras by their affine vertex subalgebras, and triality is an isomorphism between three such algebras. Special cases of our result provide new and unified proofs of many theorems and open conjectures in the literature on $\mathcal{W}$-algebras of type $A$. This includes (1) Feigin-Frenkel duality, (2) the coset realization of principal $\mathcal{W}$-algebras due to Arakawa and us, (3) Feigin and Semikhatov's conjectured triality between subregular $\mathcal{W}$-algebras, principal $\mathcal{W}$-superalgebras, and affine vertex superalgebras, (4) the rationality of subregular $\mathcal{W}$-algebras due to Arakawa and van Ekeren, (5) the identification of Heisenberg cosets of subregular $\mathcal{W}$-algebras with principal rational $\mathcal{W}$-algebras that was conjectured in the physics literature over 25 years ago. Finally, we prove the conjectures of Procházka and Rapčák on the explicit truncation curves realizing the simple $Y$-algebras as $\mathcal{W}_{1+\infty}$-quotients, and on their minimal strong generating types.
