2026-03-05 清華大学
Proof of a key identity for showing rigidity, using graphical calculus from braided tensor category theory
＜関連情報＞
- https://www.tsinghua.edu.cn/en/info/1245/14723.htm
- https://link.intlpress.com/JDetail/2026366315557601282
C2-有限頂点演算子代数の合理性について On rationality for C2-cofinite vertex operator algebras
Robert McRae
Cambridge Journal of Mathematics
DOI:https://dx.doi.org/10.4310/CJM.260225023743
Abstract
Let V be an N-graded, simple, self-contragredient,C2 -cofinite vertex operator algebra. We show that if the -transformation of the character of V is a linear combination of characters of V-modules, then the category C of grading-restricted generalized V-modules is a rigid tensor category. We further show, without any assumption on the character of V but assuming that C is rigid, that C is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of V is semisimple, then C is semisimple and thus V is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated toV .
We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that C2-cofinite affine W-algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such W-algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the “coset rationality problem” to the problem of C2-cofiniteness for the coset. That is, given a vertex operator algebra inclusion U ⊗ V→ A with A, U strongly rational and a U ,V pair of mutual commutant subalgebras in A, we show that V is also strongly rational provided it is C2-cofinite.
