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Register a new userRationality and $C_2$-cofiniteness of certain diagonal coset vertex operator algebras
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Xingjun Lin
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In this paper, it is shown that the diagonal coset vertex operator algebra $C(L_{mathfrak{g}}(k+2,0),L_{mathfrak{g}}(k,0)otimes L_{mathfrak{g}}(2,0))$ is rational and $C_2$-cofinite in case $mathfrak{g}=so(2n), ngeq 3$ and $k$ is an admissible number for $hat{mathfrak{g}}$. It is also shown that the diagonal coset vertex operator algebra $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)otimes L_{sl_2}(4,0))$ is rational and $C_2$-cofinite in case $k$ is an admissible number for $hat{sl_2}$. Furthermore, irreducible modules of $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)otimes L_{sl_2}(4,0))$ are classified in case $k$ is a positive odd integer.
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Let $V$ be an $mathbb{N}$-graded, simple, self-contragredient, $C_2$-cofinite vertex operator algebra. We show that if the $S$-transformation of the character of $V$ is a linear combination of characters of $V$-modules, then the category $mathcal{C}$ 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 $mathcal{C}$ is rigid, that $mathcal{C}$ 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 $mathcal{C}$ 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 to $V$. We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that $C_2$-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 $C_2$-cofiniteness for the coset. That is, given a vertex operator algebra inclusion $Uotimes Vhookrightarrow A$ with $A$, $U$ strongly rational and $U$, $V$ a pair of mutual commutant subalgebras in $A$, we show that $V$ is also strongly rational provided it is $C_2$-cofinite.
In this paper, irreducible modules of the diagonal coset vertex operator algebra $C(L_{mathfrak{g}}(k+l,0),L_{mathfrak{g}}(k,0)otimes L_{mathfrak{g}}(l,0))$ are classified under the assumption that $C(L_{mathfrak{g}}(k+l,0),L_{mathfrak{g}}(k,0)otimes L_{mathfrak{g}}(l,0))$ is rational, $C_2$-cofinite and certain additional assumption. An explicit modular transformation formula of traces functions of $C(L_{mathfrak{g}}(k+l,0),L_{mathfrak{g}}(k,0)otimes L_{mathfrak{g}}(l,0))$ is obtained. As an application, the fusion rules of $C(L_{E_8}(k+2,0), L_{E_8}(k,0)otimes L_{E_8}(2,0))$ are determined by using the Verlinde formula.
We prove a general mirror duality theorem for a subalgebra $U$ of a simple vertex operator algebra $A$ and its coset $V=mathrm{Com}_A(U)$, under the assumption that $A$ is a semisimple $Uotimes V$-module. More specifically, we assume that $Acongbigoplus_{iin I} U_iotimes V_i$ as a $Uotimes V$-module, where the $U$-modules $U_i$ are simple and distinct and are objects of a semisimple braided ribbon category of $U$-modules, and the $V$-modules $V_i$ are semisimple and contained in a (not necessarily rigid) braided tensor category of $V$-modules. We also assume that $U$ and $V$ form a dual pair in $A$, so that $U$ is the coset $mathrm{Com}_A(V)$. Under these conditions, we show that there is a braid-reversing tensor equivalence $tau: mathcal{U}_Arightarrowmathcal{V}_A$, where $mathcal{U}_A$ is the semisimple category of $U$-modules with simple objects $U_i$, $iin I$, and $mathcal{V}_A$ is the category of $V$-modules whose objects are finite direct sums of the $V_i$. In particular, the $V$-modules $V_i$ are simple and distinct, and $mathcal{V}_A$ is a rigid tensor category.
In this paper, I investigate the ascending chain condition of right ideals in the case of vertex operator algebras satisfying a finiteness and/or a simplicity condition. Possible applications to the study of finiteness of orbifold VOAs is discussed.
It is proved that the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A_1^{(1)} of level k coincides with a certain W-algebra. In particular, a set of generators for the parafermion vertex operator algebra is determined.
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