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Register a new userOn rationality for $C_2$-cofinite vertex operator algebras
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Authors
Robert McRae
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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.
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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.
Let $Vsubseteq A$ be a conformal inclusion of vertex operator algebras and let $mathcal{C}$ be a category of grading-restricted generalized $V$-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which $mathcal{C}$ inherits semisimplicity from the category of grading-restricted generalized $A$-modules in $mathcal{C}$, and vice versa. The most important condition is that $A$ be a rigid $V$-module in $mathcal{C}$ with non-zero categorical dimension, that is, we assume the index of $V$ as a subalgebra of $A$ is finite and non-zero. As a consequence, we show that if $A$ is strongly rational, then $V$ is also strongly rational under the following conditions: $A$ contains $V$ as a $V$-module direct summand, $V$ is $C_2$-cofinite with a rigid tensor category of modules, and $A$ has non-zero categorical dimension as a $V$-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that $A$ is a vertex operator superalgebra.
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.
We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $mathcal{M}(p)$, $pgeq 2$. The first category consists of all finite-length $mathcal{M}(p)$-modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra $mathcal{W}(p)$. We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers.
We provide a ribbon tensor equivalence between the representation category of small quantum SL(2), at parameter q=exp($pi$ i/p), and the representation category of the triplet vertex operator algebra at integral parameter p>1. We provide similar quantum group equivalences for representation categories associated to the Virasoro, and singlet vertex operator algebras at central charge c=1-6(p-1)^2/p. These results resolve a number of fundamental conjectures coming from studies of logarithmic CFTs in type A_1.
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