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Register a new userQuantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge
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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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We introduce a subalgebra $overline F$ of the Clifford vertex superalgebra ($bc$ system) which is completely reducible as a $L^{Vir} (-2,0)$-module, $C_2$-cofinite, but it is not conformal and it is not isomorphic to the symplectic fermion algebra $mathcal{SF}(1)$. We show that $mathcal{SF}(1)$ and $overline{F}$ are in an interesting duality, since $overline{F}$ can be equipped with the structure of a $mathcal{SF}(1)$-module and vice versa. Using the decomposition of $overline F$ and a free-field realization from arXiv:1711.11342, we decompose $L_k(mathfrak{osp}(1vert 2))$ at the critical level $k=-3/2$ as a module for $L_k(mathfrak{sl}(2))$. The decomposition of $L_k(mathfrak{osp}(1vert 2))$ is exactly the same as of the $N=4$ superconformal vertex algebra with central charge $c=-9$, denoted by $mathcal V^{(2)}$. Using the duality between $overline{F}$ and $mathcal{SF}(1)$, we prove that $L_k(mathfrak{osp}(1vert 2))$ and $mathcal V^{(2)}$ are in the duality of the same type. As an application, we construct and classify all irreducible $L_k(mathfrak{osp}(1vert 2))$-modules in the category $mathcal O$ and the category $mathcal R$ which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra $N_{-3/2}(mathfrak{osp}(1vert 2))$ as a $N_{-3/2}(mathfrak{sl}(2))$-module. We extend this example, and for each $p ge 2$, we introduce a non-conformal vertex algebra $mathcal A^{(p)}_{new}$ and show that $mathcal A^{(p)}_{new} $ is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra $ mathcal V^{(p)} _{new}$ which is isomorphic to the logarithmic vertex algebra $mathcal V^{(p)}$ as a module for $widehat{mathfrak{sl}}(2)$.
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.
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.
Let $mathcal{O}_c$ be the category of finite-length central-charge-$c$ modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that $mathcal{O}_c$ admits vertex algebraic tensor category structure for any $cinmathbb{C}$. Here, we determine the structure of this tensor category when $c=13-6p-6p^{-1}$ for an integer $p>1$. For such $c$, we prove that $mathcal{O}_{c}$ is rigid, and we construct projective covers of irreducible modules in a natural tensor subcategory $mathcal{O}_{c}^0$. We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that $mathcal{O}_c$ has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine $mathfrak{sl}_2$ at levels $-2+p^{pm 1}$. Next, as a straightforward consequence of the braided tensor category structure on $mathcal{O}_c$ together with the theory of vertex operator algebra extensions, we rederive known results for triplet vertex operator algebras $mathcal{W}(p)$, including rigidity, fusion rules, and construction of projective covers. Finally, we prove a recent conjecture of Negron that $mathcal{O}_c^0$ is braided tensor equivalent to the $PSL(2,mathbb{C})$-equivariantization of the category of $mathcal{W}(p)$-modules.
The rational and C_2-cofinite simple vertex operator algebras whose effective central charges and the central charges c are equal and less than 1 are classified. Such a vertex operator algebra is zero if c<0 and C if c=0. If c>0, it is an extension of discrete Virasoro vertex operator algebra L(c_{p,q},0) by its irreducible modules. It is also proved that for any rational and C_2-cofinite simple vertex operator algebra whose effective central charge and central charge are equal, the vertex operator subalgebra generated by the Virasoro vector is simple.
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