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Register a new userTrace functions and fusion rules of diagonal coset vertex operator algebras
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Xingjun Lin
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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.
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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.
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.
It is proved that any vertex operator algebra for which the image of the Virasoro element in Zhus algebra is algebraic over complex numbers is finitely generated. In particular, any vertex operator algebra with a finite dimensional Zhus algebra is finitely generated. As a result, any rational vertex operator algebra is finitely generated.
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