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Register a new userTowards Vertex Algebras of Krichever-Novikov Type, Part I
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Authors
K. J. Linde
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It is shown that a certain representation of the Heisenberg type Krichever-Novikov algebra gives rise to a state field correspondence that is quite similar to the vertex algebra structure of the usual Heisenberg algebra. Finally a definition of Krichever-Novikov type vertex algebras is proposed and its relation to vertex algebras is discussed.
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Let $R := R_{2}(p)=mathbb{C}[t^{pm 1}, u : u^2 = t(t-alpha_1)cdots (t-alpha_{2n})] $ be the coordinate ring of a nonsingular hyperelliptic curve and let $mathfrak{g}otimes R$ be the corresponding current Lie algebra. color{black} Here $mathfrak g$ is a finite dimensional simple Lie algebra defined over $mathbb C$ and begin{equation*} p(t)= t(t-alpha_1)cdots (t-alpha_{2n})=sum_{k=1}^{2n+1}a_kt^k. end{equation*} In earlier work, Cox and Im gave a generator and relations description of the universal central extension of $mathfrak{g}otimes R$ in terms of certain families of polynomials $P_{k,i}$ and $Q_{k,i}$ and they described how the center $Omega_R/dR$ of this universal central extension decomposes into a direct sum of irreducible representations when the automorphism group was the cyclic group $C_{2k}$ or the dihedral group $D_{2k}$. We give examples of $2n$-tuples $(alpha_1,dots,alpha_{2n})$, which are the automorphism groups $mathbb G_n=text{Dic}_{n}$, $mathbb U_ncong D_n$ ($n$ odd), or $mathbb U_n$ ($n$ even) of the hyperelliptic curves begin{equation} S=mathbb{C}[t, u: u^2 = t(t-alpha_1)cdots (t-alpha_{2n})] end{equation} given in [CGLZ17]. In the work below, we describe this decomposition when the automorphism group is $mathbb U_n=D_n$, where $n$ is odd.
We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which plays an important role in the classification of lattices. We derive a formula relating the mass for vertex algebras to that for lattices, and then give a new characterization of some holomorphic vertex operator algebras.
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 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.
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 $widehat{mathfrak{sl}}(2)$.
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