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Equivalence between chain categories of representations of affine \(\text{sl}(2)\) and \(N=2\) superconformal algebras. (English) Zbl 0935.17011

From the text: The authors study highest-weight-type modules over the \(N=2\) superconformal algebra and the affine \(sl(2)\) algebra and demonstrate the equivalence of the two representation theories modulo the respective spectral flows. The motivation comes partly from the analysis of \(\widehat{sl}(2)\) fusion rules, which suggests the occurrence of more general modules than the usual Verma modules. The \(N=2\) superconformal algebra is known to have various relations to \(\widehat{sl}(2)\). A prerequisite for constructing complete \(N=2\) fusion rules is a better understanding of modules over the \(N=2\) algebra, and, in particular, in view of a recent progress in constructing \(\widehat{sl}(2)\) conformal blocks, of their relation to \(\widehat{sl}(2)\) modules.
Here the authors “compare” \(\widehat{sl}(2)\) and \(N=2\) representation theories and then find certain categories built out of representations of these algebras that are equivalent. In particular, they give a complete proof of the conjecture of A. M. Semikhatov [Mod. Phys. Lett. A 9, 1867-1896 (1994; Zbl 0895.17021)] that singular vectors in the “topological” (“chiral”) Verma modules over the \(N=2\) algebra are isomorphic to singular vectors in \(\widehat{sl}(2)\) Verma modules.
Then they extend this result along two directions. First, the relation between topological \(N=2\) Verma modules and the standard \(\widehat{sl}(2)\) Verma modules is extended to the equivalence of certain categories of representations of the two algebras, which are essentially the respective twisted \({\mathcal O}\) categories, in fact, the categories of chains of such modules related by the respective spectral flows. Hence the importance of twisted (spectral-flow-transformed) modules. In general, the spectral flow transformations and the canonical involution constitute the automorphism group of \(\widehat{sl}(2)\) as well as of the \(N=2\) algebra; the action of automorphisms on representations can be defined in a natural way, hence the appearance of twisted modules.
Second, the authors extend the correspondence between \(\widehat{sl}(2)\) and \(N=2\) Verma modules to representations of a different highest-weight type. For the \(N=2\) algebra, one should distinguish between two types of Verma-like objects: the topological Verma modules mentioned above, and the so-called massive \(N=2\) Verma modules. The latter are often considered as the “standard \(N=2\) Verma modules”. Their \(\widehat{sl}(2)\) counterpart, are the relaxed Verma modules, which differ from the usual \(\widehat{sl}(2)\) Verma modules by somewhat “relaxed” highest-weight conditions, possessing infinitely many highest-weight vectors. On the \(\widehat{sl}(2)\) side, the standard \(\widehat{sl}(2)\) Verma modules may appear as submodules of the relaxed Verma modules. On the \(N=2\) side, the situation is similar: a massive Verma module may contain submodules that are topological Verma modules (in fact, twisted topological Verma modules; such are, in particular, the submodules generated from the charged \(N=2\) singular vectors).

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B81 Applications of Lie (super)algebras to physics, etc.

Citations:

Zbl 0895.17021
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References:

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