Abstract
Families of vertex algebras associated to nilpotent elements of simply-laced Lie algebras are constructed. These algebras are close cousins of logarithmic W-algebras of Feigin and Tipunin and they are also obtained as modifications of semiclassical limits of vertex algebras appearing in the context of S-duality for four-dimensional gauge theories. In the case of type A and principal nilpotent element the character agrees precisely with the Schur-Index formula for corresponding Argyres-Douglas theories with irregular singularities. For other nilpotent elements they are identified with Schur-indices of type IV Argyres-Douglas theories. Further, there is a conformal embedding pattern of these vertex operator algebras that nicely matches the RG-flow of Argyres-Douglas theories as discussed by Buican and Nishinaka.
Highlights
I find it very interesting that the vertex operator algebras appearing in this gauge theory context are often of logarithmic type, that is they allow for indecomposable but reducible representations
There are various difficult questions in the context of logarithmic vertex operator algebras, see [27] for an introduction, and it is good that they allow for gauge theory interpretations
The best studied logarithmic vertex operator algebras are the triplet algebras [28,29,30,31], the fractional level WZW theories of sl2 [32,33,34], the logarithmic B(p)-algebras [35] and some progress is currently made on higher rank cases [36]
Summary
I find it very interesting that the vertex operator algebras appearing in this gauge theory context are often of logarithmic type, that is they allow for indecomposable but reducible representations. There is a conformal embedding pattern of these vertex operator algebras that nicely matches the RG-flow of Argyres-Douglas theories as discussed by Buican and Nishinaka. 1. The paper [16] observes for some eamples of type A1 Argyres-Douglas theories that a limit of wild Hitchin characters can be expressed in terms of the modular data of the semisimplification of the module category of the corresponding vertex operator algebra.
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