Vertex algebras and TKK algebras

In this paper, we associate the TKK algebra G ($) over cap (J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number l, the category of restricted G ($) over capG(J)-modules of level l is canonically isomorphic to the category of sigma-twisted V C-G ($) over capg( l, 0)-modules, where V (G ($) over capg)( l, 0) is a vertex algebra arising from the toroidal Lie algebra of type C2 and sis an isomorphism of V C-G ($) over capg( l, 0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer l, the integrable restricted C-G ($) over capg( l, 0)-modules of level l are exactly the s-twisted modules for the quotient vertex algebra L C-G ($) over capg( l, 0) of VCG ($) over capg( l, 0). Finally, we classify the irreducible 12N- graded s-twisted L C-G ($) over capg( l, 0)-modules. (c) 2023 Elsevier Inc. All rights reserved.