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Vector valued modular forms in Vertex Operator Algebras Auger, Jean


Any vertex operator algebra that is \(C_2\)-cofinite and non-rational has both a finite number of simple modules and a non-semisimple representation theory. Such VOAs have some of the simpler non-obvious structures one can think of. However, only one such family of examples is known. With a mid term aim of exposing new examples, we come across modular objects by looking at certain characters. Consider an affine VOA at rational admissible negative level. Then some extension of a coset sub-VOA is believed to provide new families of examples. In the case of the affine \(sl_2\) simple VOA, we show that the linear span of the characters of the lifted simple coset-modules give rise to a vector valued modular form. This non-obvious result is to be expected if one has faith in the conjectured properties of the extended coset VOA in question.

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