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arXiv · 1109.4065

A commutant realization of W^(2)_n at critical level

Thomas Creutzig·Peng Gao·Andrew R. Linshaw
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Abstract

For n\geq 2, there is a free field realization of the affine vertex superalgebra A associated to psl(n|n) at critical level inside the bcβγsystem W of rank n^2. We show that the commutant C=Com(A,W) is purely bosonic and is freely generated by n+1 fields. We identify the Zhu algebra of C with the ring of invariant differential operators on the space of n\times n matrices under SL_n \times SL_n, and we classify the irreducible, admissible C-modules with finite dimensional graded pieces. For n\leq 4, C is isomorphic to the W_n^{(2)}-algebra at critical level, and we conjecture that this holds for all n.

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Thomas Creutzig, Peng Gao, Andrew R. Linshaw. 2012-10-05. A commutant realization of W^(2)_n at critical level. https://doi.org/10.1093/imrn%2Frns229

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