The Heisenberg category of a category

Gyenge, Ádám and Clemens, Koppensteiner and Timothy, Logvinenko (2026) The Heisenberg category of a category. MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY, 320 (1633). pp. 1-131. ISSN 0065-9266

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Abstract

Starting with a k-linear or DG category admitting a (homotopy) Serre functor, we construct a k-linear or DG 2-category categorifying the Heisenberg algebra of the numerical K-group of the original category. We also define a 2-categorical analogue of the Fock space representation of the Heisenberg algebra. Our construction generalises and unifies various categorical Heisenberg algebra actions appearing in the literature. In particular, we give a full categorical enhancement of the action on derived categories of symmetric quotient stacks introduced by Krug, which itself categorifies a Heisenberg algebra action proposed by Grojnowski.

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