November 9, 2025Open Access

The Heisenberg algebra of a vector space and Hochschild homology

Key Points

Hochschild homology connects noncommutative geometry with the Heisenberg algebra framework.
The approach emphasizes properties of graded vector spaces alongside their algebraic structures.
Computational methods shed light on the complex interplay in noncommutative varieties and cohomology.
Insights into higher algebraic structures may enhance understanding of mathematical phenomena and applications.

Abstract

We decategorify the Heisenberg 2-category of Gyenge-Koppensteiner-Logvinenko using Hochschild homology. We use this to generalise the Heisenberg algebra action of Grojnowski and Nakajima to all smooth and proper noncommutative varieties in the noncommutative geometry setting proposed by Kontsevich and Soibelman. For ordinary commutative varieties, we compute the resulting action on Chen-Ruan orbifold cohomology. As tools, we prove results about Heisenberg algebras of a graded vector space which might be of independent interest.

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Gyenge et al. (Wed,) studied this question.

synapsesocial.com/papers/690fdcdaf60c54d04ea38288 https://doi.org/https://doi.org/10.48550/arxiv.2511.03649

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