Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Heisenberg algebra of a vector space and Hochschild homology

Ádám Gyenge, Timothy Logvinenko

TL;DR

This work completes a Hochschild-homology–driven decategorification of the Heisenberg 2-category from Gyenge–Koppensteiner–Logvinenko and extends the Grojnowski–Nakajima Heisenberg action to all smooth and proper noncommutative varieties in Kontsevich–Soibelman’s noncommutative geometry framework. It constructs an action on the direct sum of Hochschild homologies of symmetric powers HH_∗(S^n 𝒱) for a smooth and proper DG category 𝒱, identifying the total Hochschild homology with the Fock space of a Heisenberg algebra H_{HH_∗(𝒱), χ} via the Euler pairing χ. The paper develops two equivalent generator systems (A- and PQ-generators) for the Heisenberg algebra in vector- and lattice-settings, proves basis-reduction and χ-independence results, and shows how Baranovsky-type decompositions connect HH_∗(S^n 𝒱) to tensor powers of HH_∗(𝒱). The main decategorification map π injects the Heisenberg algebra on HH_∗(𝒱) into the Hochschild-algebra of the categorified symmetric powers, with the Baranovsky decomposition showing the induced Fock-space embedding is bijective; this provides a universal HH-based decategorification of the Heisenberg 2-category and a noncommutative analogue of the Grojnowski–Nakajima action. The framework unifies noncommutative and commutative cases, and in the commutative setting yields an action on Chen–Ruan orbifold cohomology via the noncommutative Baranovsky decomposition and HKR isomorphism.

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.

The Heisenberg algebra of a vector space and Hochschild homology

TL;DR

This work completes a Hochschild-homology–driven decategorification of the Heisenberg 2-category from Gyenge–Koppensteiner–Logvinenko and extends the Grojnowski–Nakajima Heisenberg action to all smooth and proper noncommutative varieties in Kontsevich–Soibelman’s noncommutative geometry framework. It constructs an action on the direct sum of Hochschild homologies of symmetric powers HH_∗(S^n 𝒱) for a smooth and proper DG category 𝒱, identifying the total Hochschild homology with the Fock space of a Heisenberg algebra H_{HH_∗(𝒱), χ} via the Euler pairing χ. The paper develops two equivalent generator systems (A- and PQ-generators) for the Heisenberg algebra in vector- and lattice-settings, proves basis-reduction and χ-independence results, and shows how Baranovsky-type decompositions connect HH_∗(S^n 𝒱) to tensor powers of HH_∗(𝒱). The main decategorification map π injects the Heisenberg algebra on HH_∗(𝒱) into the Hochschild-algebra of the categorified symmetric powers, with the Baranovsky decomposition showing the induced Fock-space embedding is bijective; this provides a universal HH-based decategorification of the Heisenberg 2-category and a noncommutative analogue of the Grojnowski–Nakajima action. The framework unifies noncommutative and commutative cases, and in the commutative setting yields an action on Chen–Ruan orbifold cohomology via the noncommutative Baranovsky decomposition and HKR isomorphism.

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.

Paper Structure

This paper contains 44 sections, 47 theorems, 326 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalised binomial coefficients
  4. Partitions
  5. The existing definitions of the Heisenberg algebra
  6. A-generator definition
  7. PQ-generator definition
  8. Relation between A- and PQ-generators
  9. Heisenberg algebra of a graded vector space
  10. Basis reduction for the lattice PQ Heisenberg algebras
  11. Vector space definition
  12. $A$-generator definition
  13. $PQ$-generator definition
  14. Equivalence of $A$- and $PQ$-generator definitions
  15. Independence of $\chi$
  16. ...and 29 more sections

Key Result

Theorem 1

Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $X^{[n]}$ be the Hilbert scheme of $n$ points on $X$. Let $\chi$ be the pairing on $H^\bullet(X,\mathbb{Q})$ given by the cup product and then the direct image along $X \rightarrow \text{pt}$. For each $\alpha \in H^*(X,\mathbb{Q})$ and $ and thus define an action of the Heisenberg algebra $\underline{H}_{H^\bullet(X,\mathbb{Q}), \chi}$

Theorems & Definitions (137)

  • Theorem : see nakajima1997heisenberg, Theorem 3.1, grojnowski1995instantons, Theorem 7, and nakajima1999lectures, Theorem 8.13
  • Theorem 1.1: see Theorem \ref{['theorem-noncommutative-grojnowski-nakajima-action']}
  • Theorem 1.2: see Theorem \ref{['theorem-generalised-nakajima-grojnowski-heisenberg-action']}
  • Theorem 1.3: Theorem \ref{['theorem-construction-of-the-decategorification-map']} and Prop. \ref{['prps-decategorification-map-pi-is-injective']}
  • Conjecture 1.4
  • Theorem 1.5: see Defns. \ref{['defn-the-heisenberg-algebra-of-a-graded-vector-space-a-gen']}, \ref{['defn-the-heisenberg-algebra-of-a-graded-vector-space-pq-gen']}, Prop. \ref{['theorem-basis-reduction-for-heisenberg-algebra-graded-vector-space-pq-gen']}, and Theorems \ref{['theorem-basis-reduction-for-graded-vector-space-heisenberg-algebra']}, \ref{['theorem-A-and-PQ-generator-definition-equivalence-graded']}, \ref{['theorem-independence-of-chi-for-nondegenerate-chi-graded']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 127 more