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Hamiltonian elements in algebraic K-theory

Yasha Savelyev

TL;DR

The work develops a framework marrying Hamiltonian symplectomorphism actions on Fukaya categories with categorified algebraic K-theory, via monotone and Calabi–Yau regimes, producing $K^{Cat}(k)$ and $K^{Cat,\mathbb{Z}_2}(k)$ and a Hochschild map to $K^{\mathbb{Z}_2}(k)$. It defines geometric elements from Hamiltonian fibrations as maps into these K-theory spaces and shows they induce homotopy classes on classifying spaces, yielding algebraic K-theory classes in both the categorified and classical worlds, including $K_k(\Lambda)$ in the Calabi–Yau case. A mirror-symmetry interpretation is proposed, linking A-model Hamiltonian K-theory elements with a B-model analogue via Langlands dual data, suggesting a common image in $K^{Cat,\mathbb{Z}_2}_*(k)$ for dual groups. The construction leverages Waldhausen K-theory of $A_\infty$-categories, Hochschild homology, and Fukaya categories, and opens a path to a gauged/homological mirror symmetry program within algebraic K-theory. Practical impact lies in providing a rigorous K-theoretic framework for gauged symmetries, mirror symmetry, and Hamiltonian dynamics in a homotopy-theoretic setting.

Abstract

Recall that topological complex $K$-theory associates to an isomorphism class of a complex vector bundle $E$ over a space $X$ an element of the complex $K$-theory group of $X$. Or from algebraic $K$-theory perspective, one assigns a homotopy class $[X \to K (\mathcal{K})]$, where $\mathcal{K}$ is the ring of compact operators on the Hilbert space. We show that there is an analogous story for algebraic $K$-theory of a general commutative ring $k$, replacing, and in a sense generalizing complex vector bundles by certain monotone/Calabi-Yau Hamiltonian fiber bundles. (In Calabi-Yau setting $k$ must be restricted.) In suitable cases, we may first assign elements in a certain categorified algebraic $K$-theory, analogous to Toën's secondary $K$-theory of $k$. And there is a natural ``Hochschild'' map from this categorified algebraic $K$-theory to the classical variant. In particular, if $k$ is regular and $G$ is a compact Lie group we obtain a natural group homomorphism $π_{m} (BG) \to K _{m}(k) \oplus K _{m-1} (k) $. This story leads us to formulate a generalization of the homological mirror symmetry phenomenon to the algebraic $K$-theory context, based on ideas of gauged mirror symmetry of Teleman, and the formalism of Langlands dual groups.

Hamiltonian elements in algebraic K-theory

TL;DR

The work develops a framework marrying Hamiltonian symplectomorphism actions on Fukaya categories with categorified algebraic K-theory, via monotone and Calabi–Yau regimes, producing and and a Hochschild map to . It defines geometric elements from Hamiltonian fibrations as maps into these K-theory spaces and shows they induce homotopy classes on classifying spaces, yielding algebraic K-theory classes in both the categorified and classical worlds, including in the Calabi–Yau case. A mirror-symmetry interpretation is proposed, linking A-model Hamiltonian K-theory elements with a B-model analogue via Langlands dual data, suggesting a common image in for dual groups. The construction leverages Waldhausen K-theory of -categories, Hochschild homology, and Fukaya categories, and opens a path to a gauged/homological mirror symmetry program within algebraic K-theory. Practical impact lies in providing a rigorous K-theoretic framework for gauged symmetries, mirror symmetry, and Hamiltonian dynamics in a homotopy-theoretic setting.

Abstract

Recall that topological complex -theory associates to an isomorphism class of a complex vector bundle over a space an element of the complex -theory group of . Or from algebraic -theory perspective, one assigns a homotopy class , where is the ring of compact operators on the Hilbert space. We show that there is an analogous story for algebraic -theory of a general commutative ring , replacing, and in a sense generalizing complex vector bundles by certain monotone/Calabi-Yau Hamiltonian fiber bundles. (In Calabi-Yau setting must be restricted.) In suitable cases, we may first assign elements in a certain categorified algebraic -theory, analogous to Toën's secondary -theory of . And there is a natural ``Hochschild'' map from this categorified algebraic -theory to the classical variant. In particular, if is regular and is a compact Lie group we obtain a natural group homomorphism . This story leads us to formulate a generalization of the homological mirror symmetry phenomenon to the algebraic -theory context, based on ideas of gauged mirror symmetry of Teleman, and the formalism of Langlands dual groups.
Paper Structure (10 sections, 11 theorems, 54 equations)

This paper contains 10 sections, 11 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Building a Waldhausen category with objects $A _{\infty}$ categories
  4. Categorified algebraic $K$-theory
  5. Some computation, and relation with Toën's secondary $K$-theory
  6. $|\mathcal{A} _{k}|$, $|\mathcal{A} _{k} ^{\mathbb{Z}_{n}}|$ as representing spaces of Fukaya field theories.
  7. Hamiltonian elements in categorified algebraic $K$-theory
  8. Hamiltonian elements in algebraic $K$-theory
  9. Mirror Hamiltonian algebraic $K$-theory elements.
  10. Acknowledgments

Key Result

Theorem 1.2

There is a natural homotopy class $[g: BPU (n) \to K ^{Cat, \mathbb{Z} _{2}}]$ and so natural group homomorphisms: for each commutative ring $k$. Furthermore, for any compact Lie group $G$, there is a homotopy natural map:

Theorems & Definitions (25)

  • Remark 1.1
  • Theorem 1.2
  • Example 1
  • Example 2
  • Remark 3.4
  • Lemma 3.5
  • proof : Proof
  • Lemma 4.5: Toën cite_ToenLectures
  • Theorem 4.6: Saito cite_SaitoPeriodicDerived
  • Corollary 4.9
  • ...and 15 more