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Relating Hodge Atoms, Spectral Triples, and BPS Flows

Mark Raugas

Abstract

We compare algebraic and analytic pictures relevant to the study of birational invariants. Motivated by recent advances in the development of non-commutative Hodge structures, we examine their implication for quiver gauge field theory on the cubic fourfold. By interpreting the semiorthogonal property as a dynamical selection rule, we conjecture that the K3 Hodge atom of the cubic fourfold represents a protected quantum phase whose spectra remain invariant under non-perturbative tunneling processes.

Relating Hodge Atoms, Spectral Triples, and BPS Flows

Abstract

We compare algebraic and analytic pictures relevant to the study of birational invariants. Motivated by recent advances in the development of non-commutative Hodge structures, we examine their implication for quiver gauge field theory on the cubic fourfold. By interpreting the semiorthogonal property as a dynamical selection rule, we conjecture that the K3 Hodge atom of the cubic fourfold represents a protected quantum phase whose spectra remain invariant under non-perturbative tunneling processes.
Paper Structure (26 sections, 24 equations, 1 figure)

This paper contains 26 sections, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. From Static Hodge Theory to F-bundles
  4. U-plane Duality and Mirror Symmetry
  5. The Calabi-Yau Quintic and Wall-Crossing
  6. The Kuznetsov Component and Categorical Rigidity
  7. Categorical Filtering and Semiorthogonal Decompositions
  8. HKR and the JLO Cocycle
  9. Quantum Blow-up Theorem and Weight Filtration
  10. Numerical Invariants and the Mukai Lattice
  11. Spectral Networks and the Geometry of Vacua
  12. BPS Wall-Crossing and Global Stability
  13. Explicit Spectral Triples for LG Mirrors
  14. The Spectral Triple for $\mathbb{P}^1$
  15. The Spectral Triple for $\mathbb{P}^2$
  16. ...and 11 more sections

Figures (1)

  • Figure 1: Ext-quiver of the cubic fourfold under blow-up. Solid arrows denote non-zero $\operatorname{Ext}^\bullet$ groups (BPS solitons); numerical labels give $\dim\operatorname{Hom}$. Dashed arrows indicate vanishing $\operatorname{RHom}$ forced by semiorthogonality. Blow-up appends $E_1, E_2, E_3$ to the right of the SOD; the K3 block $S_{\mathcal{A}}$ remains unperturbed.

Theorems & Definitions (1)

  • Conjecture 8.1