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Fusion Trees and Homological Representations

Sung Kim

TL;DR

The paper builds a bridge between $α$-fusion trees in non-semisimple topological quantum computation and the Lawrence homological representations specialized at roots of unity, establishing a projective isomorphism between the braid group action on fusion-tree spaces $\mathcal{H}_{n,m,α}^r$ and the $q$-specialized Lawrence representations $\mathcal{L}_{n,m,α}^r$ and enabling a graphical calculus approach. It uses this identification to give a new, diagrammatic proof of Ito's colored Alexander invariant formula and introduces special $α$-fusion trees with a non-degenerate Hermitian pairing inspired by Anghel, yielding a fusion-tree formulation of non-semisimple quantum knot invariants. The work highlights Hermitian structures in non-semisimple categories, analyzes the density and unitarity properties of the representations, and discusses potential NSS TQC architectures, including connections to surface braid group representations and mapping class groups. Collectively, the results provide a concrete, calculable framework that encodes non-semisimple quantum invariants in the language of fusion trees, with implications for both topological quantum computation and low-dimensional topology.

Abstract

We establish an identification between the spaces of $α$-fusion trees in non-semisimple topological quantum computation (NSS TQC) and a family of homological representations of the braid group known as the Lawrence representations specialized at roots of unity. Leveraging this connection, we provide a new proof of Ito's colored Alexander invariant formula using graphical calculus. Inspired by Anghel's topological model, we derive a formula involving the Hermitian pairing of fusion trees. This formula verifies that non-semisimple quantum knot invariants can be explicitly encoded via the language of fusion trees in the NSS TQC mathematical architecture.

Fusion Trees and Homological Representations

TL;DR

The paper builds a bridge between -fusion trees in non-semisimple topological quantum computation and the Lawrence homological representations specialized at roots of unity, establishing a projective isomorphism between the braid group action on fusion-tree spaces and the -specialized Lawrence representations and enabling a graphical calculus approach. It uses this identification to give a new, diagrammatic proof of Ito's colored Alexander invariant formula and introduces special -fusion trees with a non-degenerate Hermitian pairing inspired by Anghel, yielding a fusion-tree formulation of non-semisimple quantum knot invariants. The work highlights Hermitian structures in non-semisimple categories, analyzes the density and unitarity properties of the representations, and discusses potential NSS TQC architectures, including connections to surface braid group representations and mapping class groups. Collectively, the results provide a concrete, calculable framework that encodes non-semisimple quantum invariants in the language of fusion trees, with implications for both topological quantum computation and low-dimensional topology.

Abstract

We establish an identification between the spaces of -fusion trees in non-semisimple topological quantum computation (NSS TQC) and a family of homological representations of the braid group known as the Lawrence representations specialized at roots of unity. Leveraging this connection, we provide a new proof of Ito's colored Alexander invariant formula using graphical calculus. Inspired by Anghel's topological model, we derive a formula involving the Hermitian pairing of fusion trees. This formula verifies that non-semisimple quantum knot invariants can be explicitly encoded via the language of fusion trees in the NSS TQC mathematical architecture.

Paper Structure

This paper contains 25 sections, 24 theorems, 99 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Unitary TQFTs and Topological Phases of Matter
  3. The Non-semisimple Paradigm
  4. Non-semisimple TQFTs and TQC
  5. Homological Representations
  6. Fusion Trees and Homological Representations
  7. Organization
  8. Acknowledgments
  9. Homological Representations
  10. Lawrence representations
  11. Truncated Lawrence representations
  12. The unrolled quantum ${\mathfrak{sl}(2)}$
  13. The Drinfel'd-Jimbo quantum group
  14. A modified version of ${U_q{\mathfrak{sl}(2)}}$
  15. Fusion Trees
  16. ...and 10 more sections

Key Result

Theorem 1

The braid group representation on $\mathcal{H}_{n,m,\alpha}^r$ (the ${\rm Hom}$-space spanned by $\alpha$-fusion trees; see Section sec:fusion-trees) is projectively isomorphic to the Lawrence representation specialized at a root of unity $\mathcal{L}_{n,m,\alpha}^{r}$.

Figures (3)

  • Figure 1: An example of quantum states encoded as fusion trees (see neglectonNSSTQC).
  • Figure 2: Fusion trees involved in this study.
  • Figure 3: Visual depiction of the loops $\sigma_i$ and $\delta$ in $\pi_1(C_{n,m})$.

Theorems & Definitions (51)

  • Theorem : Theorem \ref{['thm:fusiontree-homological']}
  • Theorem : Theorem \ref{['thm:Ito']}, Ito's formula Ito16
  • Theorem : Theorem \ref{['thm:special-pairing']}
  • Proposition 1
  • Remark 2
  • Remark 3
  • Proposition 4
  • Proposition 5
  • Remark 6
  • Definition 7
  • ...and 41 more