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Fibonacci-type orbifold data in Ising modular categories

Vincentas Mulevicius, Ingo Runkel

TL;DR

This work develops a concrete framework for constructing orbifold data in modular fusion categories under simplifying assumptions, reducing the defining conditions to polynomial equations in a finite scalar data set. Focusing on Ising-type categories, the authors exhibit Fibonacci-type orbifold data, yielding 32 pairwise non-equivalent modular orbifold categories $\\mathcal{D}=(\\mathcal{I}_{h^3,\\epsilon})_{\\mathbb{A}_{h,\\epsilon}}$, each with 11 simple objects and an explicit global dimension $24\,(h^2+h^{-2})^{-2}$. These orbifold categories realign with known extensions, notably inverting the $E_6$-invariant extension of $\\mathcal{C}(sl(2),10)$ to $\\mathcal{C}(sp(4),1)$, and their data are conjectured to correspond to Galois conjugates of $\\mathcal{C}(sl(2),10)$. The paper also develops adjunction-based techniques to read off simple objects and dimensions of $\\mathcal{C}_{\\mathbb{A}}$ from underlying bimodules, illustrating how the Fibonacci-type construction emerges from the interplay between algebraic data and TQFT invariants. Overall, the results contribute to modular-category classification, illuminate generalized orbifold/gauging procedures in topological phases, and provide a concrete path to new Witt-class explorations within small representative categories.

Abstract

An orbifold datum is a collection $\mathbb{A}$ of algebraic data in a modular fusion category $\mathcal{C}$. It allows one to define a new modular fusion category $\mathcal{C}_{\mathbb{A}}$ in a construction that is a generalisation of taking the Drinfeld centre of a fusion category. Under certain simplifying assumptions we characterise orbifold data $\mathbb{A}$ in terms of scalars satisfying polynomial equations and give an explicit expression which computes the number of isomorphism classes of simple objects in $\mathcal{C}_{\mathbb{A}}$. In Ising-type modular categories we find new examples of orbifold data which - in an appropriate sense - exhibit Fibonacci fusion rules. The corresponding orbifold modular categories have 11 simple objects, and for a certain choice of parameters one obtains the modular category for $sl(2)$ at level 10. This construction inverts the extension of the latter category by the $E_6$ commutative algebra.

Fibonacci-type orbifold data in Ising modular categories

TL;DR

This work develops a concrete framework for constructing orbifold data in modular fusion categories under simplifying assumptions, reducing the defining conditions to polynomial equations in a finite scalar data set. Focusing on Ising-type categories, the authors exhibit Fibonacci-type orbifold data, yielding 32 pairwise non-equivalent modular orbifold categories , each with 11 simple objects and an explicit global dimension . These orbifold categories realign with known extensions, notably inverting the -invariant extension of to , and their data are conjectured to correspond to Galois conjugates of . The paper also develops adjunction-based techniques to read off simple objects and dimensions of from underlying bimodules, illustrating how the Fibonacci-type construction emerges from the interplay between algebraic data and TQFT invariants. Overall, the results contribute to modular-category classification, illuminate generalized orbifold/gauging procedures in topological phases, and provide a concrete path to new Witt-class explorations within small representative categories.

Abstract

An orbifold datum is a collection of algebraic data in a modular fusion category . It allows one to define a new modular fusion category in a construction that is a generalisation of taking the Drinfeld centre of a fusion category. Under certain simplifying assumptions we characterise orbifold data in terms of scalars satisfying polynomial equations and give an explicit expression which computes the number of isomorphism classes of simple objects in . In Ising-type modular categories we find new examples of orbifold data which - in an appropriate sense - exhibit Fibonacci fusion rules. The corresponding orbifold modular categories have 11 simple objects, and for a certain choice of parameters one obtains the modular category for at level 10. This construction inverts the extension of the latter category by the commutative algebra.

Paper Structure

This paper contains 10 sections, 13 theorems, 113 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction and summary
  2. Orbifold data via polynomial equations
  3. The category $\mathcal{C}_\mathbb{A}$ and some of its basic properties
  4. Ising-type modular categories
  5. Fibonacci-type solutions inside Ising categories
  6. Analysing $\mathcal{C}_\mathbb{A}$ via adjunctions
  7. Appendix
  8. Labelling convention for polynomial equations
  9. Deriving the pentagon condition
  10. Evaluating the $T^3$-invariant

Key Result

Theorem 1.1

For each primitive $48^\text{th}$ root of unity $h \in \mathbb{C}$ and for each $\epsilon \in \{\pm 1\}$ we obtain an orbifold datum $\mathbb{A}_{h,\epsilon}$ in $\mathcal{I}_{\zeta,\epsilon}$, where $\zeta = h^3$ and where $A$ and $T$ are as above.

Figures (1)

  • Figure A.1: Three-torus with a series of embedded ribbon graphs as used in the calculation in \ref{['eq:T3calc2']}.

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • ...and 16 more