Fibonacci category

Template:Multiple issues In mathematics, the Fibonacci category is a certain modular tensor category. Due to its connections with quantum field theory and its particularly simple structure, the Fibonacci category was among the first modular tensor categories to be considered. It was developed in the early 2000s by Michael Freedman, Zhenghan Wang, and Michael Larsen in the context of topological quantum computation via Fibonacci anyons. The term 'Fibonacci category' was coined by Greg Kuperberg, in reference to the fact that its fusion rules are described by Fibonacci numbers.

Definition

The Fibonacci category $𝐅𝐢𝐛$ is defined as follows. The set of simple objects of $𝐅𝐢𝐛$ has size two, and is denoted $ℒ=\{1,\tau \}$. Its non-trivial fusion rule is given by $\tau \otimes \tau =1\oplus \tau$. The other fusion rules are $\tau \otimes 1=1\otimes \tau =\tau$ and $1\otimes 1=1$. The twist values are ${\theta }_1=1$ and ${\theta }_{\tau }=e^{4\pi i/5}$. The R-symbols are $R_1^{\tau ,\tau }=e^{-4\pi i/5}$, $R_{\tau }^{\tau ,\tau }=e^{3\pi i/5}$, and $R_1^{1,1}=R_{\tau }^{1,\tau }=R_{\tau }^{\tau ,1}=1$. All non-zero F-symbols are all equal to 1, except for the symbols $F_{\tau ;1,1}^{\tau ,\tau ,\tau }={\phi }^{-1}$, $F_{\tau ;\tau ,1}^{\tau ,\tau ,\tau }=F_{\tau ;1,\tau }^{\tau ,\tau ,\tau }={\phi }^{-1/2}$, and $F_{\tau ;\tau ,\tau }^{\tau ,\tau ,\tau }=-{\phi }^{-1}$ where $\phi$ is the golden ratio.

Algebraic properties

The Fibonacci category has several notable algebraic properties.

• Taking the trace of the identity $\tau \otimes \tau =1\oplus \tau$, one arrives at the formula $d_{\tau }^2=1+d_{\tau }$ where $d_{\tau }$ is the quantum dimension of $\tau$. Seeing as the Fibonacci category is unitary all of its quantum dimensions are positive, and so $d_{\tau }=\phi$ is the Golden ratio, the unique positive solution to the equation $x^2=1+x$. It is a theorem that any simple object in unitary modular tensor category whose quantum dimension $d$ satisfies $1\le d<2$ must be of the form $d=2\cos (\pi /n)$ for some $n\ge 3$. This theorem is consistent with the Fibonacci category, since $\phi =2\cos (\pi /5)$.
• The Fibonacci category is the unique unitary modular tensor category with exactly one non-trivial simple object, such that this non-trivial object is non-abelian (in the sense that is quantum dimension is greater than one). There is one other unitary modular tensor category with exactly one non-trivial simple object, known as the semion category, but its non-trivial object is abelian.
• There is a fusion relation ${\tau }^{\otimes n}=F_{n-1}\oplus F_n\tau$, where $F_n$ is the $n$th Fibonacci number, normalized so that $F_0=0$ and $F_1=1$. Here, ${\tau }^{\otimes n}$ denotes the $n$-fold tensor product of $\tau$ with itself, and $F_n\tau$ denotes the $F_n$-fold direct sum of $\tau$ with itself. This relation can be proved using a simple induction. It is from this relation that the Fibonacci category gets its name.

Relationship to topological quantum field theory

In the context of topological quantum field theory, the Fibonacci category corresponds to the quantum Chern–Simons theory with gauge group $G=SO(3)$ at level $k=5$. Seeing as $\text{SO}(3)$ is a double cover of $\text{SU}(2)$, the Fibonacci category can alternatively be described as the even sectors in the Chern-Simons theory with gauge group $G=\text{SU}(2)$ at level $k=3$.

From this perspective, one can see a connection between Fibonacci anyons and the Jones polynomial polynomial using the classical techniques of Edward Witten. In his seminal 1989 paper, Witten demonstrated that the link and manifold invariants of quantum Chern–Simons theory with gauge group $G=SU(2)$ are related intimately to the Jones polynomial evaluated at roots of unity. Since the Fibonacci category corresponds to $G=\text{SU}(2)$ Chern-Simons theory, this means that the Fibonacci category will necessarily be related to the Jones polynomial.

A key insight of Michael Freedman in 1997 was to compare Witten's results with the fact that the evaluation of the Jones polynomial at $k$th roots of unity is a computationally difficult problem. In particular, evaluating the Jones polynomial exactly is an NP-hard problem whenever $k=5$ and $k\ge 7$, and giving an additive approximation of the Jones polynomial is BQP-complete whenever $k=5$ and $k\ge 7$. Under Witten's correspondence, the Fibonacci theory ($G=\text{SU}(2)$ at level $k=3$) is related to the Jones polynomial evaluated at 5th roots of unity, and thus when appropriately used can allow one to resolve BQP-complete problems.

Relationship to the Yang-Lee edge theory

The Fibonacci modular category is related to a separate model from non-unitary conformal field theory, known as the Yang-Lee theory. This theory describes the behavior of the two-dimensional Ising model in its paramagnetic phase at its critical imaginary value of magnetic field. It was shown by John Cardy that the Yang-Lee theory has two primary fields, denoted $𝕀$ and $\mathrm{\Phi }$, and that they satisfy the non-trivial fusion relation $\mathrm{\Phi }\otimes \mathrm{\Phi }=𝕀\oplus \mathrm{\Phi }$. This is the same fusion relation of the Fibonacci category. The Yang-Lee theory is related to a non-unitary conformal field theory, and as such it corresponds to a non-unitary modular tensor category.

Despite having the same fusion rules, the modular tensor category associated to the Yang-Lee theory is not the same as the Fibonacci modular category. The difference between these two categories is present in their associativity and braiding rules. The relationship between these two theories is that the Yang-Lee theory is the Galois conjugate of the Fibonacci theory. Namely, there exists an automorphism $\sigma :\bar{\mathbb{Q}}\to \bar{\mathbb{Q}}$ living in the absolute Galois group of the rational numbers such that applying $\sigma$ to all of the data of the Fibonacci theory recovers the data of the Yang-Lee theory. This means that for any F-symbol $F_{d;e,f}^{a,b,c}$ or R-symbol $R_c^{a,b}$ of the Fibonacci theory, the corresponding F-symbol or R-symbol of the Yang-Lee theory is $\sigma (F_{d;e,f}^{a,b,c})$ or $\sigma (R_c^{a,b})$.

Relationship to Jones polynomial

The Fibonacci category is related to the Kauffman bracket by the fact that the Reshetikhin–Turaev invariant of framed links associated to $𝐅𝐢𝐛$ is equal to the Kauffman bracket with parameter $A=e^{3\pi i/5}$. Since the Kauffman bracket is related to the Jones polynomial via a change of normalization, there is also a close relationship between $𝐅𝐢𝐛$ and the Jones polynomial.

The technical insight which relates the framed link invariants associated in

$𝐅𝐢𝐛$

to the Kauffman bracket is the low-dimensionality of the hom-spaces in the Fibonacci category, which implies the existence many linear relationships between its morphisms. In particular, the hom-space

${\text{Hom}}_{𝐅𝐢𝐛}(\tau \otimes \tau ,\tau \otimes \tau )$

is two-dimensional since

$\tau =1\oplus \tau$

. Using standard techniques to compute its coefficients, the following linear relationship is seen to be true:

Skein relation satisfied by the unique non-trivial simple object in the Fibonacci category.

This can be compared with the usual Skein relation for the Kauffman bracket, with

$A=e^{3\pi i/5}$

.

The Skein relation for the Kauffman bracket.

As an extended invariant

Due to the existence of a morphism $\tau \otimes \tau \to \tau$, the Fibonacci category naturally also lends itself to defining invariants of a generalization of links that allows for degree 3 vertices ("branchings"). These invariants can also be defined using generalized Skein relations. To do this, one chooses some distinguished morphisms $\tau \otimes \tau \to \tau$ and $\tau \to \tau \otimes \tau$, depicted visually below.

String diagrams for some distinguished morphisms $\tau \otimes \tau \to \tau$ and $\tau \to \tau \otimes \tau$ in the Fibonacci category.

Choosing these distinguished morphisms so that

A choice of normalization

Then the following generalized Skein relation holds:

Generalized Skein relation for the Fibonacci category, with a branching

Note that to make a proper topological invariant it is necessary to keep track of more structure on the links, such as orientations on the strands.