Fibonacci Non-Invertible Symmetry

• Fibonacci non-invertible symmetry is a concept that describes algebraic and categorical structures governed by fusion rules based on the Fibonacci sequence.
• It manifests in matrix theory and conformal field theories through non-invertible inversion properties and fusion rules that dictate strict spectral and combinatorial constraints.
• Applications extend from automata-based Fibonacci numeration systems to topologically ordered quantum phases, illuminating novel pathways in symmetry and dualities.

Fibonacci Non-Invertible Symmetry is a multifaceted concept at the intersection of algebra, combinatorics, category theory, and mathematical physics. It refers broadly to symmetry phenomena governed by the Fibonacci sequence and, more precisely, to algebraic structures—such as defects, fusion rules, and matrix identities—whose fusion, composition, or summation laws echo the characteristic non-invertible pattern of the Fibonacci category. Unlike conventional (group-like) symmetries where elements have inverses, Fibonacci non-invertible symmetries lack global invertibility, resulting instead in fusion or sum rules of the form $W \otimes W = 1 + W$, which are central both in algebraic combinatorics (e.g., matrix theory) and in the categorical symmetries of modern quantum field theories.

1. Algebraic and Matrix-Theoretic Manifestations

A foundational instance of Fibonacci non-invertible symmetry arises in matrix theory, specifically among $(0,1)$ upper triangular matrices. Let $A$ be an $n \times n$ such matrix (with $n \geq 3$) and $A^{-1}$ its inverse. The sum $S = \sum_{i,j} (A^{-1})_{ij}$ is not arbitrary; rather, it satisfies the constraint

$2 - F_{n-1} \leq S \leq 2 + F_{n-1}$

where $F_m$ denotes the $m$th Fibonacci number (Farber et al., 2013, Chatterjee et al., 2020). This range is both necessary and sufficient: for each integer $S$ in this interval, there exists a suitable $A$ with $S(A^{-1}) = S$ (Farber et al., 2013). The extreme cases are achieved when the structure of $A^{-1}$ aligns with vectors whose components alternate in sign and magnitude according to $\left((-1)^i F_{i-1}\right)_{i=1}^n$.

In the singular case, provided $A$ admits a group inverse $A^\#$ (i.e., $rank(A) = rank(A^2)$), the sum $S(A^\#)$ also satisfies a Fibonacci-determined interval (with adjustments reflecting block structure) (Chatterjee et al., 2020). In both invertible and group-invertible cases, the presence of Fibonacci numbers in the sum reflects an underlying, emergent symmetry only present in the matrix inversion structure and not necessarily visible in $A$ itself. This non-invertibility is twofold: most $(0,1)$-matrices are not invertible (hence the "non-invertible" class is much larger), and the sign-alternating Fibonacci structure emerges in the inversion, not the matrix itself.

2. Categorical Symmetry and Fusion Categories

In 1+1-dimensional conformal field theories (CFTs), non-invertible symmetries are encoded in topological line defects subject not to group structure but to fusion rules given by braided fusion categories. The Fibonacci category is the paradigmatic example, featuring a single nontrivial generator $W$ with the fusion rule

$W \otimes W = 1 + W$

(Lin et al., 2023). This fusion rule exemplifies non-invertibility: $W$ lacks a two-sided inverse, and its iterated fusion produces a Fibonacci sequence of line defect types.

This categorical symmetry constrains possible operator spectra and appears in a wide range of CFTs, including the $(\mathfrak{g}_2)_1$ and $(\mathfrak{f}_4)_1$ WZW models, tricritical Ising and three-state Potts models, and, in the lattice, in anyonic "golden chains" (Lin et al., 2023, Cordova et al., 2023, Cordova et al., 13 Mar 2024). The Fibonacci fusion rule organizes both the operator content and the transformation properties of sectors under the action of topological defects, enforcing spectral degeneracies (e.g., threefold particle-soliton degeneracy in deformed tricritical Ising flows (Cordova et al., 13 Mar 2024)) and constraining renormalization group flows such that robust gapless phases protected solely by Fibonacci non-invertible symmetry cannot exist within certain bounds (Lin et al., 2023).

3. Sums, Finite Symmetries, and Generalized Recurrences

Non-invertible Fibonacci symmetry manifests in the context of finite sums and generalized Fibonacci numbers as invariance under reindexing transformations that are not bijections at the term level. For example, consider sums of the form

$\sum_{k=1}^{n} (-1)^{pk}\, G_{pk}\, G_{pk+pq}$

where $G_n$ is a generalized Fibonacci sequence and $p,q \in \mathbb{Z}$. A fundamental identity shows that exchanging $q$ and $n$ leaves the sum invariant—even though the index-level map is not invertible (Adegoke et al., 2017). This is a precise instance of a global symmetry (invariance of the total sum) that is not realized by an invertible map on the individual summands—a combinatorial version of non-invertible symmetry.

Such properties extend beyond classic Fibonacci and Lucas sequences to Horadam sequences and demonstrate that non-invertible symmetry is a robust consequence of underlying recurrence relations. The widespread applicability of these identities gives alternative approaches to combinatorial and number-theoretic problems.

4. Fibonacci Non-Invertible Symmetry in Quantum Field Theory

Fibonacci non-invertible symmetries form a central structural element in topological quantum field theories (TQFTs) and finite-group gauge theories. In 2+1-dimensional gauge theory with dihedral group $\mathbb{D}_4$, explicit domain walls exhibit the Fibonacci fusion law: $$\mathcal{D}_{\mathbb{D}_4 \times \mathbb{D}_4, (1, 1)} \times \mathcal{D}_{\mathbb{D}_4 \times \mathbb{D}_4, (1, 1)} = 1 + \mathcal{D}_{\mathbb{D}_4 \times \mathbb{D}_4, (1, 1)}$$ (Cordova et al., 10 Jul 2024). Such domain walls, and more generally categorical symmetries, mediate non-invertible dualities, act non-trivially on Wilson lines and magnetic fluxes, and shape the possible topological orders and phase transitions in lattice and continuum models. These structures also play a significant role in more abstract contexts, such as the paper of anomalies and symmetry topological field theories ("SymTFTs"), where categorical symmetries constrain the dynamics and phase diagrams both in gapped and gapless regimes (Kaidi et al., 2023, Lin et al., 2023, Cordova et al., 2023).

5. Automata, Representation Theory, and Computational Aspects

Non-invertible symmetry is also reflected in the combinatorics of numeration systems and automata. In the paper of Fibonacci numeration and the Fibonacci analogue of the two's complement system, addition and related operations may be implemented by finite-state transducers; these devices enforce the sum and carry rules dictated by the Fibonacci recurrence and echo the collapse of information in non-invertible settings (Labbé et al., 2022).

Similar phenomena appear in partition function calculations, such as the Robbins–Ardila result that the coefficients $a(n)$ in the expansion

$\prod_{n\ge2} (1-X^{F_n}) = \sum_{n\ge0} a(n) X^n$

are always $-1, 0,$ or $1$, a property explainable by the automaton's semigroup structure "collapsing" under the Fibonacci recurrence to a triplet of possible outcomes (Shallit, 2020). The same non-invertibility surfaces in the fact that the mapping from "illegal" to "legal" Fibonacci representations via transducers is surjective but not injective.

6. Generalizations, Extensions, and Physical Interpretation

These themes extend into continuous (differential) analogues of the Fibonacci sequence. Continuous Fibonacci functions—solutions of $F''(x) - \tilde\varphi^2 F(x) = 0$—can be Darboux-deformed, yielding a one-parameter family of "Fibonacci numbers." The associated Darboux (supersymmetric) transformation is intrinsically non-invertible: from a given seed, infinitely many deformed partners arise, and the symmetry of the initial sequence is broken and replaced with a parametrized family of invariants (Rosu et al., 2022). The analogy is deepened by connections to the Friedmann equation in cosmology, where the exponential stretching of the Fibonacci sequence mirrors scale factor growth.

Non-invertible symmetries impact quantum information, topological phases, and the characterization of entanglement. The symmetry-resolved entanglement entropy for non-invertible symmetries (e.g., in Potts or tricritical Ising models) must account for modifications of projector constructions in the presence of boundaries, since the defect fusion algebra is sensitive to boundary conditions—again reflecting the failure of invertibility in the symmetry sector splitting and fusion (Heymann et al., 3 Sep 2024).

7. Structural Features and Broader Implications

The essence of Fibonacci non-invertible symmetry lies in algebraic and categorical structures—found in matrix theory, quantum field theory, and automata—where symmetries act globally rather than locally invertibly, are tied to the combinatorics of the Fibonacci numbers, and enforce rigid constraints on spectra, degeneracies, and observable features. These symmetries are encoded in fusion or sum rules of the Fibonacci type, produce non-trivial degeneracies (such as the threefold multiplets in soliton/particle spectra (Cordova et al., 13 Mar 2024)), and govern recurrences and matrix identities with striking regularity.

The ubiquity of the Fibonacci category in categorical symmetry, the rigidity of its fusion rules, and its emergence from both algebraic and physical systems exemplify the deep interplay between combinatorics, algebra, topology, and physics that defines the field of non-invertible symmetries. The paper of Fibonacci non-invertible symmetry continues to find new applications in condensed matter, quantum information, categorical quantum mechanics, and mathematical physics, driven by both its combinatorial richness and its explanatory power for non-classical symmetry phenomena.