Fibonacci, golden ratio, and vector bundles

There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional group $G_2$ case of these bundles in three different ways, we derive a family of summation formulas for Fibonacci numbers in terms of the golden ratio.


Introduction
The purpose of this note is to establish the following formulas: Theorem 1.1. For all integers n ≥ 0 and g ≥ 2, (1) where F k is the k th Fibonacci number, ϕ = 1+ √ 5 2 is the golden ratio and ϕ = 1− √ 5 2 is the conjugate root of its minimal polynomial x 2 − x − 1.
While these formulas could be proven using elementary means, our emphasis in this paper is how naturally they emerge from an ostensibly unrelated setting. Indeed, by computing the rank of a certain vector bundle, for each n and g, in three different ways we arrive at the above formulas.
The relevant vector bundles here are the exceptional Lie algebra g 2 case of a more general construction, occurring for each simple complex Lie algebra, of the so-called vector bundle of conformal blocks on the Deligne-Mumford moduli space M g,n of stable n-pointed genus g curves. These vector bundles first appeared in mathematical physics [KZ84,Ver88] and they have been studied from the perspectives of both infinite-dimensional representation theory [TUY89,Uen97,Loo05] and algebraic geometry [NR93,BL94,Fal94,Pau96,Fak12,MOP15]; these citations are just a sample of the vast literature. One of the most significant properties of these bundles is their recursive structure reflecting the boundary stratification of M g,n . This property, known as factorization, was cleverly used by Mukhopadhyay to compute the rank of the g 2 , level 1, case of these bundles and the formula he obtained [Muk16] is the middle expression in Equation (1) involving the golden ratio; by using factorization in two different ways we obtain the two Fibonacci summation expressions in that equation. The details of this assertion are the content of this paper.
Remark 1.2. The author learned from the On-Line Encyclopedia of Integer Sequences that in the case n = 0, the formulas in Theorem 1.1 count the number of no-leaf edge-subgraphs in the Möbius ladder M g−1 [OEIS] (see [McS98,Theorem 16.1]).

Conformal blocks vector bundles
2.1. Notation. Given a simple complex Lie algebra g, an integer ℓ ≥ 1 called the level, and an n-tuple µ = (µ 1 , . . . , µ n ) ∈ P ℓ of dominant integral weights for g of level ℓ, there is a vector bundle of conformal blocks on the moduli stack M g,n of stable n-pointed genus g curves that we denote by V g,ℓ g,n (µ), or by V g,ℓ g when n = 0 [TUY89, Uen97,Fak12]. The construction of this vector bundle is immaterial for our purposes, we shall only need some standard properties well-established in the literature.
2.2. The Verlinde formula. The main property of conformal blocks vector bundles we are concerned with is their rank. There is a famous direct formula for this, the Verlinde formula (see [Sor96] for a nice survey), which for n = 0 and g of classical type or g 2 reads as follows (see [Gre11,Proposition B.1]): where P ℓ is the set of dominant integral weights for g of level ℓ, ∆ + is the set of positive roots, ·, · is the Killing form, ρ = 1 2 α j ∈∆ + α j , h ∨ is the dual Coxeter number, P is the weight lattice, Q is the root lattice, and Q lg is the long root lattice.
Gregoire has evaluated this explicitly in the case that is relevant for this paper, g = g 2 and ℓ = 1 [Gre11, Appendix B] (see also [GP13]): This is the middle expression in Equation (1) when n = 0.
Another ingredient we shall need is the propagation of vacua property (also first proven in [TUY89] then expressed in modular terms in [Fak12, Proposition 2.4]), which says that if a weight µ i in µ is zero then V g,ℓ g,n (µ) π * i V g,ℓ g,n−1 (µ 1 , . . . , µ i , . . . , µ n ), where π i is the map forgetting the i th marked point. This says that all points marked by the zero weight can be ignored when computing the rank of a conformal blocks vector bundle.
The final ingredient we shall need is the fusion rules, which are computations of the rank when g = 0 and n = 3. The special case of this that we require is the following [Uen97, Corollary 3.5.2]: 2.4. Specializing to an exceptional Lie algebra. Our focus is on the conformal blocks vector bundles for the exceptional Lie algebra g = g 2 and level ℓ = 1. Here there is only one nontrivial representation, and it is self-dual, so we can write P 1 = {0, µ} where µ * = µ. The Verlinde formula was proven using factorization, and Mukhopadhyay used factorization and induction to generalize Gregoire's Verlinde evaluation formula (3) for rk V g 2 ,1 g from n = 0 to all n ≥ 0 as follows [Muk16]: where µ n := (µ, . . . , µ n ) ∈ P n 1 .
This provides the middle expression in Equation (1) for all n ≥ 0 and g ≥ 2.
This completes the proof of Theorem 1.1.