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points of the circle, then 

The computation of the determinant is done simultane- ously with the calculation of the curvature, which also requires the walk on the same circle. The n umerical cal- culations give a direct confirmation that F $^{q}$( P )=1 for all points of the sphere. 

## VII. QUANTUM GEOMETRY FOR TWO FLUXES ADDED 

In this case we have 4 anyons and the zero modes space is 2-dimensional. We will keep 3 anyons fixed and move the forth one along different braiding paths. 

For the Non-Abelian case, there is no simple Stokes theorem, 33,34 which means the monodromy can not be simply computed from the curvature as we did for the previous case. Even so, mapping the curvature provides a clear picture of the non-comutative and topological prop- erties of the states. 

The parameter space remains 2-dimensional. As be- fore, a point in this parameter space indicates the posi- tion of the mobile anyon. Thus dF = ˆ Fdw $^{1}$∧ dw $^{2}$, but ˆ F is now a 2 × 2 matrix. We compute ˆ F ( P ) using the same algorithm (see Eq. 48). Using the Pauli’s matrices, σ$_{i}$ , i=1, 2, 3, ˆ F ( P ) can be uniquely decomposed as: 

where f$_{0}$ ( P )= 1 $_{2}$Tr ˆ F ( P ) and f ( P ) is a 3-component vec- tor. We will refer to f$_{0}$ as the Abelian and to f · σ as the non-Abelian part of the curvature. It is important to notice how different quantities behave when changing the gauge, i.e. the basis in the 2-dimensional zero modes space. We have: f$_{0}$ ( P ) is gauge independent; the mag- nitude of f ( P ) is gauge independent; the orientation of f ( P ) is gauge dependent. 

Fig. 9 shows plots of Tr ˆ F ( P ) for different system sizes. For each size, we show Tr ˆ F ( P ) calculated with the stan- dard and with the quantum metric tensor. The numerics confirm again the theoretical prediction that Tr ˆ F ( P )=1. 

To demonstrate that the braid group is non- comutative, we need first to show that f ( P ) is non-zero. This, however, is not enough. We need also to rule out the existence of a particular gauge in which the adiabatic connection becomes diagonal at every point of the param- eter space. If such a gauge exists, the fiber bundle of the zero-modes degenerates into a trivial U (1) × U (1) fiber bundle, in which case there will be two 1-dimensional fibers that do not mix during the adiabatic braiding. Consequently, all monodromies will take a diagonal form in this gauge and they will commute with each other. 

As already mentioned, the magnitude of f ( P ) is gauge independent. Thus we can see if f ( P ) is zero or not by 

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simply plotting its magnitude, which is shown in Fig. 10 for different sizes. The graph clearly demonstrates that f ( P ) is non-zero and it appears to be concentrated near the positions of the fixed anyons. We will further discuss Fig. 10 in the next Section. 

Next, we introduce a scalar function which we call the Twist density ρ$_{TW}$ , which gives a measure of how much are the fibers twisted during the adiabatic parallel trans- port. We start the construction from the following 2- form: 

where D$_{µ}$ denotes the covariant derivative correspond- ing to the adiabatic connection and [ , ] denotes the usual commutator. This form is invariant to coordinate trans- formations, thus well defined. The coefficients of this form are 2 × 2 matrices. In the special case of a two dimensional parameter space, the form reduces to: 

Based on the above observations, we construct the fol- lowing 2-form, 

whose coefficient is a pseudo-scalar function. Since d ˆ ρ is invariant to coordinate transformations, dρ$_{TW}$ is also invariant and hence well defined. The density 

is gauge invariant and it is identically zero for trivial fiber bundles, in particular for U (1) × U (1) fiber bundle over our parameter space. Thus, if we show that ρ$_{TW}$ is non-zero, that will be equivalent to demonstrating that the zero modes fiber bundle is non-trivial. 

Let us now give the physical interpretation of our con- struction. For this we consider three points on the sphere: w , w +Δ w$_{1}$ and w + i Δ w$_{2}$ , as in Fig. 11. Assume that we 

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