A novel conformal geometric algebraic construction of the modular and braid groups

Clifford algebra provides a simple construction of the orthogonal groups via spin double covers, which often simplify the algebraic treatment of these groups. Using a homomorphism between the conformal group in $(p, q)$ dimensions and the special orthogonal group in $(p+1, q+1)$ dimensions, this construction can be extended to construct the conformal group in terms of spinors in a conformal model.The modular group is a subgroup of the 2D conformal group and has many important applications, for instance in string theory and modular form theory.Here we therefore construct the modular group in this conformal model.Such a spinorial approach could open up novel applications in many areas. In turn, the braid group in three strands $B_3$ is a double cover of the modular group. Our Clifford spinor construction of course yields a double cover of the modular group, which is exactly this braid group.In the conformal picture, the two braid group operations of swapping pairs of strands have a nice new geometric interpretation as translations with respect to the origin and the point at infinity