Roughly speaking, a (mathematical) genus-$0$ conformal field theory (CFT) is a projective symmetric monoidal functor $Z$ from $C$ to $GrVec$ [1], where $GrVec$ is the category of graded complex vector spaces, and $C$ is roughly a category with a typical object being parametrized closed $1$-manifold, and morphisms being bordisms with complex structures (up to biholomorphisms). Hence, morphism space in $C$ is naturally identified as the moduli space of complex structures of genus-$0$ Riemann surfaces with suitable amount of punctures. The term "projective" here means that the image of each morphism is a linear operator modulo a non-zero scalar. In particular, this gives a graded vector space $V = Z(S^{1})$ and a vertex operator algebra structure on $V$ [1][2].
To remove the term "projective" in the definition of $Z$, we can redefine the domain category: Instead of having the morphism spaces identified as the moduli spaces, have each of them identified as a certain holomorphic line bundle (with the zero section removed) over each of them. Indeed, to each fixed Riemann surface, we only need to specify a non-zero scalar, the space must be a (holomorphic) crossed product of the moduli space itself and $\mathbb{C}^{\star}$.
Question Why must these holomorphic line bundles be the Quillen bundles (which are abstractly defined as the determinant line bundle of some Fredholm operators)?
Reference
- [1] L. Kong, Conformal field theory and a new geometry
- [2] Y. Huang, Two-dimensional conformal geometry and vertex operator algebras
