Super Riemann Surface, super derivatives and connections

For a N = 1 SRS S σ, it is convenient to denote the Jacobian of the superconformal change of coordinates z→ z′ (z) by e^−w ≡ D θ′.

Superconformal field. Let n be an integer. Then, a superconformal field of weight on S σ is a superfield with the transformation property with respect to a superconformal change of coordinates [2]. The field C _n is taken to have Grassmann parity (−)ⁿ and the space of these fields is denoted by _n .

Superaffine connection. A superaffine connection on S σ is a collection of odd superfields which are locally superanalytic (i.e. ) and which transform under a superconformal change of coordinates as

Supercovariant derivative. Given a superaffine connection, one can locally define a supercovariant derivative which maps superconformal fields to superconformal fields:

Super Schwarzian derivative. The super Schwarzian derivative of the coordinate transformation z→z′(z) is defined by [2]

Under the composition of superconformal transformations, z→ z′ → z″, it transforms...