$(0,4)$ Projective Superspaces I: Interacting Linear Sigma Models

We describe the projective superspace approach to supersymmetric models with off-shell $(0,4)$ supersymmetry in two dimensions. In addition to the usual superspace coordinates, projective superspace has extra bosonic variables – one doublet for each $\text{SU}(2)$ in the R-symmetry $\text{SU}(2)\times \text{SU}(2)$ which are interpreted as homogeneous coordinates on ${\mathbf{C}\mathbf{P}}^1\times {\mathbf{C}\mathbf{P}}^1$. The superfields are analytic in the ${\mathbf{C}\mathbf{P}}^1$ coordinates and this analyticity plays an important role in our description. For instance, it leads to stringent constraints on the interactions one can write down for a given superfield content of the model. As an example, we describe in projective superspace Witten’s ADHM sigma model – a linear sigma model with non-derivative interactions whose target is ${\mathbf{R}}^4$ with a Yang-Mills instanton solution. The hyperkähler nature of target space and the twistor description of instantons by Ward, and Atiyah, Hitchin, Drinfeld and Manin are natural outputs of our construction.