Two-dimensional superconformal field theories from Riemann surfaces with a boundary

We consider a 2-dimensional conformal field theory (CFT) obtained from twisted compactification of the 4-dimensional $\mathcal{N}=4$ super Yang-Mills theory on a Riemann surface with a boundary. We find the boundary conditions for preserving some of the supersymmetry. In particular, an $\mathcal{N}=(2,2)$ superconformal field theory is obtained from supersymmetry breaking due to the boundary from $\mathcal{N}=(4,4)$. In this case we calculate the central charge of the CFT and show its dependence on the topology of the Riemann surface.

Figure 1

(Doubling trick) We construct a closed surface $\mathrm{\Sigma }$ by taking ${\mathrm{\Sigma }}_o$ and its a copy with the opposite orientation ${\overline{\mathrm{\Sigma }}}_o$.