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We calculate the trace and axial anomalies of $\mathcal{N}=(2,2)$superconformal theories with exactly marginal deformations, on a surface withboundary. Extending recent work by Gomis et al, we derive the boundarycontribution that captures the anomalous scale dependence of the one-pointfunctions of exactly marginal operators. Integration of the bulk super-Weylanomaly shows that the sphere partition function computes the K\"ahlerpotential $K(\lambda, \bar\lambda)$ on the superconformal manifold. Likewise,our results confirm the conjecture that the partition function on thesupersymmetric hemisphere computes the holomorphic central charge,$c^\Omega(\lambda)$, associated with the boundary condition $\Omega$. Theboundary entropy, given by a ratio of hemispheres and sphere, is thereforefully determined by anomalies.

Boundary Weyl anomaly of $\mathcal{N}=(2,2)$ superconformal models