Exact results for boundaries and domain walls in 2d supersymmetric theories

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We apply supersymmetric localization to N=(2,2) gauged linear sigma
models on a hemisphere, with boundary conditions, i.e., D-branes,
preserving B-type supersymmetries. We explain how to compute the
hemisphere partition function for each object in the derived category of
equivariant coherent sheaves, and argue that it depends only on its K
theory class. The hemisphere partition function computes exactly the
central charge of the D-brane, completing the well-known formula
obtained by an anomaly inflow argument. We also formulate supersymmetric
domain walls as D-branes in the product of two theories.  We exhibit
domain walls that realize the sl(2) affine Hecke algebra.  Based on