Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

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We compute the
partition function on the hemisphere of a class of two-dimensional (2,2)
supersymmetric field theories including gauged linear sigma models. The
result provides a general exact formula for the central charge of the
D-brane placed at the boundary. It takes the form of Mellin-Barnes
integral and the question of its convergence leads to the grade
restriction rule concerning branes near the phase boundaries. We find
expressions in various phases including the large volume formula in
which a characteristic class called the Gamma class shows up. The two
sphere partition function factorizes into two hemispheres glued by
inverse to the annulus. The result can also be written in a form
familiar in mirror symmetry, and suggests a way to find explicit mirror
correspondence between branes.