Chiral BRST Cohomology of N=2 Strings at Arbitrary Ghost and Picture Number

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We compute the BRST cohomology of the holomorphic part of the N=2 string at arbitrary ghost and picture number. We confirm the expectation that the relative cohomology at non-zero momentum consists of a single massless state in each picture. The absolute cohomology is obtained by an independent method based on homological algebra. For vanishing momentum, the relative and absolute cohomologies both display a picture dependence -- a phenomenon discovered recently also in the relative Ramond sector of N=1 strings by Berkovits and Zwiebach.