G/G TOPOLOGICAL FIELD-THEORIES BY COSETTING G(K)

G/G topological field theories based on G(k) WZW models are constructed and studied. These coset models are formulated as complex BRST cohomology in G(k)c, the complexified level k current algebra. The finite physical spectrum corresponds to the conformal blocks of G(k). The amplitudes for G/G theories are argued to be given in terms of the G(k) fusion rules. The G(k)/G(k) character is the Kac-Weyl numerator of G(k) and is interpreted as an index. The complex BRST cohomology is found to contain states of arbitrary ghost number. Intriguing similarities of G/G to c less-than-or-equal-to 1 matter systems coupled to two dimensional gravity are pointed out.