Duality and modular form in topological gauge theory

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From the viewpoint of non-perturbative dynamics of supersymmetric gauge theory and dualities of superstring theory we have investigated the geometry of instanton moduli space in higher dimensions and its relation to supersymmetric cycles, which are naturally introduced on the manifold of special holonomy.We first argued that supersyrnmetric cycles play a significant role in the problem of compactifying the instanton moduli space in higher dimensions. As an explicit example we constructed the octonionic instanton solutions on a non-compact eight-dimensional manifold with Spin (7) holonomy. However, our understanding of the geometry of its moduli space is still incomplete and it is an open problem to construct a new kind of topological invariants based on the octonionic instanton moduli space.Quite recently a substantial progress has been made in the five dimensional supersymmetric gauge theory compactified on a circle, which was the research subject in 1998. It seems possible to understand the space-time instanton in topological gauge theory from the world-sheet instanton of string theory in which the mirror symmetry gives a powerful tool.