Abstract
Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg–Witten moduli space and the moduli space of pseudoholomorphic curves.
Highlights
Moduli spaces parametrize solutions of partial differential equations up to some natural notion of equivalence
The topological and inherently non-perturbative aspects of the physical system are often manifested in the geometry of the moduli space
We investigate the local structure of the moduli space obtained by taking the quotient of a level set of the equivariant map by the group action
Summary
Moduli spaces parametrize solutions of partial differential equations up to some natural notion of equivalence. The normal form theorems are phrased and proved in a modularized way, leading to a flexible general framework in which other analytical setups can be included in a “plug and play” fashion based on other inverse function theorems As another approach to the issues that composition is not smooth relative to a fixed Sobolev regularity, Hofer, Wysocki and Zehnder have introduced the scale calculus, see, e.g., [40, 41] and references therein. Under the assumption that the map can be brought into an equivariant normal form, the corresponding moduli space has the structure of a Kuranishi space, which roughly speaking means that it can be locally identified with the quotient of the zero set of a smooth map with respect to the linear action of a compact group. Most of the material first appeared in the first author’s thesis [13]
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