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A space M is a fine moduli space for the functor F if M represents F, i.e., ... Moduli spaces also appear in physics in topological field theory, ...
The total moduli space is locally a product of these two branches, as nonrenormalization theorems imply that the metric of each is independent of the fields of the other multiplet.(See for example Argyres, Non-Perturbative Dynamics Of Four-Dimensional Supersymmetric Field Theories, pp. 6–7, for further discussion of the local product structure.)
There the moduli space obtains an alternative description as a moduli space of holomorphic vector bundles. This is the Narasimhan–Seshadri theorem, which was proved in this form relating Yang–Mills connections to holomorphic vector bundles by Donaldson. [5] In this setting the moduli space has the structure of a compact Kähler manifold.
The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.
Moduli theory is a branch of the fields of algebraic geometry, complex manifolds and singularity theory.It aims to construct and study moduli spaces, which are the parameter spaces encoding the continuous variation of geometric structures (for example Riemann surfaces of fixed genus, vector bundles with holomorphic structure, singularities in families).
The moduli space is usually a manifold. For generic metrics, after gauge fixing, the equations cut out the solution space transversely and so define a smooth manifold. The residual U(1) "gauge fixed" gauge group U(1) acts freely except at reducible monopoles i.e. solutions with =. By the Atiyah–Singer index theorem the moduli space is finite ...
Many examples of noncompact hyperkähler manifolds arise as moduli spaces of solutions to certain gauge theory equations which arise from the dimensional reduction of the anti-self dual Yang–Mills equations: instanton moduli spaces, [9] monopole moduli spaces, [10] spaces of solutions to Nigel Hitchin's self-duality equations on Riemann ...
In theoretical physics, Seiberg–Witten theory is an = supersymmetric gauge theory with an exact low-energy effective action (for massless degrees of freedom), of which the kinetic part coincides with the Kähler potential of the moduli space of vacua.