Complex Numbers and Functions

Doubly Periodic Monopole Dynamics and Crystal Volumes

Unitary anti-self-dual connections on Asymptotically Locally Flat (ALF) hyperkahler spaces are constructed in terms of data organized in a bow. Bows generalize quivers, and the relevant bow gives rise to the underlying ALF space as the moduli space of its particular representation -- the small representation. Any other representation of that bow gives rise to anti-self-dual connections on that ALF space. 
We prove that each resulting connection has finite action, i.e. it is an instanton. Moreover, we derive the asymptotic form of such a connection and compute its topological class.

Instantons on multi-Taub-NUT Spaces II: Bow Construction

The low velocity dynamic of a doubly periodic monopole, also called a monopole wall or monowall for short, is described by geodesic motion on its moduli space. This moduli space is hyperkaehler and non-compact. We establish a relation between the Kaehler potential of this moduli space and the volume of a region in Euclidean three-space cut out by a plane arrangement associated with each monowall.

Crystal Volumes and Monopole Dynamics

The low velocity dynamic of a doubly periodic monopole, also called a monopole wall or monowall for short, is described by geodesic motion on its moduli space. This moduli space is hyperkahler and non-compact. We establish a relation between the Kahler potential of this moduli space and the volume of a region in Euclidean three-space cut out by a plane arrangement associated with each monowall.

We determine the asymptotic dynamics of the U(N) doubly periodic BPS monopole in Yang-Mills-Higgs theory, called a monopole wall, by exploring its Higgs curve using the Newton polytope and amoeba. In particular, we show that the monopole wall splits into subwalls when any of its moduli become large. The long-distance gauge and Higgs field interactions of these subwalls are abelian, allowing us to derive an asymptotic metric for the monopole wall moduli space.

Asymptotic dynamics of monopole walls

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Asymptotic dynamics of monopole walls