4 dimensional Hyperk\"{a}hler manifolds with $L^2$ Riemann curvature and geodesic volume growth like $r^{4/3}$ are called ALH* Tesserons. The recent work (2021) of Sun, Zhang and Collins, Jacob, Lin and Hein, Sun, VIaclovsky, Zhang has classified ALH* Tesserons into 10 types, which we will denote $E_0, \tilde{E}_1, E_1, E_2, E_3, E_4, E_5, E_6, E_7, E_8$.

Through the work of Cherkis, Cross and Ward, it was shown that $E_0, \tidle{E}_1, \ldots, E_6$ ALH* Tesserons arise as Doubly Periodic Monopole Moduli Spaces. 

 

Part of my PhD thesis is showing that $E_7, E_8$ ALH* Tesserons also arise from Doubly Periodic Monopole Moduli Spaces.

 

In this talk, I will explain the following: 

1. How to understand the coordinates of the moduli space as a combination of holonomies and monopole positions.

2. Why deforming the holonomies and monopoles positions leads to ALH* style geometry.

3. Why each of the ALH* tesserons arise from a certain doubly periodic monopole moduli space.