Each $ALH^*$ Tesseron from Doubly Periodic Monopoles

Abstract

We complete a program in the study of $ALH^*$ tesserons by constructing the previously missing $E_7$ and $E_8$ tesserons as $L^2$ moduli spaces of monowalls - singular monopoles on $\R \times S^1 \times S^1$ with prescribed boundary and singularity conditions. First, we establish a Kobayashi–Hitchin type correspondence between monowalls and para-spectral data: each monowall is uniquely determined by para-spectral data comprising a complex algebraic curve, a degree-zero holomorphic line bundle on its toric compactification, and parabolic weights at punctures summing to zero. Next, using the Kobayashi-Hitchin correspondence, we identify and construct candidate monowall moduli spaces for the $E_7$ and $E_8$ tesserons. We determine that these candidate monowall moduli spaces are indeed the missing $E_7$ and $E_8$ tesserons by analyzing their ends using a gluing construction and then matching them to the model $E_7$ and $E_8$ tesseron ends. Since all other $ALH^*$ tesserons, $E_0 \ldots E_6$, were previously known to arise as monowall moduli spaces, we thus complete the program of realizing all $ALH^*$ tesserons as monowall moduli spaces. In addition to the above two main results, we establish a few auxiliary results. First is that our definition of para-spectral data is invariant with respect to the Nahm transform on monowalls. Next, we establish an index theorem for a Dirac operator coupled to a monowall and equate its index to either the width (for the Dirac operator in the fundamental representation) or number of interior coefficients (for the Dirac operator in the adjoint representatio) of a Newton polygon associated to this monowall.