arXiv subjects

Clarence Kineider

Publications and source records attributed to Clarence Kineider.

3 recordsLinked to original sources

arXiv2026

Spectral Networks: Bridging higher-rank Teichmüller theory and BPS states

This book offers a comprehensive introduction to spectral networks from a unified viewpoint that bridges geometry with the physics of supersymmetric gauge theories. It provides the foundational background needed to approach the frontiers of this rapidly evolving field, treating geometric and physical aspects in parallel. After surveying fundamental topics in algebra and geometry, a detailed introduction to higher-rank Teichmüller theory is developed, including Fock-Goncharov theory for Hitchin representations, maximal representations and the more recent notion of $Θ$-positivity. Spectral networks are subsequently introduced, emphasizing their utility in the study of character varieties via the abelianization and non-abelianization maps they define. In parallel, key aspects of four-dimensional gauge dynamics with eight supercharges are explored, including electric-magnetic duality, Seiberg-Witten theory, and class $\mathcal S$ theories. The role of spectral networks as a framework for determining and analyzing BPS spectra in class $\mathcal S$ theories is then examined. The final chapter outlines recent applications of spectral networks across a range of contemporary research areas. This volume is intended for researchers and advanced students in either mathematics or physics who wish to enter the field.

math-ph↗

arXiv2025

Connected components of the space of flags of $\mathrm{SO}_0(p,q)$ transverse to a fixed pair and restrictions on Anosov subgroups

We count and give a parametrization of connected components in the space of flags transverse to a given transverse pair in every flag varieties of $\mathrm{SO}_0(p,q)$. We compute the effect the involution of the unipotent radical has on those components and, using methods of Dey--Greenberg--Riestenberg, we show that for certain parabolic subgroups $P_Θ$, any $P_Θ$-Anosov subgroup is virtually isomorphic to either a surface group of a free group. We give examples of Anosov subgroups which are neither free nor surface groups for some sets of roots which do not fall under the previous results. As a consequence of the methods developed here, we get an explicit computation of some Plücker coordinates to check if a unipotent matrix in $\mathrm{SO}_0(p,q)$ belong to the $Θ$-positive semigroup $U_Θ^{>0}$ when $p\neq q$.

math.DG↗

arXiv2022

On partial abelianization of framed local systems

D.~Gaiotto, G.~W.~Moore and A.~Neitzke introduced spectral networks to understand the framed $G$-local systems over punctured surfaces for $G$ a split Lie group via a procedure called abelianization. We generalize this construction to groups $G$ of the form $\mathrm{GL}_2(A)$, where $A$ is a unital associative ring, and to some of its subgroups. This relies on a precise analysis of the degree 2 ramified coverings associated with spectral networks and triangulations and on a matrix reinterpretation of their path lifting rules; along the way we provide another proof of the Laurent phenomenon brought to light by A.~Berenstein and V.~Retakh. The partial abelianization enables us to gives parametrizations of the moduli spaces of decorated $G$-local systems and of framed $G$-local systems over punctured surfaces. For $(A, σ)$ a Hermitian involutive $\mathbf{R}$-algebra the group $G=\mathrm{Sp}_2(A, σ)$ is a classical Hermitian Lie group of tube type, and we are able to identify and parametrize the moduli space of maximal framed $G$-local systems.

math.DG↗