Spectral Networks: Bridging higher-rank Teichm\"uller theory and BPS states

TL;DR

This book introduces spectral networks, connecting higher-rank Teichmüller theory with supersymmetric gauge theories, and explores their applications in character varieties and BPS spectra.

Contribution

It provides a unified, comprehensive framework linking geometric, algebraic, and physical aspects of spectral networks and their role in modern mathematical physics.

Findings

Develops a detailed introduction to higher-rank Teichmüller theory.

Explains the utility of spectral networks in character varieties.

Examines spectral networks' role in analyzing BPS spectra in class S theories.

Abstract

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 Teichm\"uller theory is developed, including Fock-Goncharov theory for Hitchin representations, maximal representations and the more recent notion of $\Theta$-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.