This open access 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 FockGoncharov theory for Hitchin representations, maximal representations and the more recent notion of -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 electricmagnetic duality, SeibergWitten theory, and class S theories. The role of spectral networks as a framework for determining and analyzing BPS spectra in class 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.
Preface.-Motivation and Historical Perspective.-I. Mathematical
Foundation.- 1.The Fundamental Group.-2.Covering Spaces.-3.Bundles and
Connections.-4.Hyperbolic Geometry.- 5.Cluster Varieties.- II Higher-rank
Teichmüller Theory.-6. The Teichmüller Space.-7.Coordinates on Teichmüller
Space.-8.differentials on a Riemann Surface.-9.Higher-rank Teichmüller
Spaces.- 10.Hitchin Representations.-11.Maximal
Representations.-12.Positivity.- III Spectral Networks in
Geometry.-13.Non-degenerate Spectral Networks.-14.Combinatorial and Analytic
Construction.-15.Non-abelianization and Abelianization.-16.WKB Method and
Stokes Graphs.- IV Counting BPS States.-17.Four-dimensional Supersymmetric
Quantum Field Theories.-18.Four-dimensional N=2 Gauge
Theories.-19.Electric-magnetic Duality.-20.Seiberg-Witten Theory.-21.Theories
of Class S.-22.BPS States in Class S Theories.- V
Generalizations.-23.Exponential Networks.-24.Three-dimensional
Networks.-25.WKB Cameral Networks.-26.Nonabelianization for Conformal
Virasoro Blocks.
Clarence Kineider studied mathematics at the École Normale Supérieure de Rennes and the University of Strasbourg between 2016 and 2019. He completed his Ph.D. at the University of Strasbourg, where his research focused on higher Teichmüller spaces, FockGoncharov coordinates, and spectral networks. Since 2024, he has been a postdoctoral fellow at the Max Planck Institute for Mathematics in the Sciences in Leipzig.
Georgios Kydonakis studied Mathematics (B.Sc.) at the University of Athens in Greece and obtained his Ph.D. degree from the University of Illinois at Urbana-Champaign in the USA. He has held postdoctoral positions in Strasbourg, Bonn, and Heidelberg as an Alexander von Humboldt fellow. Since 2023, he has been an Assistant Professor at the University of Patras in Greece. His research interests lie on the crossroad between complex geometry and mathematical physics.
Eugen Rogozinnikov studied mathematics at the University of Heidelberg, where he completed his master's degree in 2016 and received his Ph.D. in mathematics in 2020. After that, he held a postdoctoral position at the Université de Strasbourg. In 20232024, he was a DAAD fellow at the University of Notre Dame (USA). He later worked at the Max Planck Institute for Mathematics in the Sciences in Leipzig and is currently a research fellow at the Korea Institute for Advanced Study (KIAS) in Seoul. His research interests include differential geometry, geometric topology, and higher Teichmüller theory.
Valdo Tatitscheff studied mathematics and physics at the École normale supérieure de la rue d'Ulm in Paris from 2014 to 2018, and earned a Ph.D. in mathematics from the University of Strasbourg and the Institut de Recherche Mathématique Avancée (IRMA) in 2022. Currently a Postdoctoral Fellow at the Institute for Mathematics at Heidelberg University, Tatitscheff's main research interests lie at the interface of geometry and gauge theories.
Alexander Thomas studied mathematics at the École Normale Supérieure de Lyon from 2013 to 2017 and completed his Ph.D. at the University of Strasbourg in 2020. He held postdoctoral positions at the Max Planck Institute for Mathematics in Bonn (20202022) and at the University of Heidelberg (20222024). Since 2024, he has been a Maître de Conférences (Associate Professor) at Université Claude Bernard Lyon 1. His research focuses primarily on gauge theory and quantum algebra.
