Spectral Networks : Bridging Higher-rank Teichmüller Theory and BPS States (Lecture Notes in Mathematics)

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Spectral Networks : Bridging Higher-rank Teichmüller Theory and BPS States (Lecture Notes in Mathematics)

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  • 製本 Paperback:紙装版/ペーパーバック版
  • 言語 ENG
  • 商品コード 9783032092182

Full Description

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 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.

Contents

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.