Spectral networks (1204.4824v2)

Abstract: We introduce new geometric objects called spectral networks. Spectral networks are networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional N=2 theories coupled to surface defects, particularly the theories of class S. In these theories spectral networks provide a useful tool for the computation of BPS degeneracies: the network directly determines the degeneracies of solitons living on the surface defect, which in turn determine the degeneracies for particles living in the 4d bulk. Spectral networks also lead to a new map between flat GL(K,C) connections on a two-dimensional surface C and flat abelian connections on an appropriate branched cover Sigma of C. This construction produces natural coordinate systems on moduli spaces of flat GL(K,C) connections on C, which we conjecture are cluster coordinate systems.

• The paper introduces spectral networks as a novel method to compute BPS degeneracies in 4D N=2 theories with surface defects.
• It develops a mapping between flat GL(K, C) connections and flat abelian connections on branched covers, yielding new coordinate systems on moduli spaces.
• The study details the evolution of networks via critical phase shifts (K-walls), providing an algorithm to track 2D-4D BPS state transformations.

Spectral Networks: A Synopsis

The paper by Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke introduces a mathematical framework for what they term "spectral networks." These networks arise naturally in the context of four-dimensional $\mathcal{N}=2$ supersymmetric theories coupled to surface defects, particularly in theories of class $\mathcal{S}$. Their work provides a new approach to computing BPS (Bogomol'nyi–Prasad–Sommerfield) degeneracies, offering a significant toolset for both mathematical and physical explorations within this domain.

Core Concepts and Definitions

Spectral networks are defined as collections of trajectories (or "walls") on Riemann surfaces, which obey specific local interaction rules and are associated with the branched covers of these surfaces. Notably, the walls carry additional discrete data, namely pairs of sheets from the covering and integers that involve BPS soliton degeneracies. These constructs lead to the emergence of a rich tapestry of interconnections between geometry, topology, and quantum field theory.

Spectral Networks and BPS Degeneracies

For $\mathcal{N}=2$ supersymmetric theories, the spectral networks directly determine the degeneracies of solitons living on the surface defect. These solitons, in turn, inform us about the degeneracies of particles living in the four-dimensional bulk. The paper elucidates this relationship by developing a precise machinery for reading off these degeneracies from the geometrical characteristics of the spectral networks.

Mathematical and Physical Implications

The work extends to a mathematical correspondence between flat $GL(K,\mathbb{C})$ connections on the base Riemann surface and flat abelian connections on branched covers. Specifically, the authors construct a novel map that provides natural coordinate systems on moduli spaces of such connections, conjecturing that these coordinate systems form cluster varieties. This has deep implications for understanding the structures of moduli spaces in relation to quantum field theories and their BPS spectra.

Computational Framework

Through the lens of spectral networks, Gaiotto, Moore, and Neitzke provide a comprehensive methodology to compute framed 2D-4D BPS indices. These indices are arranged into generating functions, which are defined by sum expansions based on allowed soliton paths—offering a computational recipe that remains invariant under homotopy.

Variational Analysis and Wall Crossings

The authors explore how these networks evolve when the phase parameter $\vartheta$ is varied, highlighting "critical phases" where topology changes occur, known as “$\mathcal{K}$-walls.” At these intersections, they observe changes in the framed BPS state spectra, providing an algorithm to derive universal transformations that describe these spectral jumps and thereby infer the 4D BPS particle content of the theory.

Open Problems and Future Directions

The paper opens several directions for future research, including generalizations to other Lie algebras beyond type A, incorporating more intricate variants of defects, investigating the implications for cluster coordinates, and potential categorifications of BPS degeneracies. Additionally, they suggest further exploration into the connections between their spectral networks and the broader mathematical context of cluster varieties and higher Teichmüller theory.

Conclusion

This paper establishes spectral networks as a powerful conceptual and computational tool in the paper of both mathematical structures and physical theories. The synthesis of geometric, topological, and quantum mechanical ideas encapsulated in these networks offers new pathways for understanding the dynamics and spectrum of $\mathcal{N}=2$ theories, with broader implications for both pure mathematics and theoretical physics.