Cluster-like coordinates in supersymmetric quantum field theory

Recently it has become apparent that [Formula] supersymmetric quantum field theory has something to do with cluster algebras. I review one aspect of the connection: supersymmetric quantum field theories have associated hyperkähler moduli spaces, and these moduli spaces carry a structure that looks l...

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Published inProceedings of the National Academy of Sciences - PNAS Vol. 111; no. 27; pp. 9717 - 9724
Main Author Neitzke, Andrew
Format Journal Article
LanguageEnglish
Published United States National Academy of Sciences 08.07.2014
National Acad Sciences
SeriesFrom the Cover
Subjects
Algebra
Class theory
cluster analysis
Coordinate systems
equations
Gauge theory
MANY FACETS OF CLUSTER ALGEBRAS SPECIAL FEATURE
mathematical models
Mathematics
Mutation
Physical Sciences
Physics
Quantum field theory
Topology
Trajectories
Triangulation
variance
supersymmetry
character varieties
Hitchin systems
Online AccessGet full text
ISSN0027-8424
1091-6490
1091-6490
DOI10.1073/pnas.1313073111

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Summary:Recently it has become apparent that [Formula] supersymmetric quantum field theory has something to do with cluster algebras. I review one aspect of the connection: supersymmetric quantum field theories have associated hyperkähler moduli spaces, and these moduli spaces carry a structure that looks like an extension of the notion of cluster variety. In particular, one encounters the usual variables and mutations of the cluster story, along with more exotic extra variables and generalized mutations. I focus on a class of examples where the underlying cluster varieties are moduli spaces of flat connections on surfaces, as considered by Fock and Goncharov [Fock V, Goncharov A (2006) Publ Math Inst Hautes Études Sci 103:1–211]. The work reviewed here is largely joint with Davide Gaiotto and Greg Moore.
Bibliography:http://dx.doi.org/10.1073/pnas.1313073111
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Author contributions: A.N. wrote the paper.
Edited by Lauren K. Williams, University of California, Berkeley, CA, and accepted by the Editorial Board April 30, 2014 (received for review November 20, 2013)
ISSN:0027-8424
1091-6490
1091-6490
DOI:10.1073/pnas.1313073111