2013/04/02 by David Li-Bland, Li-Bland, David, Pavol Ševera +1 · 1 citation
Mathematics · #53D17 #53D20 #53D30 #81T45 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1304.0737
openalex publication_date 2013/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the symplectic and Poisson geometry of moduli spaces\nof flat connections over quilted surfaces. These are surfaces where the\nstructure group varies from region to region in the surface, and where a\nreduction (or relation) of structure occurs along the boundaries of the\nregions. Our main theoretical tool is a new form moment-map reduction in the\ncontext of Dirac geometry. This reduction framework allows us to use very\ngeneral relations of structure groups, and to investigate both the symplectic\nand Poisson geometry of the resulting moduli spaces from a unified perspective.\n The moduli spaces we construct in this way include a number of important\nexamples, including Poisson Lie groups and their Homogeneous spaces, moduli\nspaces for meromorphic connections over Riemann surfaces (following the work of\nPhilip Boalch), and various symplectic groupoids. Realizing these examples as\nmoduli spaces for quilted surfaces provides new insights into their geometry.\n