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Title: K-decompositions and 3d gauge theories

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K, C)-connections on a large class of 3-manifolds M with boundary. We introduce a moduli space $$\mathcal{L}$$ K(M) of framed flat connections on the boundary ∂M that extend to M. Our goal is to understand an open part of $$\mathcal{L}$$ K(M) as a Lagrangian subvariety in the symplectic moduli space X un K(∂M) of framed flat connections on the boundary — and more so, as a “K 2-Lagrangian,” meaning that the K 2-avatar of the symplectic form restricts to zero. We construct an open part of $$\mathcal{L}$$ K(M) from elementary data associated with the hypersimplicial K-decomposition of an ideal triangulation of M, in a way that generalizes (and combines) both Thurston’s gluing equations in 3d hyperbolic geometry and the cluster coordinates for framed flat PGL(K, C)-connections on surfaces. By using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of $$\mathcal{L}$$ K(M) is K 2-isotropic as long as ∂M satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that $$\mathcal{L}$$ K(M) is K 2-Lagrangian. For general M, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K 2-Lagrangian property to a combinatorial statement. Physically, we translate the K-decomposition of an ideal triangulation of M and its symplectic properties to produce an explicit construction of 3d N = 2 superconformal field theories T K [M] resulting (conjecturally) from the compactification of K M5-branes on M. This extends known constructions for K = 2. Just as for K = 2, the theories T K [M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N f = 1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T K [M] grow cubically in K.
Authors:
 [1] ;  [2] ;  [3]
  1. Inst. for Advanced Study, Princeton, NJ (United States); Univ. of California, Davis, CA (United States). Dept. of Mathematics. Center for Quantum Mathematics and Physics
  2. Inst. for Advanced Study, Princeton, NJ (United States); Alternative Energies and Atomic Energy Commission (CEA), Gif-sur-Yvette (France). Inst. of Theoretical Physics
  3. Yale Univ., New Haven, CT (United States). Mathematics Dept.
Publication Date:
Grant/Contract Number:
FG02-90ER40542; DMS-1059129; DMS-1301776; 259133
Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2016; Journal Issue: 11; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Research Org:
Inst. for Advanced Study, Princeton, NJ (United States); Yale Univ., New Haven, CT (United States); Alternative Energies and Atomic Energy Commission (CEA), Gif-sur-Yvette (France)
Sponsoring Org:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); National Science Foundation (NSF); European Research Council (ERC)
Contributing Orgs:
Univ. of California, Davis, CA (United States)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Differential and Algebraic Geometry; Supersymmetric gauge theory; Supersymmetry and Duality
OSTI Identifier:
1368083