Abstract
This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)connections on a large class of 3manifolds M with boundary. We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2Lagrangian, meaning that the K_2avatar of the symplectic form restricts to zero. We construct an open part of L_K(M) from data assigned to a hypersimplicial Kdecomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)connections on surfaces. Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2isotropic if the boundary satisfies some topological constraints (Theorem 4.2). In some cases this implies that L_K(M) is K_2Lagrangian. For general M, we extend a classic result of NeumannZagier on symplectic properties of PGL(2) gluing equations to reduce the K_2Lagrangian property to a combinatorial claim.
Physically, we use the symplectic properties of Kdecompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5branes 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 ChernSimonsmatter theories. Changes of triangulation (23 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.
Physically, we use the symplectic properties of Kdecompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5branes 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 ChernSimonsmatter theories. Changes of triangulation (23 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.
Original language  English 

Article number  151 
Number of pages  148 
Journal  Journal of High Energy Physics 
Volume  2016 
Issue number  11 
DOIs  
Publication status  Published  24 Nov 2016 
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Tudor Dimofte
 School of Mathematics  Reader in Algebra, Geometry & Topology and related fields
Person: Academic: Research Active (Teaching)