Tudor Dimofte, Maxime Gabella, Alexander B. Goncharov
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 of framed flat connections on the boundary of M that extend to M. Our goal is to understand it as a Lagrangian subvariety in the symplectic moduli space of framed flat connections on the boundary, and to provide the data needed to quantize it. We call this the Lagrangian pair associated to M. The basic example, an elementary Lagrangian pair, is for M a tetrahedron and K=2. Our essential geometric tool is the hypersimplicial K-decomposition of M related to an ideal triangulation. By restricting framed flat connections to tetrahedra, and localizing further to octahedra of the K-decomposition, we associate an elementary Lagrangian pair to every octahedron. Going back, we reconstruct the Lagrangian pair associated to M by "symplectic gluing" of elementary pairs. We study the 2-3 Pachner moves by decomposing them into elementary cobordisms. We prove that our Lagrangian subvarieties are K_2-Lagrangians: the K_2-avatar of the symplectic form restricts on them to zero. Physically, we translate the data of Lagrangian pairs into an explicit construction of 3d N=2 superconformal field theories T_K[M] resulting from the compactification of K M5 branes on M (generalizing known results in the case 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 strong hints that the degrees of freedom of T_K[M] grow like K^3.
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http://arxiv.org/abs/1301.0192
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