K-decompositions and 3d gauge theories

Journal Article · Thu Nov 24 00:00:00 EST 2016 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/JHEP11(2016)151· OSTI ID:1368083

Dimofte, Tudor ^[1]; Gabella, Maxime ^[2]; Goncharov, Alexander B. ^[3]

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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.

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 “K₂-Lagrangian,” meaning that the K₂-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 K₂-isotropic as long as ∂M satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that $$\mathcal{L}$$_K(M) is K₂-Lagrangian. For general M, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K₂-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.

• Research Organization: Institute 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 Organization: USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF); European Research Council (ERC)
• Contributing Organization: Univ. of California, Davis, CA (United States)
• Grant/Contract Number: FG02-90ER40542; DMS-1059129; DMS-1301776; 259133
• OSTI ID: 1368083
• Journal Information: Journal of High Energy Physics (Online), Vol. 2016, Issue 11; ISSN 1029-8479
• Publisher: Springer Berlin
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
Differential and Algebraic Geometry
Supersymmetric gauge theory
Supersymmetry and Duality