Kdecompositions 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 3manifolds 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 Kdecomposition 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 Kdecomposition 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 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.
 Authors:

^{[1]};
^{[2]};
^{[3]}
 Inst. for Advanced Study, Princeton, NJ (United States); Univ. of California, Davis, CA (United States). Dept. of Mathematics. Center for Quantum Mathematics and Physics
 Inst. for Advanced Study, Princeton, NJ (United States); Alternative Energies and Atomic Energy Commission (CEA), GifsurYvette (France). Inst. of Theoretical Physics
 Yale Univ., New Haven, CT (United States). Mathematics Dept.
 Publication Date:
 Grant/Contract Number:
 FG0290ER40542; DMS1059129; DMS1301776; 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 10298479
 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), GifsurYvette (France)
 Sponsoring Org:
 USDOE Office of Science (SC), High Energy Physics (HEP) (SC25); 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
Dimofte, Tudor, Gabella, Maxime, and Goncharov, Alexander B. Kdecompositions and 3d gauge theories. United States: N. p.,
Web. doi:10.1007/JHEP11(2016)151.
Dimofte, Tudor, Gabella, Maxime, & Goncharov, Alexander B. Kdecompositions and 3d gauge theories. United States. doi:10.1007/JHEP11(2016)151.
Dimofte, Tudor, Gabella, Maxime, and Goncharov, Alexander B. 2016.
"Kdecompositions and 3d gauge theories". United States.
doi:10.1007/JHEP11(2016)151. https://www.osti.gov/servlets/purl/1368083.
@article{osti_1368083,
title = {Kdecompositions and 3d gauge theories},
author = {Dimofte, Tudor and Gabella, Maxime and Goncharov, Alexander B.},
abstractNote = {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 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 XunK(∂M) of framed flat connections on the boundary — and more so, as a “K2Lagrangian,” meaning that the K2avatar of the symplectic form restricts to zero. We construct an open part of $\mathcal{L}$K(M) from elementary data associated with the hypersimplicial Kdecomposition 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 K2isotropic as long as ∂M satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that $\mathcal{L}$K(M) is K2Lagrangian. For general M, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K2Lagrangian property to a combinatorial statement. Physically, we translate the Kdecomposition of an ideal triangulation of M and its symplectic properties to produce an explicit construction of 3d N = 2 superconformal field theories TK [M] resulting (conjecturally) from the compactification of K M5branes on M. This extends known constructions for K = 2. Just as for K = 2, the theories TK [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 Nf = 1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of TK [M] grow cubically in K.},
doi = {10.1007/JHEP11(2016)151},
journal = {Journal of High Energy Physics (Online)},
number = 11,
volume = 2016,
place = {United States},
year = {2016},
month = {11}
}