Tangles, generalized Reidemeister moves, and three-dimensional mirror symmetry

• Published in: The journal of high energy physics Vol. 2014; no. 5; pp. 1 - 70
• Main Authors: Córdova, Clay, Espahbodi, Sam, Haghighat, Babak, Rastogi, Ashwin, Vafa, Cumrun
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2014; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Descriptions; Elementary Particles; Field theory; High energy physics; Infrared; Mathematics; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Relativity Theory; Singularities; String Theory; Symmetry; Ultraviolet; Field Theories in Lower Dimensions; Supersymmetry and Duality; M-Theory; Extended Supersymmetry
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP05(2014)014

A bstract Three-dimensional = 2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle , a one-dimensional knotted submanifold of the ultraviolet geometry. A choice of branch sheet for this cover yields a Lagrangian for the theory, and varying the branch sheet provides dual descriptions. Massless matter arises from vanishing size M2-branes and appears as singularities of the tangle where branch lines collide. Massive deformations of the field theory correspond to resolutions of singularities resulting in distinct smooth manifolds connected by geometric transitions. A generalization of Reidemeister moves for singular tangles captures mirror symmetries of the underlying theory yielding a geometric framework where dualities are manifest.