4d Quantum Geometry from 3d Supersymmetric Gauge Theory and Holomorphic Block

Abstract

A class of 3d $\mathcal{N}=2$ supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction in 3d/3d correspondence to certain graph complement 3-manifolds. Given a gauge theory in this class, the massive supersymmetric vacua of the theory contain the classical geometries on a 4d simplicial complex. The corresponding 4d simplicial geometries are locally constant curvature (either dS or AdS), in the sense that they are made by gluing geometrical 4-simplices of the same constant curvature. When the simplicial complex is sufficiently refined, the simplicial geometries can approximate all possible smooth geometries on 4-manifold. At the quantum level, we propose that a class of holomorphic blocks defined in arXiv:1211.1986 from the 3d $\mathcal{N}=2$ gauge theories are wave functions of quantum 4d simplicial geometries. In the semiclassical limit, the asymptotic behavior of holomorphic block reproduces the classical action of 4d Einstein-Hilbert gravity in the simplicial context.

35+13 pages, 9 figures, presentation improved, reference added

References in corpus (10)

• LQG vertex with finite Immirzi parameter
• A New Spin Foam Model for 4d Gravity
• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• On the characteristic and deformation varieties of a knot
• Exact Half-BPS Flux Solutions in M-theory II: Global solutions asymptotic to AdS_7 x S^4
• Four-dimensional Quantum Gravity with a Cosmological Constant from Three-dimensional Holomorphic Blocks
• Holography of 3d-3d correspondence at Large N
• Quantum spin dynamics
• Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence
• A Spectral Perspective on Neumann-Zagier

Cited by in corpus (14)

• Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy
• Aspects of Defects in 3d-3d Correspondence
• Spinfoam on Lefschetz Thimble: Markov Chain Monte-Carlo Computation of Lorentzian Spinfoam Propagator
• A slow review of the AGT correspondence
• Einstein Equation from Covariant Loop Quantum Gravity in Semiclassical Continuum Limit
• Group Field theory and Tensor Networks: towards a Ryu-Takayanagi formula in full quantum gravity
• Four-dimensional Spinfoam Quantum Gravity with Cosmological Constant: Finiteness and Semiclassical Limit
• On Spinfoams Near a Classical Curvature Singularity
• Spinfoams and high performance computing
• SU(2) Flat Connection on Riemann Surface and Twisted Geometry with Cosmological Constant
• The thermodynamics of quantum spacetime histories
• Holographic relations in loop quantum gravity
• Loop-Quantum-Gravity Simplicity Constraint as Surface Defect in Complex Chern-Simons Theory
• Spin foam propagator: A new perspective to include a cosmological constant