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

TL;DR

This paper constructs 3d supersymmetric gauge theories that encode 4d simplicial geometries, linking quantum gauge theory wave functions to classical 4d gravity actions, thus bridging quantum and classical geometry.

Contribution

It introduces a novel class of 3d $ ext{N}=2$ gauge theories that encode 4d simplicial geometries and connects their holomorphic blocks to quantum 4d gravity.

Findings

Massive vacua correspond to classical 4d geometries.

Holomorphic blocks act as wave functions for quantum 4d geometries.

Semiclassical limit reproduces 4d Einstein-Hilbert action.

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.