Quantized K\"ahler Geometry and Quantum Gravity

TL;DR

This paper explores the deep connection between K"ahler geometry and $U(1)$ gauge theory, proposing a quantized, noncommutative framework that offers new insights into quantum gravity and emergent spacetime.

Contribution

It introduces a novel approach linking K"ahler geometry with $U(1)$ gauge theory and develops a quantized, noncommutative model leading to a background-independent formulation of quantum gravity.

Findings

Quantized K"ahler geometry can be described by a noncommutative $U(1)$ gauge theory.

The model reduces to a zero-dimensional matrix model.

This framework offers new perspectives on fundamental problems in physics.

Abstract

It has been often observed that K\"ahler geometry is essentially a $U(1)$ gauge theory whose field strength is identified with the K\"ahler form. However it has been pursued neither seriously nor deeply. We argue that this remarkable connection between the K\"ahler geometry and $U(1)$ gauge theory is a missing corner in our understanding of quantum gravity. We show that the K\"ahler geometry can be described by a $U(1)$ gauge theory on a symplectic manifold with a slight generalization. We derive a natural Poisson algebra associated with the K\"ahler geometry we have started with. The quantization of the underlying Poisson algebra leads to a noncommutative $U(1)$ gauge theory which arguably describes a quantized K\"ahler geometry. The Hilbert space representation of quantized K\"ahler geometry eventually ends in a zero-dimensional matrix model. We then play with the zero-dimensional matrix model to examine how to recover our starting point--K\"ahler geometry--from the background-independent formulation. The round-trip journey suggests many remarkable pictures for quantum gravity that will open a new perspective to resolve the notorious problems in theoretical physics such as the cosmological constant problem, hierarchy problem, dark energy, dark matter and cosmic inflation. We also discuss how time emerges to generate a Lorentzian spacetime in the context of emergent gravity.