# Noncommutative Geometry, Hodge Theorem and Holography

## TL;DR

This paper proposes a noncommutative version of the Hodge Theorem and defines a noncommutative free bosonic propagator, aiming to support the development of noncommutative topological quantum field theories relevant to quantum gravity and holography.

## Contribution

It introduces a noncommutative Hodge Theorem and a noncommutative free bosonic propagator, advancing the mathematical framework for noncommutative topological quantum field theories.

## Key findings

- Established a noncommutative Hodge Theorem
- Defined a noncommutative free bosonic propagator
- Provided a mathematical foundation for noncommutative TQFTs

## Abstract

Some time ago we presented an article (which was in fact the outline of a research programme) in which we argued for the need to develop a nonommutative version of topological quantum field theories (NCTQFT for short). Recent work by C.J. Hogan et all, has demonstrated the possibility to get experimental verification of holography; if this comes true, then that would indicate that quantum gravity is indeed a TQFT. On the other hand there is accumulating evidence that the underlying geometry of spacetime is a noncommutative (abreviated to nc in the sequel) space, hence if one wants a unified theory of all physical interactions including gravity that would mean that the right framework would be NCTQFT. Towards this goal we present a modest achievement which is a nc version of Hodge Theorem and the definition of the nc free bosonic propagator.