Anti-selfdual Connections on the Quantum Projective Plane: Monopoles

TL;DR

This paper explores the geometry of the quantum projective plane CP2q, providing explicit algebraic generators, differential calculus, anti-selfdual connections, and quantum invariants, advancing the understanding of noncommutative geometric structures.

Contribution

It introduces explicit generators for K-theory and K-homology, constructs anti-selfdual connections with monopole and instanton charges, and computes quantum invariants on CP2q.

Findings

Explicit K-theory and K-homology generators

Construction of anti-selfdual connections and monopoles

Diagonalization of gauged Laplacians and quantum invariants

Abstract

We present several results on the geometry of the quantum projective plane CP2q. They include: explicit generators for the K-theory and the K-homology; a real calculus with a Hodge star operator; anti-selfdual connections on line bundles with explicit computation of the corresponding 'monopoles' and 'instanton' charges; complete diagonalization of gauged Laplacians on these line bundles; quantum invariants via equivariant K-theory and q-indices.