# On the Octonionic Self Duality equations of 3-brane Instantons

## TL;DR

This paper investigates octonionic self-duality equations for 3-branes, constructs explicit instanton solutions in various dimensions, and explores their implications for brane topology change and non-perturbative interactions.

## Contribution

It extends previous work on membrane self-duality to 3-branes, providing explicit solutions and analyzing their topological and calibration properties in higher dimensions.

## Key findings

- Constructed explicit instanton solutions for 3-branes in multiple dimensions.
- Reduced self-duality equations to integrable systems with elliptic function solutions.
- Identified differences in calibration topology between spherical and toroidal cases.

## Abstract

We study the octonionic selfduality equations for $p=3$-branes in the light cone gauge and we construct explicitly, instanton solutions for spherical and toroidal topologies in various flat spacetime dimensions $(D=5+1,7+1,8+1,9+1)$, extending previous results for $p=2$ membranes. Assuming factorization of time we reduce the self-duality equations to integrable systems and we determine explicitly periodic, in Euclidean time, solutions in terms of the elliptic functions. These solutions describe 4d associative and non-associative calibrations in $D=7,8$ dimensions. It turns out that for spherical topology the calibration is non compact while for the toroidal topology is compact. We discuss possible applications of our results to the problem of 3-brane topology change and its implications for a non-perturbative definition of the 3-brane interactions.