String theory does not just include strings but higher-dimensional objects known as membranes. Physically, they provide a useful framework for particle physics model building and applications of the AdS/CFT duality.
A fundamental realisation of the mid 1990s was that the fundamental degrees of freedom of string theory include not just strings, but objects with extra spatial dimensions known as membranes. These extended objects can fluctuate, have tension and can move around in space-time.
As membranes have mass, they interact with gravity acting as gravitational sources. When one considers a theory containing multiple membranes intersecting in a non-trivial manner, the gravitational interactions rapidly become complicated, which makes solving Einstein's equations hard.
However, finding solutions to Einstein's equations are incredibly useful. Physically, they provide a useful framework for particle physics model building and applications of the AdS/CFT duality. Mathematically, these are examples of solutions to a non-linear partial different equation of Monge—Ampère type, of which there few, if any, known solutions.
One feature of the membranes known as D-branes is that open strings can have their endpoints attached to these hyperplanes. The dynamics of the open-string endpoints informs us of the degrees of freedom living on the world volume of the D-brane. Systems of intersecting D-branes give rise to particular low-energies effective theories which can be realised in terms of gravitational duals. The symmetry groups governing such brane configurations, which have a direct correspondence with the gauge groups of the low-energy theories, are determined by the number of coincident branes, likewise the number of supersymmetries, which are preserved by each particular brane setup. Amongst other things, we develop techniques to construct these intersecting membrane solutions and study their physical and mathematical properties.