Black holes are possibly the most intriguing objects in the Universe. They deform the space-time around them and prevent everything, including light, from escaping. Black holes are important objects for studying quantum gravity.
Hawking discovered that holes satisfy black hole laws of thermodynamics, analogous to the laws satisfied by, for example, gases. In the case of a gas, the entropy is derived by examining statistical averages of the microstates, which are associated with the molecules of the gas. From the perspective of general relativity, it is unclear what the corresponding black hole microstates are. String theory, which is the most promising approach to quantum gravity, provides us with insight into black hole microstates — they are associated with higher-dimensional branes. By superposing many branes in certain well-defined configurations, one can obtain black holes. The branes are therefore analogous to the molecules of a gas, and counting the branes provides a route to evaluate the black hole entropy.
Higher-dimensional black holes have interesting geometric properties. In four dimensions, there are particularly strong uniqueness theorems, which constrain the types of black hole solutions that can exist. For example, four dimensional black hole event horizons must have spherical topology and the geometry is typically specified uniquely by a small number of parameters, such as the mass and angular momentum. However, these uniqueness theorems are known to break down in higher dimensions, and more unusual solutions exist. For example, in five dimensions, there are black rings, and even black Saturn solutions, and more complicated geometries such as black holes with non-trivial topology outside the event horizon, which is supported by magnetic charges.
It is to be expected that more exotic black objects exist in higher dimensions, which is particularly relevant in the context of higher-dimensional supergravity theories associated with string theory. Notably, interesting geometric structures exist near to the event horizons of such black holes. The topology of these solutions plays a key role in the analysis. For event horizons which are smooth, compact (i.e. have a finite area) and have no boundary, it can be shown that such solutions have certain symmetries. Amongst other things, we analyse these types of solutions to gain new insights into the relationship between geometry and black hole physics, and there may also be interesting applications to condensed matter systems using conjectured gauge/string dualities.