Many reasons for doing string theory involve physics. However strings have also had an important impact on mathematics. Ideas from string theory have an independent existence in the mathematical world, separate from any question about their relationship to reality.
This is particularly true for a duality called mirror symmetry. Remember that string theory requires six extra dimensions, which must be compactified (curled-up) very small. The extra dimensions can form many different types of shape. The most important type of compactification is called a Calabi-Yau manifold (pictured right).
The esoteric Calabi-Yau manifolds are of independent interest in mathematics: both Calabi and Yau are mathematicians. They are studied in a branch of maths called algebraic geometry.
Mirror symmetry provides unprecedented relationships between pairs of different Calabi-Yau manifolds. It allows enigmatic properties of manifold A to be determined by obvious properties of manifold B and vice-versa. This suggests the existence of a deep mathematical structure giving rise to mirror symmetry.
Incredibly it was string theorists who first came across mirror symmetry, not mathematicians. Their motivating arguments were based on physical intuition, but gave little insight into the mathematical foundations. Currently many mathematicians are working to understand mirror symmetry more fully.
A second example comes from the concept of vertex operators. These technical constructs are essential in string theory for computing how particles interact: one vertex operator is needed for each particle.
Vertex operators have an standalone significance in mathematics. They played a crucial role in Richard Borcherds’ proof of the wonderfully named Monstrous Moonshine conjecture, for which he won the Fields Medal in 1998.
Why has string theory been useful in mathematics? You could look at it like this. Much of mathematics is the study of structure. Whatever else it may provide, studying strings equips us with a large conceptual framework. So perhaps it’s not surprising that string theory contains deep mathematics.