13 - D-branes and geometry I

Summary

In previous chapters we became increasingly aware of the intimate relation of D-branes to both spacetime geometry and to gauge theory, via the collective description of their low energy dynamics. In fact, we have already seen that we can reinterpret many aspects of the spacetime geometry in which the brane moves by reference to the vevs of scalars in the world-volume gauge theory. In this chapter we explore this in much more detail, by using D-branes to probe a number of string theory backgrounds, and find that they allow us to get a new handle on quite detailed properties of the geometry. In addition, we will find that D-branes can take on the properties of a variety of familiar objects, such as monopoles and instantons, depending upon the situation.

D-branes as probes of ALE spaces

One of the beautiful results which we uncovered soon after constructing the type II strings was that we can ‘blow-up’ the 16 fixed points of the T⁴/ℤ₂ ‘orbifold compactification’ to recover string propagation on the smooth hyper-Kähler manifold K3. (We had a lot of fun with this in section 7.6.) Strictly speaking, we only recovered the algebraic data of the K3 manifold this way, and it seemed plausible that the full metric geometry of the space is recovered, but how can we see this directly?

We can recover the metric data by using a brane as a short distance ‘probe’ of the geometry.