However, in the last two years we have witnessed dramatic developments bringing for the first time nonperturbative questions into reach. In a confluence of a wide variety of ideas, many of them dating back to the 70s and 80s, the structure, internal consistency and beauty of string theory has greatly improved. We have now a much better and clearer picture what the theory is about: it is well-defined and unique!

Crucial in all this has been the concept of *string duality.* In fact, duality has been a powerful idea in physics and mathematics for a long time. A very elementary illustration can be given by considering simple linear duality of a vector space, or even better, of a lattice . The theta-function

satisfies by Poisson's formula the relation with the dual lattice. So the expansion of for small values of , can be directly translated into an expansion of the dual function for large values of and *vice versa.*

Physically, the transformation to a dual set of variables can translate a difficult question (such as strong-coupling behaviour with strong quantum fluctuations for large ) into a much more accessible one (weak-coupling behaviour where the semi-classical approximation for small makes sense). String duality is the statement that the above string theory functors have precisely the same properties as the simple theta-function we discussed above! Furthermore, all five diffferent perturbative strings are related in such a fashion and are just expansions of one single unified theory around different backgrounds.

The mathematical applications of string duality are just in their infancy. For example, one of the consequences is the realisation that string theory does not only include strings but also various higher dimensional objects, known as D-branes. A D-brane is simply a place in space-time where the string can begin or end. So, instead of considering maps of a Riemann surface into a space-time manifold *M*, we consider a submanifold (the D-brane) and maps with the property that the boundary of the surface lies on the brane, as seen in figure 2. These cycles satisfy special conditions on the embedding and define a moduli space (typically a moduli space of sheaves). The D-brane quantum states should then roughly correspond to the homology of this moduli space. String duality can now be used to argue that in certain cases the Hilbert spaces of D-branes are isomorphic to the Hilbert spaces of strings that we introduced earlier. Thus two very different subjects are related: moduli spaces of vector bundles or sheaves and infinite-dimensional algebras! These connections indicate beautiful new mathematical structures and are now actively investigated.

Fri Mar 20 16:01:06 MET 1998