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String duality and D-branes

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 tex2html_wrap_inline3946 . The theta-function


satisfies by Poisson's formula the relation tex2html_wrap_inline3950 with tex2html_wrap_inline3952 the dual lattice. So the expansion of tex2html_wrap_inline3954 for small values of tex2html_wrap_inline3938 , can be directly translated into an expansion of the dual function tex2html_wrap_inline3958 for large values of tex2html_wrap_inline3938 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 tex2html_wrap_inline3938 ) into a much more accessible one (weak-coupling behaviour where the semi-classical approximation for small tex2html_wrap_inline3938 makes sense). String duality is the statement that the above string theory functors tex2html_wrap_inline3942 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 tex2html_wrap_inline3914 into a space-time manifold M, we consider a submanifold tex2html_wrap_inline3976 (the D-brane) and maps tex2html_wrap_inline3978 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 tex2html_wrap_inline3922 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.

next up previous contents
Next: String theory in the Up: String Theory: A Happy Previous: Perturbative strings and Riemann

Fri Mar 20 16:01:06 MET 1998