Integrability, Duality and Strings †

After the two years ago work of Seiberg and Witten [1, 2], the Pandora box of string theory has been opened once more. The magic opening word has been duality. A web of exciting interrelated results it contains has appeared during the last months: string-string duality (U duality) [3]; the physical interpretation of the conifold singularity [4]; heterotic-type II dual pairs [3, 5, 6, 7, 8]; R-R states and Dirichlet-branes [9, 10]; derivation from M-theory of dualities in string theory; and non-perturbative enhancement of symmetries [11, 12, 13]. String-string duality was first pointed out between heterotic strings compactified on T 4 , and type II A strings on K3 [5]. The first check of this duality relation between these two different approaches to string theory is of course trying to understand the equivalence, for the type II A string on K3, of the well known enhancement of symmetry for the heterotic string on T 4 at certain points in the moduli space. This problem was beatifully solved in [5], where the enhanced non abelian gauge symmetries appear associated to the orbifold singularities of K3. These singularities are of AD -E type, and its combinaton corresponds to a group of total rank ≤ 20. Geometrically, these singularities arise from the collapse of a set of 2-cycles; the number of collapsing cycles equals the rank of the gauge group, and the intersection matrix is given by the Dynkin diagram of the singularity. This geometry can be directly connected with Strominger's suggestion for the interpretation, in Calabi-Yau threefolds, of conifold singularities, where a 3-cycle collapses to a point; in this case, and for the type II B , a massless soliton with R-R charge can be interpreted in terms of an appropiated 3-brane wrapping around the 3-cycle. For the case of K3, the enhancement of symmetry is interpreted in terms of 2-branes wrapping around 2-cycles that collapse at the orbifold point. String-string duality in six dimensions can now be used to produce dual heterotic-type II pairs in four dimensions, by compactifying on a 2-torus. This is the point where string-string duality and type II T-duality produces the desired S duality for the heterotic string [14]. The dual pairs become then the string analog of the celebrated Seiberg-Witten solution of N = 2 gauge theories. In fact, in the simpler case of supersymmetric gauge theories, the quantum moduli of a particular theory with some …