Mysterious duality
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In theoretical physics, mysterious duality is a set of mathematical similarities between some objects and laws (and perhaps all of them, if the conjecture is extended appropriately) describing M-theory on k-dimensional tori (i.e. type II superstring theory on <math>T^{k-1}<math> if k is positive) on one side, and geometry of del Pezzo surfaces (for example, the cubic surfaces) on the other side.
The main observation is that the large diffeomorphisms of del Pezzo surfaces match the Weyl group of the U-duality group of the corresponding compactification of M-theory. The elements of the second homology of the del Pezzo surfaces are mapped to various BPS objects of different dimensions in M-theory.
The complex projective plane <math>CP^2<math> is related to M-theory in 11 dimensions. When k points are blown-up, the del Pezzo surface describes M-theory on a k-torus, and the exceptional del Pezzo surface, namely <math>CP^1 \times CP^1<math>, is connected with type IIB string theory in 10 dimensions.
This conjecture was developed by Cumrun Vafa, Amer Iqbal, and Andrew Neitzke from Harvard University.