nLab mysterious duality

Idea

What is called the mysterious duality in Vafa 00, Iqbal-Neitzke-Vafa 01 is a kind of “duality in string theory” in the form of a correspondence between toroidal KK-compactifications of M-theory and del Pezzo surfaces: Here M-theory on the $k$-torus $T^k$ corresponds to the complex projective space $\mathbb{C}P^2$ blown up at $k$ generic points. In particular, for $k = 1$ this corresponds to type IIA string theory (via duality between M-theory and type IIA string theory). Type IIB corresponds to $\mathbb{P}^1 \times \mathbb{P}^1$.

Moreover, the moduli of KK-compactifications of M-theory on rectangular tori are mapped to Kähler moduli of del Pezzo surfaces. The U-duality group of M-theory corresponds to a group of classical symmetries of the del Pezzo represented by global diffeomorphisms. The $\frac{1}{2}$-BPS brane charges of M-theory correspond to spheres in the del Pezzo, and their tension to the exponentiated volume of the corresponding spheres.

The S-duality of type IIB in 10 dimensions corresponds to the exchange of the two complex projective curves $\mathbb{C}P^1$s in $\mathbb{C}P^1 \times \mathbb{C}P^1$.

References

The original observation:

• Cumrun Vafa, from slide 42 in: Mirror symmetry, Talk at String Theory at the Millennium, Caltech, January 2000 (slides html)

• Amer Iqbal, Andrew Neitzke, Cumrun Vafa, A mysterious duality, (arXiv:hep-th/0111068)

Relation to Borcherds algebras:

• Pierre Henry-Labordere, Bernard Julia, Louis Paulot, Borcherds symmetries in M-theory, (arXiv:hep-th/0203070)

Relation to Hypothesis H via automorphisms of iterated cyclic loop spaces of the 4-sphere:

• Hisham Sati, Alexander Voronov, Mysterious Triality and Rational Homotopy Theory, Comm. Math. Phys. 400 (2023) 1915-1960 [arXiv:2111.14810, doi:10.1007/s00220-023-04643-7]

• Hisham Sati, Alexander Voronov, Mysterious Triality and M-Theory [arXiv:2212.13968]

• Hisham Sati, Alexander A. Voronov: Mysterious Triality and the Exceptional Symmetry of Loop Spaces [arXiv:2408.13337]

review:

• Alexander Voronov (joint with Hisham Sati), Mysterious Duality, talk at Texas Tech 2021 (abstract pdf pdf, slides pdf, pdf)

• Alexander Voronov: The $E_k$ symmetry of dimensional reductions of M-theory, talk at M-Theory and Mathematics 2023, NYU Abu Dhabi (2023) [web]

See also at:

• U-duality