# nLab mysterious duality

Contents

### Context

#### Duality in string theory

duality in string theory

general mechanisms

string-fivebrane duality

string-string dualities

M-theory

F-theory

string-QFT duality

QFT-QFT duality:

# Contents

## 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:

Relation to Borcherds algebras:

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

review: