A Mysterious Duality

TL;DR

The paper unveils a deep duality between M-theory compactifications on rectangular tori and del Pezzo surfaces, identifying M-theory moduli with Kahler data on del Pezzo surfaces and realizing the U-duality group as diffeomorphisms that preserve the canonical class. It builds a precise dictionary in which 1/2-BPS branes correspond to rational curves, with tensions given by $T=2\pi\exp\omega(C)$ and charges tied to the intersection properties of curves with the canonical class $K$. The framework explains electric/magnetic pairings as $C_E+C_M=-K$ and bound-state conditions via curve intersections, and it provides a coherent map between the M-theory moduli space and the extended Kahler moduli of $\mathbb B_k$. The authors propose speculations about the physical meaning of the duality, its potential interpretation as a probe moduli space, and avenues for extending the construction to broader compactifications and deformations, highlighting the fusion of algebraic geometry with non-perturbative string theory.

Abstract

We establish a correspondence between toroidal compactifications of M-theory and del Pezzo surfaces. M-theory on T^k corresponds to P^2 blown up at k generic points; Type IIB corresponds to P^1\times P^1. The moduli of compactifications of M-theory on rectangular tori are mapped to Kahler 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 half-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 electric/magnetic pairing of branes is determined by the condition that the union of the corresponding spheres represent the anticanonical class of the del Pezzo. The condition that a pair of half-BPS states form a bound state is mapped to a condition on the intersection of the corresponding spheres. We present some speculations about the meaning of this duality.

Figures (13)

• Figure 1: The Dynkin diagrams of exceptional Lie algebras $E_{k}$.
• Figure 2: $\mathbb{P}^{1}$ as a circle fibration over an interval.
• Figure 3: a) The toric diagram of $\mathbb{P}^{2}$, b) $\mathbb{P}^{2}$ as a $T^{2}$ fibration with collapsing fibers at the boundary, c) The three boundary components as $\mathbb{P}^{1}$'s.
• Figure 4: a) The toric diagram of $\mathbb{B}_{1}$. b) The interval which replaces the rightmost vertex of the triangle represents the exceptional curve.
• Figure 5: a) The toric diagram of $\mathbb{B}_{2}$, b) The two intervals replacing the vertex represent the two exceptional curves on the base.