Orientifold planes, affine algebras and magnetic monopoles

TL;DR

The paper demonstrates a deep correspondence between string theory backgrounds with orientifold planes and (twisted) affine Kac–Moody algebras. By analyzing D-brane setups, open-string projections, and Wilson-line deformations, it shows how electrically and magnetically charged BPS states map to affine Dynkin diagrams, with extra monopole solutions on $R^3 \times S^1$ realizing twisted affine structures. It then translates these stringy constructions into low-energy field theory, constructing twisted monopole solutions and proving their fundamental status, thereby linking twisted boundary conditions to affine-algebraic data and suggesting exact superpotential structures for $\mathcal{N}=1$ and $\mathcal{N}=1^*$ theories. The results offer a framework to explore more general compactifications and potential connections to elliptic integrable models and index theorems.

Abstract

We analyze string theory backgrounds that include different kinds of orientifold planes and map out a natural correspondence to (twisted) affine Kac-Moody algebras. The low-energy description of specific BPS states in these backgrounds leads to a construction of explicit twisted magnetic monopole solutions on R^3 x S^1. These backgrounds yield new low-energy field theories with twisted boundary conditions and the link with affine algebras yields a natural guess for the superpotentials of the corresponding pure N=1, and N=1* gauge theories.

Figures (7)

• Figure 1: $D3$-branes separated on a line with $D1$-branes stretching between them.
• Figure 2: The three types of orientifold planes and allowed fundamental $F1$-string BPS states in the covering space. The relative length of the $F1$-strings in the cases of the $\widetilde{O3}^-$ and $O3^+$ is indicated above the $F1$-strings. Note that mirror images of the horizontal strings represent the same state (and that only the string in the middle of the lowest figure is its own mirror image). The $\widetilde{O3}^+$ plane has the same electric spectrum as the $O3^+$ plane.
• Figure 3: The building blocks for Dynkin diagrams and their corresponding orientifold planes. We indicated the positions of the $D3$-branes in the $O3^-$-plane example.
• Figure 4: On the far left we denote the Dynkin diagrams of untwisted and twisted affine Kac-Moody algebras. The first 3 diagrams represent affine algebras while the last 3 represent twisted affine algebras. Their standard notation is next to the diagrams, where the superscript in the notation for the (twisted) affine algebras denotes the order of the outer automorphism that is used to construct them K (see figure \ref{['outerauto']}). The six different combinations of the orientifold planes $Op^-$, $\widetilde{Op}^-$ and $Op^+$ used to construct the Dynkin diagrams within string theory, is shown in the next column. On the far right are limitations on the rank of the affine algebras for the Dynkin diagram to make sense. The zeroth root always corresponds to the root $\alpha_{\epsilon}$ to be defined later, except for the case $A_{2k}^{(2)}$.
• Figure 5: The action of outer automorphisms on the simple roots of Lie algebras, pictured here as an action on the Dynkin diagram, from which the action on the full algebra follows. We also displayed two special low-rank cases on the right.