Freedom and Constraints in the K3 Landscape

Abstract

We consider "magnetized brane" compactifications of the type I/heterotic string on K3 with U(1) background fluxes. The nonabelian gauge group and matter content of the resulting six-dimensional vacua are parameterized by a matrix encoding a lattice contained within the even, self-dual lattice Gamma^{3,19}. Mathematical results of Nikulin on lattice embeddings make possible a simple classification of such solutions. This approach makes it possible to explicitly and efficiently construct models in this class with a particular allowed gauge group and matter content, so that one can immediately "dial-a-model" with desired properties.

Figures (8)

• Figure 1: Three stacks of D9-branes and their orientifold images. The $1$ and $2$ stacks with $N_1$ and $N_2$ branes respectively have $U(1)$ fluxes $f_1, f_2$ on the world-volume of each brane in the two stacks. These stacks carry unbroken components $U(N_1)$, $U(N_2)$ of the gauge group. The $3$ and $3'$ stacks with $2M$ branes have no background flux and their gauge group is $SO(2M)$. All three stacks and their images are on top of each other and have been separated for clarity.
• Figure 2: A lattice $\mathcal{M}$ is shown on the left, and its overlattice $\mathcal{N}$ on the right.
• Figure 3: Triangular lattice corresponding to the inner product matrix $I^\triangle$.
• Figure 4: $T^4$ represented as a product of two complex tori. Each 2-torus has four fixed points under inversion of its complex coordinate. Therefore, the product $T^4$ has sixteen fixed points under the involution.
• Figure 5: The elements of $I$ represented as the vector space $\mathbb{F}_2^4$ over the field $\mathbb{F}_2$. The point $(x_1,x_2,x_3,x_4)\in \mathbb{F}_2^4$ corresponds to the element $x_1+2x_2+2^2x_3+2^3x_4\in I$. $\mathbb{F}_2^4$ is drawn as the two hyperplanes $x_4=0$ and $x_4=1$.

Theorems & Definitions (3)

• Theorem 5.1: Nikulin, simplified
• Theorem C.1: James
• Theorem C.2