The Moduli Space and M(atrix) Theory of 9d N=1 Backgrounds of M/String Theory

TL;DR

The paper classifies nine-dimensional $ $9d $N=1$ backgrounds with reduced rank, revealing two disconnected components in the rank-$2$ sector and non-perturbative corrections to orientifold planes. It furnishes a comprehensive moduli-space map, linking M theory on a Klein bottle and a Möbius strip with DP, AOA/AOB, CHL, and asymmetric Type II orbifolds through a web of dualities, all governed by $SO(n-1,1, \mathbb{Z})$ duality groups. A central achievement is the construction of M(atrix) theory descriptions for cylinder and Klein bottle backgrounds, yielding 2+1D gauge theories with spatially varying couplings and boundary conditions that encode Möbius-strip corrections and anomaly cancellation. The work also clarifies the rank-18 sector and connects the CHL/Möbius/M"obius-strip corners to the broader moduli web, including the self-dual SU(2) enhancements and X-type backgrounds. Overall, these results illuminate the non-perturbative structure of M/string backgrounds in nine dimensions and pave the way for future probes of dualities, probe branes, and lower-dimensional generalizations.

Abstract

We discuss the moduli space of nine dimensional N=1 supersymmetric compactifications of M theory / string theory with reduced rank (rank 10 or rank 2), exhibiting how all the different theories (including M theory compactified on a Klein bottle and on a Mobius strip, the Dabholkar-Park background, CHL strings and asymmetric orbifolds of type II strings on a circle) fit together, and what are the weakly coupled descriptions in different regions of the moduli space. We argue that there are two disconnected components in the moduli space of theories with rank 2. We analyze in detail the limits of the M theory compactifications on a Klein bottle and on a Mobius strip which naively give type IIA string theory with an uncharged orientifold 8-plane carrying discrete RR flux. In order to consistently describe these limits we conjecture that this orientifold non-perturbatively splits into a D8-brane and an orientifold plane of charge (-1) which sits at infinite coupling. We construct the M(atrix) theory for M theory on a Klein bottle (and the theories related to it), which is given by a 2+1 dimensional gauge theory with a varying gauge coupling compactified on a cylinder with specific boundary conditions. We also clarify the construction of the M(atrix) theory for backgrounds of rank 18, including the heterotic string on a circle.

Figures (1)

• Figure 1: To demonstrate the identifications we propose, we show an explicit example of how it works. In this figure, the middle cell in the lowest row is the (0,0) cell. We act with the orientifold transformation to demonstrate the following equality $X_{(0,0),(1,1);1,2;i,j}=X_{(-1,2),(0,0);1,2;j,i}=X_{(0,0),(1,-2);1,2;j,i}$ where $X$ is any of the orthogonal coordinates. The dashed line separates each torus fundamental domain to two copies of the Klein bottle. The filled circles are D0-branes (and the two colors correspond to two different D0-branes). The two depicted strings are identified under the transformation.