What IIB looks IIA string: String Cobordisms via Non-Compact CFTs

Abstract

The Swampland Cobordism Conjecture predicts the existence of end-of-the-world branes for every consistent Quantum Gravity theory, and domain walls connecting the Landscape. A perturbative string worldsheet description of these objects is only expected to exist when certain worldsheet invariants are vanishing or coincide across the domain wall. In this paper, we observe that many of these worldsheet obstructions can be evaded by allowing non-compact string worldsheets as part of the bordism. Using these ideas, we provide a worldsheet QFT (flowing to a critical CFT under RG flow) that connects the worldsheets of 0A and 0B string theory, as well as those of IIA and IIB string theories. The non-compact character of these interpolations means that the description of the actual domain walls develop strongly coupled regions described by a linear dilaton background, where the worldsheet description breaks down. As a result, the IIA/IIB domain wall requires a strongly coupled region, in agreement with effective field theory considerations. Our constructions are heavily inspired by supercritical string theory, but are logically independent from it. We also analyze the fate of IIA/IIB NS5 branes as they cross the domain wall between these two theories.

Figures (4)

• Figure 1: Worldsheet for a string ending on a D-brane crossing the type 0A/0B domain wall. The worldsheet boundary is shown as the horizontal red line. The 2d green area corresponds to the theory with the stacked Arf TQFT, and has a 1d Majorana fermion localized on its boundary (green line), which includes the interface with the blue area as well as half of the worldsheet boundary. The presence of an extra fermion implies that a D$p$-brane on one side transforms into a ${\widehat{{\rm D}p}}$-brane on the other, and vice versa.
• Figure 2: On the left hand panel, we have a schematic depiction of the initial worldsheet QFT we start with, with tachyon profile as in (\ref{['ewe']}). Away from the greyed-out $X_9=0$ region, the system flows to the 0A/0B vacuum. Close to $X_9=0$ we have a mildly non-compact CFT interpolating between the two solutions, but the string dilaton is small throughout the whole picture. On the right-hand panel, we have the actual IR CFT, which describes the actual spacetime profile corresponding to this configuration. The non-compact CFT has been replaced by two light-like linear dilaton backgrounds (the coordinates $Y_0,Y_1$ have a flat Minkowski metric and have been introduced for illustration) which are back-to-back, with the strongly coupled region of the 0A in the future next to the past strongly coupled region of 0B. We also depict the linear dilaton gradient on each side of the domain wall. The strongly coupled region in between the two means that the 0A/0B domain wall configuration goes beyond a perturbative worldsheet description.
• Figure 3: The type II linear dilaton backgrounds are BPS solutions. In IIA (left panel), the strong coupling region can be uplifted to a light-like Big Bang singularity Craps:2005wd, where the M-theory circle blows up and the scale factor vanishes. In the Figure, the thick black diagonal line depicts the Big Bang, with "nothing" to the bottom-left of it. This geometry is, therefore, a true end-of-the-world boundary for IIA string theory. The arrow depicts the direction of increasing dilaton. In type IIB (right panel), a possible resolution of the IIB strong coupling region: an application of S-duality to the BPS background reverses the sign of the dilaton gradient, resulting in a second copy of a IIB weakly coupled region in the bottom left of the diagram. Therefore, in this UV completion the BPS linear dilaton background would not be a valid cobordism defect. It is conceivable that additional operators turned on in the RG flows studied in the main text turn the IIB case into something similar to IIA, but we do not know whether this is the case.
• Figure 4: Structure of $\mathbb{R}\times\mathbb{S}^1$ flattened onto the $Z$-plane and its different kinds of circles. The gray dashed lines correspond to $\mathbb{S}^1$'s at constant $Y$ in the underlying $\mathbb{R}\times\mathbb{S}^1$. The origin corresponds to $Y\to -\infty$ and the asymptotic circle corresponds to $Y\to\infty$. The blue circles describe a family of $\mathbb{S}^1$'s interpolating between one which surrounds the origin (labeled with $\lambda=0$) and another which does not (labeled with $\lambda=2$). The interpolation involves a configuration (labeled $\lambda=1$) for which the circle passes through the origin, and is hence non-compact.