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The hyperplane string, RCFTs, and the swampland

Guglielmo Lockhart, Luca Novelli

TL;DR

This work uses the hyperplane (L) string in 6d $\mathcal{N}=(1,0)$ supergravity with rank-one string lattice as a probe of the swampland and landscape. It identifies five maximally non-geometric theories where the L-string’s left-moving sector is exactly captured by a rational CFT, enabling precise computations of the elliptic genus and consistency checks via unitarity. In the concrete $E_6$ locus, the left-moving sector is $E_{6,8}\times$ three-state Potts, and the authors compute the elliptic genus, finding modularity and a path-connected Higgsing chain to geometric models, consistent with the proposed mapping between the quantum-corrected L-string moduli and regions of the 6d landscape. Overall, the L-string provides a tractable window into non-geometric regions of 6d quantum gravity and offers a concrete mechanism to relate RCFT data to swampland criteria and Higgsing connectivity.

Abstract

Six dimensional $\mathcal{N}=(1,0)$ supergravity features BPS strings whose properties encode highly nontrivial information about the parent 6d theory. We focus on a distinguished set of theories whose string charge lattice is one-dimensional. In geometric theories, the generator of the lattice arises from a D3 brane wrapping the hyperplane class in $\mathbb{P}^2$. This hyperplane string is expected to remain stable even when one ventures beyond the geometric regime where it becomes challenging to verify which candidate 6d theories belong to the swampland. We identify five 6d models which from the perspective of the hyperplane string deviate the most from being geometric. For these theories we are able to provide an exact description of the left-moving sector of the hyperplane string worldsheet theory in terms of a rational conformal field theory and provide evidence for their consistency. In one instance, using RCFT methods we are able to determine the elliptic genus and find that in the unflavored limit it matches with the elliptic genus of geometric models. We argue that the non-geometric model is connected to geometric ones via a sequence of Higgsing transitions. These results lead us to formulate a proposal relating the quantum corrected moduli space of the hyperplane string CFT with a region of the landscape of 6d $(1,0)$ quantum gravity.

The hyperplane string, RCFTs, and the swampland

TL;DR

This work uses the hyperplane (L) string in 6d supergravity with rank-one string lattice as a probe of the swampland and landscape. It identifies five maximally non-geometric theories where the L-string’s left-moving sector is exactly captured by a rational CFT, enabling precise computations of the elliptic genus and consistency checks via unitarity. In the concrete locus, the left-moving sector is three-state Potts, and the authors compute the elliptic genus, finding modularity and a path-connected Higgsing chain to geometric models, consistent with the proposed mapping between the quantum-corrected L-string moduli and regions of the 6d landscape. Overall, the L-string provides a tractable window into non-geometric regions of 6d quantum gravity and offers a concrete mechanism to relate RCFT data to swampland criteria and Higgsing connectivity.

Abstract

Six dimensional supergravity features BPS strings whose properties encode highly nontrivial information about the parent 6d theory. We focus on a distinguished set of theories whose string charge lattice is one-dimensional. In geometric theories, the generator of the lattice arises from a D3 brane wrapping the hyperplane class in . This hyperplane string is expected to remain stable even when one ventures beyond the geometric regime where it becomes challenging to verify which candidate 6d theories belong to the swampland. We identify five 6d models which from the perspective of the hyperplane string deviate the most from being geometric. For these theories we are able to provide an exact description of the left-moving sector of the hyperplane string worldsheet theory in terms of a rational conformal field theory and provide evidence for their consistency. In one instance, using RCFT methods we are able to determine the elliptic genus and find that in the unflavored limit it matches with the elliptic genus of geometric models. We argue that the non-geometric model is connected to geometric ones via a sequence of Higgsing transitions. These results lead us to formulate a proposal relating the quantum corrected moduli space of the hyperplane string CFT with a region of the landscape of 6d quantum gravity.

Paper Structure

This paper contains 20 sections, 82 equations, 2 figures, 9 tables.

Table of Contents

  1. Introduction
  2. 6d quantum gravity with $\text{rk}(\Gamma)=1$: geometric and non-geometric models
  3. Review of anomaly-free conditions
  4. Geometric vs. non-geometric
  5. Higgsing down to geometric models
  6. The hyperplane string and the 6d $\mathcal{N}=(1,0)$ landscape
  7. The hyperplane string
  8. The non-geometric region and RCFT
  9. The elliptic genus and the 6d massless spectrum
  10. Exact solution for the hyperplane string at the $G=E_6$ locus
  11. Computation of the elliptic genus
  12. Remarks on symmetries of the elliptic genus
  13. Conclusions and future directions
  14. Modular forms: basic definitions
  15. RCFT data
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Cartoon of the quantum corrected moduli space $\mathcal{M}^{[L]}_{2d}$ of the hyperplane string CFT. Higgsing transitions of the parent 6d theory are depicted by arrows and are of two kinds: the thick vertical arrows start from 2d worldsheet theories corresponding to 6d theories with nontrivial gauge group and $b_i = 1$; in this case, the 2d moduli space also includes instanton branches in addition to the geometric branch, resulting in potentially higher central charges. This sequence of Higgsings ultimately terminates on $\mathcal{M}^{[L]}_{2d}$. The thin arrows correspond to Higgsing of gauge factors for which $b_i\neq 1$. In this picture we have portrayed a series of transitions that start from the theory with $G=E_6$ and matter in the $\mathbf{351'}$ dimensional representation, and terminate on the fully Higgsed theory with trivial gauge group and 273 neutral hypermultiplets. The $E_6$ theory is portrayed as an island where the string worldsheet theory is described by an RCFT. This Higgsing chain involves passing through a number of non-geometric models (portrayed here as the sea) as well as geometric models (portrayed as the sand). Branches of the moduli space where the string becomes a composite object (for $\text{rk } \Gamma > 1$) are grayed out. We have displayed only one chain of Higgsing transitions here, but there also exist other regions (see Table \ref{['tab:clg31intro']}) with other islands connected to the geometric mainland by chains of Higgsing transitions.
  • Figure 2: The $E_6$ affine Dynkin diagram. The quantities $(j,a^\vee,\mathbf{R})$ associated to each node denote respectively its label, its comark, and the associated $E_6$ representation. Depicted here are also the actions of the $\mathbb{Z}_2$ charge conjugation symmetry and $\mathbb{Z}_3$ outer automorphism group, which together assemble into the $D(E_6) = S_3$ diagram automorphism group of affine $E_6$.