8d supergravity, reconstruction of internal geometry and the Swampland

Yuta Hamada,Cumrun Vafa

doi:10.1007/jhep06(2021)178

Abstract

We sharpen Swampland constraints on 8d supergravity theories by studying consistency conditions on worldvolume theory of 3-brane probes. Combined with a stronger form of the cobordism conjecture, this leads to the reconstruction of the compact internal geometry and implies strong restrictions on the gauge algebra and on some higher derivative terms (related to the level of the current algebra on the 1-brane). In particular we argue that 8d supergravity theories with {mathfrak{g}}_2 gauge symmetry are in the Swampland. These results provide further evidence for the string lamppost principle in 8d with 16 supercharges.

Highlights

These results support the String Lamppost Principle (SLP), which states that all consistent theories of quantum gravity are in the string landscape
Even though we start from 8d supergravity, we can reconstruct the compactified internal geometry, and obtain the bound inherited from the higher dimensional theory! We believe that the reconstruction of the internal geometry and in particular its compactness can be investigated more broadly, and is a promising new direction for the Swampland program
In particular we show that the 3-brane probe geometry leads to an elliptic K3 geometry, which is the key to the bounds we will find

Summary

String constructions

Known 8d string theory compactifications with 16 supercharges have gauge groups whose rank is 18, 10, or 2. Rank = 18 Theories with Rank = 18 are realized by F-theory on elliptic K3 [16], and type I or heterotic string on T 2. For the heterotic string on T 2, all the groups of maximal enhancement are listed in [22] which agrees with [21]. Rank = 10 Theories with Rank = 10 are realized by F-theory on elliptic K3 with one frozen singularity (O7+ plane) [23],2 CHL string [24,25,26], IIA orientifold on Mobius strip, and type I or heterotic string on T 2 without the vector structure [23]. For eight dimensional CHL string, all groups of maximal enhancement are listed in [27]. We will be able to explain this fact directly by the study of 3-brane moduli space from the perspective of the eight dimensional effective theory

Review of the Swampland bounds

Derivation of the bound on the gauge algebra

The geometry of instanton 3-brane moduli

The relation between the type of the singularity and the gauge algebra

Power of the geometry and the bound

Conclusions