Contents

# Contents

## Idea

In string theory the term swampland had been introduced (Vafa 05) to publicly highlight the basic fact that many effective quantum field theory vacua will not admit a UV-completion to a string theory vacuum, hence that admitting a completion to a string theory vacuum is a strong constraint, hence that string theory predicts many more conditions to be satisfied by gauge groups, field content and coupling constants than predicted by plain quantum field theory.

The terminology was motivated since the collection of string theory vacua had previously come to be called the landscape of string theory vacua. The idea is to imagine the remaining EFTs not “in” this landscape to form a space “away from the landscape”, whence the colorful imagery of a swampland.

More in detail, there is supposedly a map

(1)$StringVacua \overset{\;\;ppl\;\;}{\longrightarrow} eQFTVacua$

that takes string theory vacua to their low-energy approximation by vacua of effective quantum field theories. One way to possibly formalize this is to take the point-particle limit (ppl) of 2d SCFTs to obtain spectral triples (as discussed at 2-spectral triple) taking string worldsheets to Feynman diagrams:

graphics grabbed from Schubert 96

Then with language of homological algebra or more generally of category theory we may begin to formalize the situation as follows:

1. the domain of (1) is the landscape of string theory vacua;

2. the image of (1) is the landscape of corresponding eQFTs that admit stringy UV-completion;

3. the cokernel of (1) is the swampland;

4. the kernel of (1), or more generally its fiber over any EFT, is the space of different choices of stringy UV-completion of the same effective quantum field theory.

Making this fully precise requires saying more about what the domain and codomain in (1) actually are, and in which ambient category (they will be some kind of moduli stacks in an ambient (∞,1)-category which may not quite be stable, whence “cokernel” may need to be interpreted in a non-abelian sense; but such details don’t change the general idea here).

For example, part of what it means to specify a string theory vacuum is to declare the D-brane charge contained in this vacuum (subject to RR-field tadpole cancellation against O-plane-charges). In actual string theory this RR-field charge is supposed to be (see there) a cocycle in some flavour of topological K-theory (twisted equivariant differential KR-theory), while its image in the underlying effective field theory is in the corresponding flavour of ordinary cohomology/de Rham cohomology. The map that relates the two incarnations of RR-field charge is the Chern character, which is what formalizes the map (1) in in the D-brane charge “sector” of the theory

$\array{ StringVacua &\overset{\;\;ppl\;\;}{\longrightarrow}& eQFTVacua \\ \left\{ \array{ \text{D-brane RR-flux} \\ \text{in K-theory} } \right\} &\overset{\text{Chern character}}{\longrightarrow}& \left\{ \array{ \text{RR-field in} \\ \text{ordinary cohomology} } \right\} }$

The Chern character in general does have non-trivial cokernel (“swampland RR-fields”) and kernel (choices of UV-completion of the effective RR-fields). In fact it fits not just in a short exact sequence, but in the differential cohomology hexagon (see there for more) of K-theory.

In contrast to this example, the literature on the “swampland” phenomenon is currently dominated by informal hand-wavy arguments. Starting with Ooguri-Vafa 06 is an attempt to guess semi-precise rules-of-thumb for recognizing EFTs in the swampland, now known as the swampland conjectures. Motivated by the re-opening of the question whether de Sitter spacetime actually appears in string theory vacua or not (Danielsson-van Riet 18), these swampland conjecture currently revolve around bounds on the cosmological constant in relation to scalar fields in the theory.

## References

### General

The terminology originates with

Comprehensive review is in:

• Eran Palti, The Swampland: Introduction and Review, lecture notes (arXiv:1903.06239)

Further discussion includes