Contents

# Contents

## Idea

In general, a perturbative string theory vacuum is defined by a worldsheet 2d SCFT, regarded as the formal dual (“2-spectral triple”) of the corresponding emergent target spacetime.

Some such 2d SCFTs arise as sigma-models from actual differential geometric pseudo-Riemannian spacetime manifolds with differential geometric background fields on them. These are hence called the geometric vacua or geometric backgrounds.

All other vacua are then called non-geometric backgrounds. These are defined only Isbell-dually in terms of vertex operator algebra on the worldsheet. For instance a symmetry of a non-geometric background is dually an automorphism of a vertex operator algebra.

In the point-particle limit, these non-geometric string vacua correspond to spectral triples that are defined purely algebraically, such as the non-geometric fiber spaces in the Connes-Lott model.

## Examples

### Heterotic string vacua

Strictly speaking, the perturbative string theory vacua of heterotic string theory are all non-geometric, even if the apparent target spacetime is plain Minkowski spacetime:

The 2d SCFT defining the heterotic string is the direct sum of two rather different chiral halves:

1. the holomorphic part, say, is one chiral half of a superstring sigma model on 10d Minkowski spacetime

2. the antiholomorphic part is one chiral half of the bosonic string on a 26d-dimensional Minkowski spacetime that is compactified on the Leech lattice.

While the direct sum of these two super vertex operator algebras is again a consistent super vertex operator algebra, in this combination this is not a sigma-model anymore, even if its two halves do arise in (two distinct!) sigma-models.

Analogous statements hold for most rational CFT-constructions of perturbative string theory vacua. Even if parts of these algebras arise as chiral half of sigma-models (notably WZW models), the way they are put together chirally yields a non-geometric construction that defines a non-geometric perturbative string theory vacuum.

Isbell duality between algebra and geometry

$\phantom{A}$geometry$\phantom{A}$$\phantom{A}$category$\phantom{A}$$\phantom{A}$dual category$\phantom{A}$$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand-Kolmogorov}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\text{almost by def.}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$$\phantom{A}$fin. gen.$\phantom{A}$
$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$
$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$$\phantom{A}$fin. gen.
$\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$$SmoothManifolds$$\phantom{A}$$\phantom{A}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$$\phantom{A}$supercommutative$\phantom{A}$
$\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$
$\phantom{A}$(super Lie theory)$\phantom{A}$
$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$$\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$$\phantom{A}$differential graded-commutative$\phantom{A}$
$\phantom{A}$superalgebra
$\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$$\phantom{A}$cobordism representation$\phantom{A}$

## References

We list references that use the “non-geometric”-terminology. But notice that all rational 2d CFT/Gepner model compactifications discussed in the 1990s are “non-geometric”, but the term wasn’t around then. Hence see the references at Gepner model for more.

### General

• Alex Flournoy, Brian Wecht, Brook Williams, Constructing Nongeometric Vacua in String Theory, Nucl.Phys.B706:127-149, 2005 (arXiv:hep-th/0404217)

• Katrin Becker, Melanie Becker, Cumrun Vafa, Johannes Walcher, Moduli Stabilization in Non-Geometric Backgrounds, Nucl.Phys.B770:1-46,2007 (arXiv:hep-th/0611001)

• Erik Plauschinn, Non-geometric backgrounds in string theory, j.physrep.2018.12.002 (arXiv:1811.11203)

• Dan Israel, Mirrored K3 automorphisms and non-geometric compactifications 2018 (pdf)

### Heterotic string

Non-geometric heterotic string vacua

• Jock McOrist, David Morrison, Savdeep Sethi, Geometries, Non-Geometries, and Fluxes (arXiv:1004.5447)

• Anamaría Font, Christoph Mayrhofer, Non-Geometric Vacua of the $SO(32).\mathbb{Z}_2$ Heterotic String and Little String Theories (arXiv:1708.05428)

• Anamaría Font, Iñaki García-Etxebarria, Dieter Lüst, Stefano Massai, Christoph Mayrhofer, Heterotic T-fects, 6D SCFTs, and F-Theory, JHEP08(2016)175 (arXiv:1603.09361)

• Iñaki García-Etxebarria, Dieter Lüst, Stefano Massai, Christoph Mayrhofer, Ubiquity of non-geometry in heterotic compactifications, JHEP03(2017)046 (arXiv:1611.10291)

### Flux compactifications

Non-geometric flux compactification:

• Anamaria Font, Adolfo Guarino, Jesus M. Moreno, Algebras and non-geometric flux vacua, JHEP 0812:050, 2008 (arXiv:0809.3748)

• David Andriot, Andre Betz, Supersymmetry with non-geometric fluxes, or a β-twist in Generalized Geometry and Dirac operator (arXiv:1411.6640)

• Stefano Risoli, On non-geometric string vacua, 2016 (pdf)

### T-Folds

Last revised on January 12, 2021 at 23:46:04. See the history of this page for a list of all contributions to it.