nLab
non-geometric string vacuum

Contents

Context

Vacua

String theory

Geometry

Contents

Idea

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

General

Type II superstring vacua

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

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}geometryA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\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 }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}superalgebraA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A}superalgebra
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}algebraA\phantom{A}A\phantom{A}geometryA\phantom{A}
A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}AQFTA\phantom{A}A\phantom{A}FQFTA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\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

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

T-folds:

Last revised on September 23, 2019 at 00:25:39. See the history of this page for a list of all contributions to it.