nLab
Taub-NUT space

Context

Riemannian geometry

Gravity

Contents

Idea

A kind of spacetime. For the moment see at KK-monopole for more.

graphics grabbed from Acharya-Gukov 04

Consider the left invariant 1-forms on the 3-sphere \simeq SU(2), which in terms of Euler angles? are

σ 1 =sinψdθcosψsinθdϕ σ 2 =cosψdθ+sinψsinθdϕ σ 3 =dψ+cosθdϕ \begin{aligned} \sigma_1 &= \sin \psi \, d \theta - \cos \psi \sin \theta \, d \phi \\ \sigma_2 & = \cos \psi \, d \theta + \sin \psi \sin \theta \, d \phi \\ \sigma_3 &= d \psi + \cos \theta \, d \phi \end{aligned}

Then the pseudo-Riemannian metric defining the Taub-NUT geometry is

(ds) 2=14r+nrn(dr) 2+rnr+nn 2σ 3 2+14(r 2n 2)(σ 1 2+σ 2 2) (d s)^2 = \frac{1}{4} \frac{r+n}{r-n} (d r)^2 + \frac{r-n}{r+n} n^2 {\sigma_3}^2 + \frac{1}{4}(r^2 - n^2)({\sigma_1}^2 + {\sigma_2}^2)

Relation to D6-branes

from M-branes to F-branes: superstrings, D-branes and NS5-branes

M-theory on S A 1×S B 1S^1_A \times S^1_B-elliptic fibrationKK-compactification on S A 1S^1_Atype IIA string theoryT-dual KK-compactification on S B 1S^1_Btype IIB string theoryF-theory on elliptically fibered-K3 fibrationduality between F-theory and heterotic string theoryheterotic string theory on elliptic fibration
M2-brane wrapping S A 1S_A^1double dimensional reduction \mapstotype IIA superstring\mapstotype IIB superstring\mapstoheterotic superstring
M2-brane wrapping S B 1S_B^1\mapstoD2-brane\mapstoD1-brane
M2-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp strings and qq D2-branes\mapsto(p,q)-string
M5-brane wrapping S A 1S_A^1double dimensional reduction \mapstoD4-brane\mapstoD5-brane
M5-brane wrapping S B 1S_B^1\mapstoNS5-brane\mapstoNS5-brane\mapstoNS5-brane
M5-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp D4-brane and qq NS5-branes\mapsto(p,q)5-brane
M5-brane wrapping S A 1×S B 1S_A^1 \times S_B^1\mapsto\mapstoD3-brane
KK-monopole/A-type ADE singularity (degeneration locus of S A 1S^1_A-circle fibration, Sen limit of S A 1×S B 1S^1_A \times S^1_B elliptic fibration)\mapstoD6-brane\mapstoD7-branesA-type nodal curve cycle degenertion locus of elliptic fibration ADE 2Cycle (Sen 97, section 2)SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity\mapstoD6-brane with O6-planes\mapstoD7-branes with O7-planesD-type nodal curve cycle degenertion locus of elliptic fibration ADE 2Cycle (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity\mapsto\mapstoexceptional ADE-singularity of elliptic fibration\mapstoE6-, E7-, E8-gauge enhancement

(e.g. Johnson 97, Blumenhagen 10)

References

Discussion in the context of M-theory on G2-manifolds and gauge enhancement

Revised on July 15, 2017 17:24:52 by Urs Schreiber (94.220.65.168)