nLab supersymmetry and Calabi-Yau manifolds

Context

Gravity

gravity, supergravity

superalgebra

and

supergeometry

Contents

Idea

A solution to the bosonic Einstein equations of ordinary gravity – some Riemannian manifold – has a global symmetry if it has a Killing vector.

Accordingly, a configuration that solves the supergravity Euler-Lagrange equations is a global supersymmetry if it has a Killing spinor: a covariantly constant spinor.

Here the notion of covariant derivative includes the usual Levi-Civita connection, but also in general torsion components and contributions from other background gauge fields such as a Kalb-Ramond field and the RR-fields in type II supergravity or heterotic supergravity.

Of particular interest to phenomenologists around the turn of the millenium (but maybe less so today with new experimental evidence) has been in solutions of spacetime manifolds of the form ${M}^{4}×{Y}^{6}$ for ${M}^{4}$ the locally observed Minkowski spacetime (that plays a role as the background for all available particle accelerator experiments) and a small closed 6-dimensional Riemannian manifold ${Y}^{6}$.

In the absence of further fields besides gravity, the condition that such a configuration has precisely one Killing spinor and hence precisely one global supersymmetry turns out to tbe precisely that ${Y}^{6}$ is a Calabi-Yau manifold. This is where all the interest into these manifolds in string theory comes from. (Notice though that nothing in the theory itself demands such a compactification. It is only the phenomenological assumption of the factorized spacetime compactification together with $N=1$ supersymmetry that does so).

More generally, in the presence of other background gauge fields, the Calabi-Yau condition here is deformed. One also speaks of generalized Calabi-Yau spaces. (For instance (GMPT05)).

(…)

References

The original references are

and chapters 12 - 16 of

A canonical textbook reference for the role of Calabi-Yau manifolds in compactifications of 10-dimensional supergravity is

Discussion of generalized Calabi-Yau backgrounds is for instance in

• Mariana Graña, Ruben Minasian, Michela Petrini, Alessandro Tomasiello, Generalized structures of $N=1$ vacua (arXiv:hep-th/0505212)

Revised on July 31, 2011 15:19:48 by Urs Schreiber (89.204.153.71)