nLab
supersymmetry and Calabi-Yau manifolds

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 $M^4\times Y^6$ 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)).

Details

(…)

Related concepts

• supergravity, supersymmetry

• Kaluza-Klein mechanism

• spontaneously broken symmetry

References

The original references are

• Candelas, Horowitz, Andrew Strominger, and Edward Witten, Vacuum Configurations for Superstrings , Nucl. Phys. B 258 (1985), p. 46.

and chapters 12 - 16 of

• Michael Green, John Schwarz, Edward Witten, Superstring Theory , Vol. 2, Cambridge University Press, (1987)

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

• Andrew Strominger (notes by John Morgan), Kaluza-Klein compactifications, Supersymmetry and Calabi-Yau spaces , volume II, starting on page 1091 in

  Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey, David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, eds. , Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

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)