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This page reviews some of the relation between the existence of supersymmetries in a spacetime quantum field theory which arises as the effective quantum field theory of some string 2d SCFT, the special geometry of that spacetime (such as Calabi-Yau manifold structure) as well as increased worldsheet supersymmetry of the superstring.
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 KK-compactification 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. One such remaining global supersymmetry in the low energy effective field theory in 4-dimensions is or was believed to be of relevance in phenomenology, see for instance the supersymmetric MSSM extension of the standard model of particle physics. This is where all the interest into these Calaboi-Yau 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)).
Alternatively, if one starts the KK-compactification not from 10-dimensional string theory but from 11-dimensional supergravity/M-theory, then the condition for the KK-compactification to preserved precisely one global supersymmetry is that it be on a G2-manifold. For more on this see at M-theory on G2-manifolds.
On the other hand, the enhanced global supersymmetry of target space is also reflected in enhanced local supersymmetry on the worldsheet of the string. For instance for the heterotic string whose worlsheet 2d SCFT apriori has $N=(1,0)$ supersymmetry, the target space theory has $N=1$ supersymmetry precisely if the worldsheet theory’s supersymmetry enhanced to $N=(2,0)$. (BDFF 88). For more on this see at 2d (2,0)-superconformal QFT.
Similar comments apply to type II superstring theory, where $N=1$ target space supersymmetry enhanced the worldheet symmetry from $N=(1,1)$ to $N=(2,2)$. This is reflected notably in the mirror symmetry of the targt Calabi-Yau manifolds.
Original references include
Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten, Vacuum Configurations for Superstrings , Nucl. Phys. B 258 (1985), p. 46.
Tom Banks, Lance Dixon, Dan Friedan, Emil Martinec, Phenomenology and Conformal Field Theory or Can String Theory Predict the Weak Mixing Angle?, Nucl. Phys. B299 (1988) 613. (pdf)
Jacques Distler, Brian Greene, Aspects Of $(2,0)$ String Compactifications, Nucl. Phys. B304 (1988)
Andrew Strominger, Special Geometry, Comm. Math. Phys. 133 (1990) 163.
Philip Candelas and X. De la Ossa, Moduli Space of Calabi-Yau Manifolds, Nucl. Phys. B355 (1991) 455.
Edward Witten, Phases of N=2 Theories in Two Dimensions, Nucl. Phys. B403 (1993) 159 (arXiv:hep-th/9301042)
and chapters 12 - 16 of
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)
Lecure notes in a more general context of string phenomenology include
Discussion of generalized Calabi-Yau backgrounds is for instance in