nLab supersymmetry and Calabi-Yau manifolds




String theory


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics




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.

Target space perspective

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 millennium (but maybe less so today with new experimental evidence) has been in KK-compactification solutions of spacetime manifolds of the form M 4×Y 6M^4 \times Y^6 for M 4M^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 6Y^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 be precisely that Y 6Y^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 Calabi-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=1N = 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.

Worldvolume perspective

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 worldsheet 2d SCFT apriori has N=(1,0)N=(1,0) supersymmetry, the target space theory has N=1N=1 supersymmetry precisely if the worldsheet theory’s supersymmetry enhanced to N=(2,0)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=1N=1 target space supersymmetry enhanced the worldsheet symmetry from N=(1,1)N=(1,1) to N=(2,2)N=(2,2). This is reflected notably in the mirror symmetry of the target Calabi-Yau manifolds.

Non-perturbative description

The above analysis in perturbative string theory is expected to find a non-perturbative lift to M-theory/F-theory. Under this lift, compactification on a Calabi-Yau complex-3-fold (CY3) lifts to compactification on a G2-manifold/CY4-fold, respectively:

KK-compactifications of higher dimensional supergravity with minimal (N=1N=1) supersymmetry:

perspectiveKK-compactification with N=1N=1 supersymmetry
M-theoryM-theory on G2-manifolds
F-theoryF-theory on CY4-manifolds
heterotic string theoryheterotic string theory on CY3-manifolds


The idea, in the context of heterotic string theory on CY3-manifolds, originates in

where in the introduction it says the following

Recently, the discovery [6] of anomaly cancellation in a modified version of d=10d = 10 supergravity and superstring theory with gauge group O(32)O(32) or E 8×E 8E_8 \times E_8 has opened the possibility that these theories might be phenomenologically realistic as well as mathematically consistent. A new string theory with E 8×E 8E_8 \times E_8 gauge group has recently been constructed [7] along with a second O(32)O(32) theory.

For these theories to be realistic, it is necessary that the vacuum state be of the form M 4×KM_4 \times K, where M 4M_4 is four-dimensional Minkowski space and K is some compact six-dimensional manifold. (Indeed, Kaluza-Klein theory – with its now widely accepted interpretation that all dimensions are on the same logical footing – was first proposed [8] in an effort to make sense out of higher-dimensional string theories). Quantum numbers of quarks and leptons are then determined by topological invariants of KK and of an O(32)O(32) or E 8×E 8E_8 \times E_8 gauge field defined on KK [9]. Such considerations, however, are far from uniquely determining KK.

In this paper, we will discuss some considerations, which, if valid, come very close to determining KK uniquely. We require

(i) The geometry to be of the form H 4×KH_4 \times K, where H 4H_4 is a maximally symmetric spacetime.

(ii) There should be an unbroken N=1N = 1 supersymmetry in four dimensions. General arguments [10] and explicit demonstrations [11] have shown that supersymmetry may play an essential role in resolving the gauge hierarchy or Dirac large numbers problem. These arguments require that supersymmetry is unbroken at the Planck (or compactification) scale.

(iii) The gauge group and fermion spectrum should be realistic.

These requirements turn out to be extremely restrictive. In previous ten-dimensional supergravity theories, supersymmetric configurations have never given rise to chiral fermions – let alone to a realistic spectrum. However, the modification introduced by Green and Schwarz to produce an anomaly-free field theory also makes it possible to satisfy these requirements. We will see that unbroken N=1N = 1 supersymmetry requires that KK have, for perturbatively accessible configurations, SU(3)SU(3) holonomy and that the four-dimensional cosmological constant vanish. The existence of spaces with SU(3)SU(3) holonomy was conjectured by Calabi [12] and proved by Yau [13].

(Of course later it was understood that Calabi-Yau spaces, even those of complex dimension 3, are not “very close to unique”.)

Lecture notes include

Further original references include

and chapters 12 - 16 of

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

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

Mathematical background:

Last revised on December 4, 2020 at 11:08:27. See the history of this page for a list of all contributions to it.