III.6 Calabi–Yau Manifolds

Eric Zaslow

1 Basic Definition

Calabi–Yau manifolds, named after Eugenio Calabi and Shing-Tung Yau, arise in Riemannian geometry and algebraic geometry, and play a prominent role in string theory and mirror symmetry.

In order to explain what they are, we need first to recall the notion of orientability on a real MANIFOLD [I.3 §6.9]. Such a manifold is orientable if you can choose coordinate systems at each point in such a way that any two systems x = (x¹, . . . , x^m) and y = (y¹ , . . . , y^m) that are defined on overlapping sets give rise to a positive Jacobian: det (∂ yⁱ/∂ x^j) > 0. The notion of a Calabi–Yau manifold is the natural complex analogue of this. Now the manifold is complex, and for each local coordinate system z = (z¹, . . . , zⁿ) One has a HOLOMORPHIC FUNCTION [I.3 §5.6] f(z). It is vital that f should be nonvanishing: that is, it never takes the value 0. There is also a compatibility condition: if (z) is another coordinate system, then the corresponding function is related to f by the equation f = det (∂ ^a / ∂ z^b). Note that if we replace all complex terms by real terms in this definition, then we have the notion of a real orientation. So a Calabi–Yau manifold can be thought of informally as a complex manifold with complex orientation.

2 Complex Manifolds and Hermitian Structure

Before we go any further, a few words about complex and kähler geometry are in order. A complex manifold is a structure that looks locally like ⁿ, in the sense that one can find complex coordinates z = (z¹, . . . , zⁿ) near every point. Moreover, where two coordinate systems z and overlap, the coordinates ^a are holomorphic when they are regarded as functions of the z^b. Thus, the notion of a holomorphic function on a complex manifold makes sense and does not depend on the coordinates used to express the function. In this way, the local geometry of a complex manifold does indeed look like an open set in ⁿ, and the tangent space at a point looks like ⁿ itself.

On complex vector spaces it is natural to consider Hermitian INNER PRODUCTS [III.37] represented by HERMITIAN MATRICES [III.50 §3] g_ab with respect to a basis e_a. On complex manifolds, a Hermitian inner product on the tangent spaces is called a “Hermitian metric,” and is represented in a coordinate basis by a Hermitian matrix g_ab, which depends on position.¹

3 Holonomy, and Calabi–Yau Manifoldsin Riemannian Geometry

On a RIEMANNIAN MANIFOLD [I.3 §6.10] one can move a vector along a path so as to keep it of constant length and “always pointing in the same direction.” Curvature expresses the fact that the vector you wind up with at the end of the path depends on the path itself. When your path is a closed loop, the vector at the starting point comes back to a new vector at the same point. (A good example to think about is a path on a sphere that goes from the North Pole to the equator, then a quarter Of the way around the equator, then back to the North Pole again. When the journey is completed, the “constant” vector that began by pointing south will have been rotated by 90°.) With each loop we associate a matrix operator called the holonomy matrix, which sends the starting vector to the ending vector; the group generated by all of these matrices is called the holonomy group of the manifold. Since the length of the vector does not change during the process of keeping it constant along the loop, the holonomy matrices all lie in the orthogonal group of length-preserving matrices, O(m). If the manifold is oriented, then the holonomy group must lie in SO(m), as one can see by transporting an oriented basis of vectors around the loop.

Every complex manifold of (complex) dimension n is also a real manifold of (real) dimension m = 2n, which one can think of as coordinatized by the real and imaginary parts of the complex coordinates z^j. Real manifolds that arise in this way have additional structure. For example, the fact that we can multiply complex coordinate directions by i = implies that there must be an operator on the real tangent space that squares to -1. This operator has eigenvalues ±i, which can be thought of as “holomorphic” and “anti-holomorphic” directions. The Hermitian property states that these directions are orthogonal, and we say that the manifold is Kähler if they remain so after transport around loops. This means that the holonomy group is a subgroup of U(n) (which itself is a subgroup of SO(m): complex manifolds always have real orientations). There is a nice local characterization of the Kähler property: if g_ab are the components of the Hermitian metric in some coordinate patch, then there exists a function φ on that patch such that g_ab = ∂ ²φ / ∂ Z^a ∂ ^b.

Given a complex orientation—that is, the nonmetric definition of a Calabi–Yau manifold given above—a compatible Kähler structure leads to a holonomy that lies in SU(n) ⊂ U(n), the natural analogue of the case of real orientation. This is the metric definition of a Calabi–Yau manifold.

4 The Calabi Conjecture

Calabi conjectured that, for any Kähler manifold of complex dimension n and any complex orientation, there exists a function u and a new Kähler metric , given in coordinates by

that is compatible with the orientation. In equations, the compatibility condition states that

where f is the holomorphic orientation function discussed above. Thus, the metric notion of a Calabi–Yau manifold amounts to a formidable nonlinear partial differential equation for u. Calabi proved the uniqueness and Yau proved the existence of a solution to this equation. So in fact the metric definition of a Calabi–Yau manifold is uniquely determined by its Kähler structure and its complex orientation.

Yau’s theorem establishes that the space of metrics with holonomy group SU(n) on a manifold with complex orientation is in correspondence with the space of inequivalent Kähler structures. The latter space can easily be probed with the techniques of algebraic geometry.

5 Calabi–Yau Manifolds in Physics

Einstein’s theory of gravity, general relativity, constructs equations that the metric of a Riemannian space-time manifold must obey (see GENERAL RELATIVITY AND THE EINSTEIN EQUATIONS [IV.13]). The equations involve three symmetric tensors: the metric, the RICCI CURVATURE [III.78] tensor, and the energy- momentum tensor of matter. A Riemannian manifold whose Ricci tensor vanishes is a solution to these equations when there is no matter, and is a special case of an Einstein manifold. A Calabi–Yau manifold with its unique SU(n)-holonomy metric has vanishing Ricci tensor, and is therefore of interest in general relativity.

A fundamental problem in theoretical physics is the incorporation of Einstein’s theory into the quantum theory of particles. This enterprise is known as quantum gravity, and Calabi–Yau manifolds figure prominently in the leading theory of quantum gravity, STRING THEORY [IV.17 §2].

In string theory, the fundamental objects are one-dimensional “strings.” The motion of the strings in space-time is described by two-dimensional trajectories, known as worldsheets, so every point on the world-sheet is labeled by the point in space-time where it sits. In this way, string theory is constructed from a quantum field theory of maps from two-dimensional RIE-MANN SURFACES [III.79] to a space-time manifold M. The two-dimensional surface should be given a Riemannian metric, and there is an infinite-dimensional space of such metrics to consider. This means that we must solve quantum gravity in two dimensions—a problem that, like its four-dimensional cousin, is too hard. If, however, it happens that the two-dimensional worldsheet theory is conformal (invariant under local changes of scale), then just a finite-dimensional space of conforma11y inequivalent metrics remains, and the theory is well-defined.

The Calabi–Yau condition arises from these considerations. The requirement that the two-dimensional theory should be conformal, so that the string theory makes good sense, is in essence the requirement that the Ricci tensor of space-time should vanish. Thus, a two-dimensional condition leads to a space-time equation, which turns out to be exactly Einstein’s equation without matter. We add to this condition the “phenomenological” criterion that the theory be endowed with “supersymmetry,” which requires the space-time manifold M to be complex. The two conditions together mean that M is a complex manifold with holonomy group SU(n): that is, a Calabi–Yau manifold. By Yau’s theorem, the choices of such M can easily be described by algebraic geometric methods.

We remark that there is a kind of distillation of string theory called “topological strings,” which can be given a rigorous mathematical framework. Calabi–Yau manifolds are both symplectic and complex, and this leads to two versions of topological strings, called A and B, that one can associate with a Calabi–Yau manifold. Mirror symmetry is the remarkable phenomenon that the A version of one Calabi–Yau manifold is related to the B version of an entirely different “mirror partner.” The mathematical consequences of such an equivalence are extremely rich. (See MIRROR SYMMETRY [IV.16] for more details. For other notions related to those discussed in this article, see SYMPLECTIC MANIFOLDS [III.88].)

The Calculus of Variations

See VARIATIONAL METHODS [III.94]

1. The notation g_a indicates the conjugate-linear property of a Hermitian inner product.