What is the geometry of spacetime? On macroscopic scales Einstein’s theory of relativity gives a precise and well-tested answer to this question: spacetime geometry and background matter fields obey Einstein’s equations. The structure of spacetime on microscopic scales remains more mysterious, and most approaches to quantum gravity suggest that on scales near the Planck length of 10
-35cm a conventional notion of spacetime geometry based on manifolds of fixed topology and probe objects following geodesics will not yield a useful description. Does this mean that to describe the microscopic structure of spacetime we must abandon our well-honed geometric tools and hard-won insights? In the final reckoning it may well be so. However, the utility of these tools and dearth of substitutes suggest that we do so gradually, identifying which geometric aspects may be safely kept, which modified, and which must be entirely discarded. In string theory, the resulting set of ideas constitutes the notion of “stringy geometry.”

In string theory, the question of spacetime structure receives a new twist since the string length provides an additional scale in the problem. If the Planck length is much smaller than the string length––the regime where the strings are weakly coupled––then strings are a natural set of objects for probing spacetime geometry, and it is sufficient to study a first-quantized formulation of the theory. This formulation is a generalization of the path-integral describing the quantum evolution of a point particle in a fixed background: the world-line is replaced by a two-dimensional space, known as the worldsheet, and the point particle action is replaced by a two-dimensional quantum field theory of maps from the worldsheet to spacetime. When the spacetime is flat, the two-dimensional theory is free; more generally, the strength of quantum corrections is governed by the scale of curvatures of spacetime fields in units of the string length.

Invariance of the theory under worldsheet reparametrizations requires the worldsheet theory to be a conformal field theory with a certain central charge. In super­symmetric string theories, when the spacetime curvatures are small, this implies that the spacetime must be ten dimensional, and the metric and other massless background fields must obey a ten-dimensional generalization of Einstein’s equations. When the curvatures become large, this geometric interpretation must be replaced by properties of abstract conformal field theory. This quantum regime is the realm of stringy geometry. To study this regime quantitatively, it is useful to concentrate on a class of solutions with enough symmetry to make the problem tractable but not so much as to make it trivial. Such a class is provided by minimally super­symmetric four-dimensional compactifications of the heterotic string.


The massless fields of the ten-dimensional heterotic string in a weakly curved background constitute a minimal supergravity coupled to super Yang-Mills theory with gauge group E8xE8 (or SO(32)). To construct a solution with four-dimensional Poincaré invariance, a six-dimensional compact manifold replaces six of the spacetime dimensions. The consistency conditions that restrict the possible gauge groups to two choices also imply that a typical compactification geometry requires a non-trivial background gauge field. Geometrically this amounts to a choice of a vector bundle over the six-dimensional manifold obeying certain topological conditions. The simplest solution that preserves minimal supersymmetry in four dimensions selects the compactification geometry to be a Ricci-flat Kähler manifold with finite fundamental group––a space known as a Calabi-Yau manifold. In this case, the Yang-Mills vector bundle must be set equal to the tangent bundle of the Calabi-Yau space. This construction is known as the “standard embedding.”

As described so far, the standard embedding solution satisfies the string consistency conditions to first order in the background curvature. Does the background continue to make sense in the full quantum worldsheet theory? There is by now overwhelming evidence that it does. The case for this assertion rests on two important properties of the worldsheet theory. The first is (2,2) supersymmetry, a property preserved by quantum corrections. The second is that the solution depends on a number of parameters that describe the size and shape of the Calabi-Yau manifold. These parameters are known respectively as Kähler moduli and complex structure moduli. Quantum corrections depend on the former set of moduli and are independent of the latter set. By tuning the Kähler moduli so that the Calabi-Yau curvature is small in units of the string length, the quantum corrections can be made arbitrarily small; moreover, by using properties of (2,2) supersymmetry and the geometry, it is possible to show that the background can be adjusted to solve the consistency conditions to all orders in perturbation theory. Arguments can be given that extend this result beyond perturbation theory, with one of the strongest based on mirror symmetry, a remarkable feature of stringy geometry.

Mirror symmetry is the property that two topologically distinct Calabi-Yau manifolds M and W can give rise to isomorphic conformal field theories and thus exactly the same heterotic string compactification. Moreover, the isomorphism maps the Kähler moduli of M to the complex structure moduli of W and vice-versa. This exchange allows control of quantum corrections in both the M and W descriptions and provides a clear example of emergent geometry: the string probe does not distinguish between two topologically distinct geometries; the choice of one versus the other is simply one of computational convenience!

A basic lesson for stringy geometry is that while the space-time geometry is an ambiguous notion, the geometry of the Kähler and complex structure moduli spaces remains well defined. The moduli have a natural geometric structure, determined for instance by the Zamolodchikov metric of the conformal field theory. The associated geometric quantities, such as the Riemann curvature, describe the moduli dependence of terms in the effective action governing the scattering of massless string states. The (2,2) supersymmetry ensures that the moduli space splits as a product of Kähler and complex structure moduli spaces and constrains the possible quantum corrections: while the metric on the former receives an infinite sum of non-perturbative corrections, the classical metric on the latter is exact. Mirror symmetry allows both to be determined exactly by using the pair M and W, as opposed to just one of the manifolds.

Although the moduli are a boon for understanding the stringy geometry of the heterotic string, they are also the source for its great shortcoming as a viable phenomenological framework. In the standard embedding scenario, the low energy four-dimensional theory is a minimal super­gravity coupled to supersymmetric gauge theory with gauge group E6xE8 and a chiral spectrum of matter fields charged under the E6 gauge group and neutral under the E8. In addition to these desirable features, each modulus of the worldsheet theory yields in the four-dimensional theory a neutral scalar field with vanishing potential. Such “fifth-force” carriers are largely ruled out by observations.

The last five or six years have seen much effort to find mechanisms by which these undesirable moduli can obtain sufficiently large masses. The work has mostly concentrated on type II superstrings, and notions of stringy geometry and mirror symmetry have played important roles in that program. It would be very useful to apply similar ideas in the context of the heterotic string, where many features of the worldsheet theory are actually under better control than in general type II backgrounds. There is, however, a significant obstacle that must be overcome: we still lack a full description of the heterotic moduli space!


We described the Kähler and complex structure moduli and claimed that (2,2) supersymmetry and mirror symmetry lead to a quantitative description of the moduli space geometry. What is lacking? The trouble is that the background typically has additional moduli associated with the Yang-Mills bundle. Turning on these moduli necessarily renders the six-dimensional metric non-Kähler and breaks half of the (2,2) worldsheet supersymmetry, leaving a less restrictive (0,2) supersymmetry. These moduli are typically not lifted by quantum corrections.

What is the full moduli space? How does mirror symmetry exchange the bundle moduli on M with those on W? What are the implications for stringy geometry? Can the moduli space metric be computed? There is one important reason to suspect that progress can be made on these issues: although the two-dimensional theory seems to be drastically modified by the bundle moduli, the four-dimensional spacetime physics is not much altered: for small but finite deformations the theory retains exactly the same spacetime supersymmetry, gauge group, and spectrum of charged matter fields. A closer look at the worldsheet theory suggests that perhaps the modifications are less drastic than they first appear. In the last few years, a number of groups have been using the experience with (2,2) theories and their various cousins such as gauged linear sigma models, topological field theories, and Landau-Ginzburg theories, to study the more general theories with (0,2) supersymmetry.

Despite being a textbook classic of string theory, heterotic compactification with standard embedding still offers important lessons. The most interesting of these relate to generali­zations of stringy geometry and mirror symmetry. The lessons learned in these reasonably simple examples will have applications to a much wider class of heterotic solutions, and they are likely to shed light on the nature of spacetime from the point of view of the string.