**THE
STRINGY GEOMETRY OF THE HETEROTIC STRING**

**THE
NOTION OF STRINGY GEOMETRY**

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^{-35}cm
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.

**HETEROTIC
STRING BACKGROUNDS AND THEIR MODULI**

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 E_{8}xE_{8}
(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 E_{6}xE_{8}
and a chiral spectrum of matter fields charged under the E_{6}
gauge group and neutral under the E_{8}.
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!

REDUCING
WORLDSHEET SUPERSYMMETRY

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.