In ten dimensions, there is a small number of consistent string theories. When first discovered, this near-uniqueness raised the hope that, despite the inherent difficulties in extrapolating from the Planck scale to the weak scale, it might be possible to obtain testable predictions from string theory. This hope was greatly diminished by the discovery of a plethora of four-dimensional string vacua. In order to compute the low-lying spectrum and couplings it is apparently first necessary to choose among many thousands of Calabi--Yau spaces (or more general conformal field theories). The process of compactification appeared to ruin the uniqueness of ten-dimensional string theory, along with the predictive power it entailed.
In this paper we will argue that the situation is in fact much better than it appears. In the context of type II strings, we will see that many, and possibly all, of these Calabi--Yau vacua are in fact different branches of a vastly larger ``universal'' moduli space. (A previous step in this direction was taken in [1,2], where string compactifications which are topologically distinct as Calabi--Yau spaces but not as conformal field theories were smoothly connected.) The branches are connected in a smooth and calculable manner by black hole condensation which can occur at ``conifold'' points of the moduli space. This condensation can not be described in the language of conformal field theory, and so is not constrained to preserve quantities such as the number of light generations which are topological invariants of conformal field theory. Thus the number of distinct four-dimensional string vacua is much smaller than previously suspected. Indeed, it is conceivable that there is a unique four-dimensional string vacuum.
Of course even if the goal of tying together all type II string vacua is realized, we are still quite far from making testable predictions. One must extend these ideas to theories with N=1 supersymmetry in four dimensions, and then understand how a superpotential is generated which lifts the continuous vacuum degeneracy and breaks N=1 down to N=0. Even then there is no guarantee that there is a unique or small number of vacua. Nevertheless we feel the time may be ripe for progress on all these problems.
It has long been known in the mathematics literature [3,4,5,6] that it is possible to travel from one Calabi--Yau to another by degenerating certain three-cycles and then blowing them back up as two-cycles, thereby changing the Hodge numbers. This process enables one to glue together Calabi--Yau moduli spaces along the subspaces corresponding to conifolds. Indeed it has been conjectured [7] that all Calabi--Yau spaces are connected in this fashion. The relevance of these results to string theory was advocated in a series of prescient papers [8,9,10,11]. However the construction, at the level of both mathematics and conformal field theory, is singular, and no proposal was made for how string theory might physically implement the transition from one Calabi--Yau to its neighbor. The purpose of the present work is to describe such a mechanism.
The key to understanding this transition is in a recent resolution [12] of simple conifold singularities in type II string theory. Near a conifold, the moduli space metric becomes singular. At the same time, string theory contains black hole hypermultiplets which are degenerating to zero mass. It was shown [12] that the Wilsonian effective field theory including the light black holes is smooth near the conifold, and that integrating out the light black holes reproduces the singularity.
The singularities of the conifolds which glue together Calabi--Yau moduli
spaces are more
complicated than
the simple type analyzed in [12]. In this paper we shall again find that these
singularities are
resolved by light
black hole hypermultiplets, but there are in general many such hypermultiplets.
The
potential V for these hypermultiplets is determined by N=2 supersymmetry.
V has flat directions, along which the black holes can condense and give
masses to
vector multiplets. In this way one discovers a new branch of the moduli space
with
different
numbers of hypermultiplets and vector multiplets. In a IIB string theory the
number of massless
vector multiplets (hypermultiplets) is 
(
) and this new
branch
corresponds to a topologically distinct Calabi--Yau space. In this paper we
analyze
only the
transition from the quintic in 
with Hodge numbers
, to a variety in 
,with Hodge numbers
, but our construction clearly generalizes.
Further analysis will appear in a forthcoming publication [13].
Our construction has a number of implications that are worth emphasizing at the outset. First, as mentioned, type II string vacua which were previously thought to be disjoint are now seen to fit together into a connected web with string physics smoothly interpolating from one component to another. Second, as we move along such an interpolating path, the spacetime background of our string theory undergoes a drastic change in topology. Unlike the spacetime topology change of [1,2], in which Hodge numbers stay fixed while more subtle topological invariants (such as the intersection form) change, here we find that string theory is perfectly smooth even as the Hodge numbers jump. Third, using these results we can vastly extend the mirror symmetry construction of [14]. Namely, on general grounds, once one proves the existence of mirror symmetry for one point in a given moduli space, via deformation arguments one can conclude the existence of mirror symmetry at all other points lying in the same connected component. This was used in [14], for instance, to argue for mirror symmetry throughout a Calabi--Yau moduli space so long as it contains a minimal model point at which an explicit mirror partner exists. Now we see that our deformation arguments are not limited to a single Calabi--Yau moduli space, but rather extend to all Calabi--Yau manifolds connected by conifold transitions; the latter includes essentially all known Calabi--Yau manifolds. (In [9,10] it was established that all simply-connected Calabi--Yau manifolds known at that time could be connected in this way. (It is easiest to deal with multiply connected Calabi--Yau's by working with their covering spaces.) Since the time of [9,10] a number of new constructions of Calabi--Yau manifolds have been proposed, and while it is clear that most of these can be connected, no systematic study has been made of this question.) Finally, our construction provides food for thought on the fascinating interplay between strings and black holes. A degenerating black hole hypermultiplet is reinterpreted as a fundamental string excitation (corresponding to a modulus) after crossing the transition. Thus there can be no fundamental distinction between strings and black holes: they smoothly transform into one another.