Two Calabi--Yau manifolds are said to constitute a mirror pair if
their corresponding conformally invariant nonlinear sigma models are isomorphic
via a mapping that flips the sign of an eigenvalue of
a 
symmetry contained in the N = 2 superconformal algebra.
Such conformal theories can be used as the internal part of a string model,
and, as we have discussed, accurately describe string physics so long as we
are sufficiently far away from conifold points. We have learned in the above
discussion that even though the conformal field theory description of the
string model
breaks down at conifold points in the moduli space, the string description is
perfectly well behaved. Thus, as in [24], it seems worthwhile to
emphasize the notion of string equivalence:
two geometric
spaces are said to be string equivalent
if they give rise to isomorphic models when
taken as the background spacetime for string theory. Mirror symmetry is
therefore
a special case of string equivalence in which the explicit isomorphism takes
the form noted above. More precisely, this isomorphism leads one to define,
in the context of type II string theory, two Calabi--Yau spaces
as constituting a mirror pair if the type IIA string on the first is isomorphic
to the type IIB string on the second.
Away from conifold points, where
conformal
field theory is valid, this is essentially
equivalent to the standard formulation of mirror
symmetry. (Some care must be taken concerning the integral
structures [25].)
Recall that in [14] a construction of pairs of mirror manifolds was presented which relied crucially on the existence of special points in moduli space at which the associated Calabi--Yau has enhanced discrete symmetries. Nonetheless, this construction was shown to establish the existence of mirror pairs away from such special points through deformation arguments. Namely, if M and W constitute a mirror pair at one point in the moduli space, then we can generate a family of mirrors by deforming, say, the complex structure of M and, correspondingly, the Kähler structure of W (and vice versa). The details of this notion were made precise in [1], but the essential idea is simple: since M and W give isomorphic physics, whatever operation is performed on M has a physically isomorphic description as an operation on W, thereby maintaining the mirror relationship.
This argument requires that the operation, say a deformation of the theory, be smooth; otherwise we lose control and have no basis for drawing any conclusions. For this reason, such deformation arguments have only been used to establish mirror symmetry for a continuously connected (in the sense of conformal field theory) family of Calabi--Yau spaces containing at least one point at which the explicit construction of [14] could be applied. We now learn, from the results of the present work, that we can continue such deformation arguments through conifold transitions since string theory is perfectly well behaved along such a path. Thus, given one point in the web of connected Calabi--Yau spaces at which we can establish the existence of a mirror partner (the construction of [14] gives us many such points) we can use our deformation arguments to establish the existence of mirror symmetry at all points in the web. As almost all known Calabi--Yau manifolds are connected to the web this establishes mirror symmetry for most, or possibly all, Calabi--Yau spaces. (The idea of using conifold transitions to yield a more general mirror construction was proposed some time ago [26]; the present work is the physical realization of that mathematical conjecture.)
Acknowledgements We are grateful to P. Aspinwall, K. Becker, M. Becker, and J. Harvey for useful discussions. The work of B.R.G. was supported by a National Young Investigator Award, the Alfred P. Sloan Foundation, and a grant from the National Science Foundation. The work of D.R.M. was supported in part by the United States Army Research Office through the Mathematical Sciences Institute of Cornell University, contract DAAL03-91-C-0027, and in part by the National Science Foundation through grants DMS-9401447 and PHY-9258582. The work of A.S. was supported in part by DOE grant DOE-91ER40618.