So far the story is similar in spirit, although more involved in detail, to the
case of the
simple conifold with a single vanishing cycle considered in [12]. However
inspection of
the effective theory near the conifold reveals a dramatic new feature.
Each hypermultiplet 
contains two charged
complex scalars which we
denote 
where 
is the global 
index of our
N = 2 representation.
This gives a total of 32 complex scalar fields.
Supersymmetry
implies a potential for these scalar fields of the general form
[16,21]
(taking the vevs of the scalar components of the vector multiplets to
vanish)
where 
is a positive definite matrix [16,21] and
This potential has a flat direction along which
the three independent components of 
vanish,
This gives 45 real constraints on the 32 complex fields. In addition there are 15 gauge transformations which rotate the fields, leaving 4 real vacuum parameters. Up to a gauge transformation the general solution of 25 in the basis 14 is
for any complex two vector v. Moving along the flat direction, the black holes condense and we see that their moduli space is parametrized by a single hypermultiplet. The conifold point in the space of quintics (at which all 16 cycles vanish) corresponds to v = 0. Moving away from this point along the flat direction corresponds to giving a vev to the charged hypermultiplets. This vev breaks all 15 U(1)'s. Thus we have discovered a second branch of the moduli space corresponding to a charged black hole condensate. This branch has 101-15=86 massless vector multiplets, and 2+1=3 massless hypermultiplets.
Compactification of IIB string theory on a Calabi--Yau with
leads to 86 vector multiplets and 3 hypermultiplets. A space with precisely
these Hodge
numbers arises if the singular conifold with 16 degenerate cycles is resolved
with a 
, as discussed in section 3.
It is natural to identify the new branch of
the moduli space discovered in the analysis of the
Wilsonian effective field theory at the conifold with
compactification on the
Calabi--Yau. Further evidence for this identification can be obtained
by analyzing the behavior of the theory from the 
side and will
be presented in [13].
While analysis of the general case will be deferred to later work,
one salient feature is worth mentioning. A general conifold has
P vanishing cycles which obey Q homology relations of the type
6. This implies P degenerating black hole hypermultiplets
which carry charge with respect to P-Q U(1)s. The generalization of the
potential 23 will then have Q flat directions, since there are
P-Q equations of the form 25 for P hypermultiplets. Generically all
P-Q U(1)s will be broken along these flat directions. Hence the new branch
of the moduli space will have P-Q fewer vector multiplets and Q more
hypermultiplets, corresponding to a Calabi-Yau space with Hodge numbers
. This counting agrees precisely with one made
using the algebraic geometry methods of
[3,22,23]
to compute the Hodge numbers of a Calabi-Yau space
obtained by degenerating and blowing up cycles.
