String Theory and M-Theory

412 String geometry
versa. An early indication of mirror symmetry was that the space of thousands
of string theory vacua appears to be self-dual in the sense that if a
Calabi–Yau manifold with Hodge numbers (h 1,1 , h 2,1 ) exists, then another
Calabi–Yau manifold with flipped Hodge numbers (h 2,1 , h 1,1 ) also exists.
The set of vacua considered were known to be only a sample, so perfect
matching was not expected. In fact, a few examples in this set had no
candidate mirror partners. This was shown in Fig. 9.1.
These observations lead to the conjecture that the type IIA superstring
theory compactified on M is exactly equivalent to the type IIB superstring
theory compactified on W . This implies, in particular, an identification of
the moduli spaces:
M 1,1 (M) = M 2,1 (W ) and M 1,1 (W ) = M 2,1 (M). (9.162)
This is a highly nontrivial statement about how strings see the geometry of
Calabi–Yau manifolds, since M and W are in general completely different
from the classical geometry point of view. Indeed, even the most basic
topology of the two manifolds is different, since the Euler characteristics are
related by
χ(M) = −χ(W ). (9.163)
Nonetheless, the mirror symmetry conjecture implies that the type IIA theory
compactified on M and the type IIB theory compactified on W are dual
descriptions of the same physics, as they give rise to isomorphic string theories.
A second, and genuinely different, possibility is given by the type IIA
theory compactified on W , which (by mirror symmetry) is equivalent to the
type IIB theory compactified on M.
Mirror symmetry is a very powerful tool for understanding string geometry.
To see this note that the prepotential of the type IIB vector multiplets
is independent of the Kähler moduli and the dilaton. As a result, its dependence
on α ′ and gs is exact. Mirror symmetry maps the complex-structure
moduli space of type IIB compactified on W to the Kähler-structure moduli
space of type IIA on the mirror M. The type IIA side does receive corrections
in α ′ . As a result, a purely classical result is mapped to an (in
general) infinite series of quantum corrections. In other words, a classical
computation of the periods of Ω in W is mapped to a problem of counting
holomorphic curves in M. Both sides should be exact to all orders in gs,
since the IIA dilaton is not part of M 1,1 (M) and the IIB dilaton is not part
of M 2,1 (W ).
Let us start by discussing mirror symmetry for a circle and a torus. These
simple examples illustrate the basic ideas.