Consider the simple example of a circle defined as the locus of common solutions to the system of equations { z=h , x2+y2+z2=R2 }.
The space of allowed values of (h,R) is the parameter space of this sphere, wherein each point (choice of the h,R pair) represents a potentially different sphere. Note that the ‘true’ radius of the sphere satisfies r2=R2-h2, and so becomes imaginary when R < h. Furthermore, only positive values of R make sense, but h can be negative. Thus, on a diagram where h is plotted horizontally and R vertically, the allowed values of (h,R) take place in an upside-down wedge with sides at 45° and the vertex at the h=0=R origin. This is the parameter space of the sphere as the above complete intersection.
Note furthermore that the choices (-h,R) and (h,R) represent the same sphere, only the first one is centered at height h below z=0, while the other one is at the same height above. If we are unconcerned with the placement of the sphere with respect to the z=0 plane, then these points need to be identified. Furthermore, all the points in the (h,R) region for which r2=R2-h2 is constant (parabolas) ought to be identified; let us denote this identification by ‘~’. Then, the (non-redundant, complete) moduli space of our sphere becomes the quotient M=(h,R)/~. The Reader is invited to verify that this is equivalent to R+, the non-negative semi-axis of real numbers, and may well be parametrized by r itself.
Clearly, cases of real interest (gauged linear sigma models) are incomparably more complicated, so that the parametrization of the appropriate moduli spaces is far from this trivial, easy or even possible in a closed form. Locally around any smooth point, they are well approximated by cohomology rings, where the ring structure (dictated by the multiplication!) provides a good deal of information about the curvature of the moduli space.
Elements of the cohomology rings correspond to supersymmetric vacua in the underlying world-sheet field theory, the correlation functions of with, in turn, determine the couplings in the effective spacetime theory. Thus, the geometry of the moduli space determines the dynamics of the effective spacetime fields, the expectation values of which are, in turn, the parameters of the string theory.
In particular, then, the mirror and duality transforms act both on the moduli space and also on the effective spacetime field theory, but also on the coupled systems of partial differential equations the solutions of which are the supersymmetric vacua. In turn, the features of this system of PDEs reflects on the geometry of the moduli space(s) and the dynamics in the effective field theory, and so also on the geometry of the effective spacetime in the framework of variable spacetime vacua.
© Tristan Hübsch, 2008