In general, a transformation that leaves a system unchanged is called a symmetry, whereas a transformation that takes one system into another is called a transform. Examples of the first kind are the rotation of a homogeneous sphere through an arbitrary angle and about any axis that passes through the sphere's center. Examples of the second kind include the various integral transforms, which generally turn a system of linear differential equations into a corresponding system of algebraic equations.
Superstring compactifications on Calabi-Yau varieties, and more generally on gauged linear sigma models, are distinguished mainly by the effective spacetime field theory that they induce. The possibly measurable features of this theory in turn depend on the moduli space geometry of the Calabi-Yau variety, i.e., the gauged linear sigma model. In turn, this may be determined through a study of supersymmetric vacua the in the underlying 1+1-dimensional field theory of these models, and the related cohomology rings.
Given that the (effective spacetime) ‘physics’ of such models depends in a rather circuitous way on the Calabi-Yau variety, gauged linear sigma model, etc., it is clear that there will in general exist transformations of these ‘building blocks’ which leave the ‘physics’ unchanged. These then are the symmetries of string theory, and even if the transformations include rather surprising relations among Calabi-Yau varieties as is the case with the co-called mirror symmetry.
This latter transformation may in fact be understood as a complex conjugation performed exclusively on the left-moving fields in the underlying 1+1-dimensional field theory. In particular, this leaves the world-sheet (2,2)-supersymmetry algebra unchanged. In this sense, mirror symmetry is in fact a feature of all (2,2)-supersymmetric sigma models, i.e., general supersymmetric sigma models which have two independent supersymmetries both among the left-moving and, independently, also among the right-moving fields.
There exist other discrete transformations that can be performed on the underlying world-sheet field theory, such as a parity reflection, but only on the (un)conjugate fields. That is, the left- and right-moving fields are exchanged, but their complex conjugates are not. This transformation changes the (2,2)-supersymmetry algebra radically, transforming the Hamiltonian and the linear momentum densities into central charges and vice versa. As a consequence, this also changes the type of the string theory, and with it also the effective field theory; it has been identified to underlie the Type IIA <--> Type IIB duality.
Notably, the effect of these discrete transformations may also be achieved through continuous (but not analytic in some suitable sense) transformation. For example, the Type IIA <--> Type IIB duality may also be achievable through a (quite more complicated but continuous) transformation involving the unified ‘master theory’. That is, all the various (generalized) string theories (including also the M-Theory which is certainly not a string theory in the usual sense) may be limits of a universal ‘master theory’, interpolating between them.
In the framework of ‘variable spacetime vacua’ this interpolation may in fact happen as one travels from one point to another, in the actual spacetime, affording thus quite literally a trip into another Universes, at select locations within our own.
In 1+1-dimensional spacetime, the d'Alembertian (wave operator) may be rewritten as a product of two first-order derivatives, with respect to the two characteristic coordinates, x±=(x±t). A function satisfying the wave equation is then equal to fL(x+)+fR(x-) fL and fR being otherwise arbitrary functions of their arguments, The former, fL, is called a left-mover since its graph shifts to the left (negative x) as time passes; by the same token, the latter, fR, is a right-mover. Unlike in higher dimensions, in 1+1-dimensional spacetime, no Lorentz transformation (change of relativistic observer) can turn a fL into fR: they are completely independent. The left-moving fields and the right-moving fields can therefore have different dynamics, features and symmetries.
© Tristan Hübsch, 2008