Mathematical Structures 97As such the space-time metric gpv is not an invariant concept, but dependent
on the “duality frame.” For example, certain singularities can appear from the
perspective of one kind a brane, but not from another one where the geometry
seems perfectly smooth. So, even to define a space-time we have to split thetotal degrees of freedom in a large source system, that produces an emergent
geometry, and a small probe system, that measures that effective geometry.
It is in no way obvious (and most likely simplywrong) that a suitable theory of quantum gravity can be obtained as a (non-
perturbative) quantization of the metric tensor field gpv(x). The ultimate quan-
tum degrees of freedom are probably not directly related to the usual quanti-ties of classical geometry. One possible direction, as suggested for example in
the case of three-dimensional geometry [15] and the Ashtekar program of loop
quantum gravity [16], is that some form of gauge fields could be appropriate,
possible a pform generalization of that [17]. However, in view of the strong
physical arguments for holography, it is likely that this change of dynamical
variables should entail more that simply replacing the metric field with another
space-time quantum field.0 Alternative variables.4.1.3
Let us now put the mathematical structures in some perspective. For pedagogical
purposes we will consider string theory as a two parameter family of deformations
of “classical” Riemannian geometry. Let us introduce these two parameters heuris-
tically. (We will give a more precise explanation later.)
First, in perturbative string theory we study the loops in a space-time manifold.
These loops can be thought to have an intrinsic length e,, the string length. Because
of the finite extent of a string, the geometry is necessarily “fuzzy.” At least at an
intuitive level it is clear that in the limit eS 4 0 the string degenerates to a point,
a constant loop, and the classical geometry is recovered. The parameter es controls
the “stringyness” of the model. We will see how the quantity e: = a‘ plays the
role of Planck’s constant on the worldsheet of the string. That is, it controls the
quantum correction of the two-dimensional field theory on the world-sheet of the
string.
A second deformation of classical geometry has to do with the fact that strings
can split and join, sweeping out a surface C of general topology in space-time.
According to the general rules of quantum mechanics we have to include a sum over
all topologies. Such a sum over topologies can be regulated if we can introduce a
formal parameter gs, the string coupling, such that a surface of genus h gets weighted
by a factor g2h-2. Higher genus topologies can be interpreted as virtual processes
wherein strings split and join - a typical quantum phenomenon. Therefore the
parameter gs controls the quantum corrections. In fact we can equate g: with
Planck’s constant in space-time. Only for small values of gs can string theory be