The Quantum Structure of Space and Time (293 pages)

Mathematical Structures 109

through the formalism of D-branes [22] these can be analyzed exactly in string

perturbation theory. D-branes give contribution that are of order

e--llgs

and therefore complement the asymptotic string perturbation series.

Gauge theory. These D-branes are described by non-abelian gauge theories and

therefore by definition non-commutative structures. This suggests that an al-

ternative formulation of string theory makes use of non-commutative variables.

These gauge-gravity dualities are the driving force of all recent progress in string

theory [23].

Extra dimensions. As we stressed, the full quantum amplitudes F depend on

many parameters or moduli. Apart from the string coupling gs all other moduli

have a geometric interpretation, in terms of the metric and B-field on X. The

second new ingredient is the insight that string theory on X with string coupling

gs can be given a fully geometric realization in terms of a new theory called

M-theory on the manifold X x S1, where the length of the circle S1 is gs [24].

Summarizing, the moduli space of string theory solutions has a structure that

in many aspects resembles the structures that described moduli of CFT’s. In this
case there are S-dualities that relate various perturbative regimes.

I ..

self-dual theories

1 Pert. String Theory

-.*-

Dual Pert. String Theory

S-Dualities: g, e 1/g,

The moduli space of string theory vacua.

4.1.4.2

The way in which quantum geometry can emerge from a non-perturbative comple-

tion of a perturbative string theory can be nicely illustrated by a topological string

example.

Topological strings and quantum crystals