94 The Quantum Structure of Space and Time
4.1.2.1 String theory and mathematics
First of all, it must be said that the subject of Quantum Field Theory (QFT) is
already a powerful source for mathematical inspiration. There are important chal-
lenges in constructive and algebraic QFT: for example, the rigorous construction
of four-dimensional asymptotically free non-abelian gauge theories and the estab-
lishment of a mass gap (one of the seven Millennium Prize problems of the Clay
Mathematics Institute [7]).
Even in perturbative QFT there remain many beautiful mathematical structures
to be discovered. Recently, a surprisingly rich algebraic structures has been discov-
ered in the combinatorics of Feynman diagrams by Connes, Kreimer and others,
relating Hopf algebras, multiple zeta-functions, and various notions from number
theory [8]. Also the reinvigorated program of the twistor reformulation of (self-
dual) Yang-Mills and gravity theories should be mentioned [9]. This development
relates directly to the special properties of so-called MHV (maximal helicity vio-
lating) amplitudes and many other hidden mathematical structures in perturbative
gauge theory [lo].
In fact, a much deeper conceptual question seems to underlie the formulation
of QFT. Modern developments have stressed the importance of quantum dualities,
special symmetries of the quantum system that are not present in the classical
system. These dualities can relate gauge theories of different gauge groups (e.g.
Langlands dual gauge groups in the N = 4 supersymmetric Yang-Mills theory ill])
and even different matter representations (Seiberg duality [12]). All of this points
to the conclusion that a QFT is more than simply the quantization of a classical
(gauge) field and that even the path-integral formulation is at best one particular,
duality-dependent choice of parametrization. This makes one wonder whether a
formulation of QFT exist that is manifest duality invariant.
Although the mathematical aspects of quantum field theory are far from ex-
hausted, it is fair to say, I believe, that the renewed bond between mathematics
and physics has been greatly further stimulated with the advent of string theory.
There is quite a history of developing and applying of new mathematical concepts
in the “old days” of string theory, leading among others to representations of Kac-
Moody and Virasoro algebras, vertex operators and supersymmetry. But since
the seminal work of Green and Schwarz in 1984 on anomaly cancellations, these
interactions have truly exploded. In particular with the discovery of Calabi-Yau
manifolds as compactifications of the heterotic strings with promising phenomeno-
logical prospectives by the pioneering work of Witten and others, many techniques
of algebraic geometry entered the field.
Most of these developments have been based on the perturbative formulation
of string theory, either in the Lagrangian formalism in terms of maps of Riemann
surfaces into manifolds or in terms of the quantization of loop spaces. This pertur-
bative approach is however only an approximate description that appears for small
values of the quantization parameter.