(iii) Quantum anomaly cancellation. The cancellation of the quantum anom-
aly of fermions on the superstring’s worldvolume the (differential) class of their
Pfaffian line bundles on the bosonic configuration space imposes subtle conditions
on the background gauge fields on spacetime to which the string couples.
By means of the machinery of generalized differential cohomology, recently
[Bu09] makes fully precise the old argument of Killingback about the worldsheet
version of the celebrated Green-Schwarz anomaly cancellation (the effect that initi-
ated the “First superstring revolution”), using a model for twisted differential string
structures [SSS10] [FSS11] in terms of bundle gerbes, due to [Wa09]. These dif-
ferential string structures controlled by the higher Lie and Chern-Weil theory of
the smooth string 2-group [Hen08][BCSS07] are the higher superstring analogs
in higher smooth geometry [Sch11] of the spin-bundles with connection that control
the dynamics of spinning/superparticles.
(In our volume the contribution by Distler-Freed-Moore presents what is to
date the most accurate description of the conditions on the differential cohomol-
ogy classes of the superstring’s background gauge fields for general orbifold and
orientifold target spaces.)
Taken together, all these developments should go a long way towards under-
standing the fundamental nature of QFT on arbitrary cobordisms and of the string
perturbation series defined by such 2d QFTs. However, even in the light of all
these developments, the reader accustomed to the prevailing physics literature may
still complain that none of this progress in QFT on cobordisms of all genera yields
a definition of what string theory really is. Of course this is true if by “string
theory” one understands its non-perturbative definition. But this supposed non-
perturbative definition of string theory is beyond reach at the moment. Marvelling
with a certain admiration of their audacity at how ill-defined this is has made
the community forget that something much more mundane, the perturbation series
over CFT correlators that defines perturbative string theory, has been ill-defined all
along: only the machinery of full CFT in terms of cobordism representations gives
a precise meaning to what exactly it is that the string pertubation series is a se-
ries over. Perhaps it causes feelings of disappointment to be thrown back from the
realm of speculations about non-perturbative string theory to just the perturbation
series. But at least this time one lands on solid ground, which is the only ground
that serves as a good jumping-off point for further speculation.
In string theory it has been the tradition to speak of major conceptual insights
into the theory as revolutions of the theory. The community speaks of a first and a
second superstring revolution and a certain longing for the third one to arrive can
be sensed. With a large part of the community busy attacking grand structures
with arguably insufficient tools, it does not seem farfetched that when the third
one does arrive, it will have come out of mathematics departments. 1
in this context for instance the opening and closing talks at the Strings 2011 conference.
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