While these axiomatizations were known and thought of highly by a few select
researchers who worked on them, they were mostly happily ignored by the quantum
field theory and string theory community at large, and to a good degree rightly so:
nobody should trust an axiom system that has not yet proven its worth by providing
useful theorems and describing nontrivial examples of interest. But neither the
study of cobordism representations nor that of systems of algebras of observables
could for a long time apart from a few isolated exceptions claim to add much
to the world-view of those who value formal structures in physics, but not a priori
formal structures in mathematics. It is precisely this that is changing now.
Major structural results have been proven about the axioms of functorial quan-
tum field theory (FQFT) in the form of cobordism representations and dually those
of local nets of algebras (AQFT) and factorization algebras. Furthermore, classes of
physically interesting examples have been constructed, filling these axiom systems
with life. We now provide a list of such results, which, while necessarily incom-
plete, may serve to give an impression of the status of the field, and serve to put
the contributions of this book into perspective.
I. Cobordism representations
(i) Topological case. The most foundational result in TQFT is arguably the
formulation and proof [Lur09b] of the cobordism hypothesis [BaDo95] which
classifies extended (meaning: “fully local”) n-dimensional TQFT by the “fully
dualizability”-structure on the “space” of states (an object in a symmetric monoidal
(∞, n)-category) that it assigns to the point. (In this volume the contribution by
Bergner surveys the formulation and proof of the cobordism hypothesis). This
hugely facilitates the construction of interesting examples of extended n-dimensional
TQFTs. For instance
recently it was understood that the state-sum constructions of 3d TQFTs
from fusion categories (e.g. [BaKi00]) are subsumed by the cobordism
hypothesis-theorem and the fact [DSS11] that fusion categories are the
fully dualizable objects in the (∞, 3)-category of monoidal categories with
bimodule categories as morphisms;
the Calabi-Yau A∞-categories that Kontsevich conjectured [Ko95] en-
code the 2d TQFTs that participate in homological mirror symmetry
have been understood to be the “almost fully dualizable” objects (Calabi-
Yau objects) that classify extended open/closed 2-dimensional TQFTs on
cobordisms with non-empty outgoing boundary with values in the (∞, 1)-
category of chain complexes (“TCFTs” [Cos07a], [Lur09b]);
In this context crucial aspects of Witten’s observation in [Wi92] have
been made precise [Cos07b], relating Chern-Simons theory to the effec-
tive target space theory of the A- and B-model topological string, thus
providing a rigorous handle on an example of the effective background
theory induced by a string perturbation series over all genera.
(ii) Conformal case. A complete classification of rational full 2d CFTs on cobor-
disms of all genera has been obtained in terms of Frobenius algebra objects in
modular tensor categories [FRS06]. While the rational case is still “too simple”
for the most interesting applications in string theory, its full solution shows that
already here considerably more interesting structure is to be found than suggested
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