by the naive considerations in much of the physics literature. (The contributions
by Kapustin-Saulina and by Kong in this volume discuss aspects of this.)
(iii) Supergeometric case. There is now a full proof available, starting from
the axioms, that the partition function of a (2|1)-dimensional supersymmetric 2d-
QFT indeed is a modular form, as suggested by Witten’s work [Wi86] on the
partition function of the heterotic string and the index of the Dirac operator on
loop space. (A formalization and proof of this fact in terms of supergeometric
cobordism representations is described in the contribution by Stolz-Teichner to this
volume.) This suggests a deep relationship between superstrings and the generalized
cohomology theory called tmf (for topological modular forms) in a sense, the
universal elliptic cohomology theory which lifts the more familiar relation between
superparticles (spinors) and K-theory to higher categorical dimension. (This is the
content of the contribution by Douglas-Henriques in this volume.)
(iv) Boundary conditions and defects/domain walls. One simple kind of ex-
tra structure on cobordisms that is of profound importance is boundary labels and
decompositions of cobordisms into domains, meeting at domain walls (“defects”).
(The definition of QFT with defects is part of the content of the contribution
by Davydov-Runkel-Kong to this volume). That cobordism representations with
boundaries for the string encode D-branes on target space was originally amplified
by Moore and Segal [MoSe06]. Typically open-closed QFTs are entirely deter-
mined by their open sectors and boundary conditions, a fact that via [Cos07a] led
to Lurie’s proof of the cobordism hypothesis. (A survey of a list of results on pre-
sentation of 2d CFT by algebras of boundary data is in the contribution by Kong
to this volume.)
(v) Holographic principle. A striking aspect of the classification of rational CFT
mentioned above is that it proceeds rigorously by a version of the holographic
principle. This states that under some conditions the partition function and corre-
lators of an n-dimensional QFT are encoded in the states of an (n +1)-dimensional
TQFT in codimension 1. The first example of this had been the holographic re-
lation between 3-dimensional Chern-Simons theory and the 2-dimensional WZW
CFT in the seminal work [Wi89], which marked the beginning of the investigation
of TQFT in the first place. A grand example of the principle is the AdS/CFT con-
jecture, which states that type II string theory itself is holographically related to
super Yang-Mills theory. While mathematical formalizations of AdS/CFT are not
available to date, lower dimensional examples are finding precise formulations. (The
contribution by Kapustin-Saulina in this volume discusses how the construction of
rational 2d CFT by [FRS06] is naturally induced from applying the holographic
principle to Chern-Simons theory with defects).
One of the editors once suggested that, in the formalization by cobordism
representations, holography corresponds to the fact that transformations between
(n + 1)-functors are in components themselves essentially given by n-functors. A
formalization of this observation for extended 2d QFT has been given in [SP10].
(The contribution by Stolz-Teichner to this volume crucially uses transformations
between higher dimensional QFTs to twist lower dimensional QFTs.)
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