4 HISHAM SATI AND URS SCHREIBER

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.)

4