10 HISHAM SATI AND URS SCHREIBER
3. Collapsing Conformal Field Theories, spaces with non-negative Ricci
curvature and non-commutative geometry – by Yan Soibelman.
The premise of perturbative string theory is that every suitable 2d (super-)CFT
describes the quantum sigma model for a string propagating in some target space
geometry, if only we understand this statement in a suﬃciently general context of
geometry, such as spectral noncommutative geometry. In this article the author
analyzes the geometries induces from quantum strings in the point-particle limit
(“collapse limit”) where only the lowest string excitations are relevant. In the limit
the algebraic data of the SCFT produces a spectral triple, which had been shown by
Alain Connes to encode generalized Riemannian geometry in terms of the spectrum
of Hamiltonian operators. The author uses this to demonstrate compactness results
about the resulting moduli space of “quantum Riemann spaces”.
4. Supersymmetric ﬁeld theories and generalized cohomology – by Stephan
Stolz and Peter Teichner.
Ever since Witten’s derivation of what is now called the Witten genus as the
partition function of the heterotic superstring, there have been indications that su-
perstring physics should be governed by the generalized cohomology theory called
topological modular forms (tmf) in analogy to how super/spinning point particles
are related to K-theory. In this article the authors discuss the latest status of their
seminal program of understanding these cohomological phenomena from a system-
atic description of functorial 2d QFT with metric structure on the cobordisms.
After noticing that key cohomological properties of the superstring depend only
on supersymmetry and not actually on conformal invariance, the authors simplify
to cobordisms with flat super-Riemannian structure, but equipped with maps into
some auxiliary target space X. A classiﬁcation of such QFTs by generalized co-
homology theories on X is described: a relation between (1|1)-dimensional flat
Riemannian ﬁeld theories and K-theory and between (2|1)-dimensional flat Rie-
mannian ﬁeld theories and tmf.
5. Topological modular forms and conformal nets – by Christopher Douglas
and Andr´ e Henriques.
Following in spirit the previous contribution, but working with the AQFT-
description instead, the authors of this article describe a reﬁnement of conformal
nets, hence of 2d CFT, incorporating defects. Using this they obtain a tricategory
of fermionic conformal nets (“spinning strings”) which constitutes a higher analog
of the bicategory of Cliﬀord algebras. Evidence is provided which shows that these
categoriﬁed spinors are related to tmf in close analogy to how ordinary Cliﬀord
algebra is related to K-theory, providing a concrete incarnation of the principle by
which string physics is a form of categoriﬁed particle physics.
Acknowledgements. The authors would like to thank Arthur Greenspoon for his
very useful editorial input.