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.

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