(geometry Isbell duality algebra)
A 2-spectral triple is the algebraic data encoding the geometry of a Riemannian manifold with string structure.
Analogous to how a spectral triple may be thought of as characterizing the worldline theory of a superparticle propagating on a Riemannian manifold with spin structure, so a 2-spectral triple is essentially nothing but the algebraic description of a 2-dimensional SCFT describing the worldvolume theory of the fundamental super-1-brane (the string) on a target with string structure.
There are several candidate formalizations of 2-spectral triple. One that has achieved a certain formal maturity is indicated at (2,1)-dimensional Euclidean field theories and tmf.
There is at least evidence that there is a continuous path in the space of 2-spectral triples that starts and ends at a point describing the ordinary geometry of a complex 3-dimensional Calabi-Yau space but passes in between through a 2-spectral triple/2d SCFT (a Gepner model?) which is not the -model of an ordinary geometry, hence which describes “noncommutative 2-geometry” (to borrow that terminology from the situation of ordinary spectral triples). This is called the flop transition (alluding to the fact that the geometries at the start and end of this path have different topology).
Early attempts to understand the string’s worldvolume CFT as a higher spectral triple appear in secton 7.2 of
Similar considerations are in
which claims to show that the corresponding spectral action reproduces the correct effective background action known in string theory.
A detailed derivation of how ordinary spectral triples arise as point particle limits of vertex operator algebras for 2d CFTs:
A summary of this is in
Also
A brief indication of some ideas of Yan Soibelman and Maxim Kontsevich on this matter is at
Details are in
See also the references at geometric model for elliptic cohomology.