(geometry $\leftarrow$ Isbell duality $\to$ algebra)
functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
An ordinary spectral triple is, as discussed there, the abstract algebraic data characterizing supersymmetric quantum mechanics on a worldline and thereby spectrally encoding an effective (possibly non-commutative, hence “non-geometric”) target space geometry. Ordinary Riemannian geometry with spin structure is the special case of this where the Hilbert space in the spectral triple is that of square integrable sections of the spinor bundle on a spin Riemannian manifold and the operator $D$ acting on that is the standard Dirac operator, hence the “supercharge” of the worldline supersymmetry of the spinning particle.
In generalization of this, a “2-spectral triple” should be the analogous algebraic data that encodes the worldsheet theory of a superstring propagating on a target space geometry which is a generalization of Riemannian geometry with (twisted) string structure.
Of course such data is just that of a 2d superconformal field theory, realized locally by, for instance, a vertex operator algebra or by a conformal net of local observables. But for emphasis it may be useful to speak of such data as constituting a “2-spectral triple”, for emphasizing more the important and intricate relation to the concept of spectral triples, which in much of the literature seems to be unduly ignored.
quantum system | supercharge | formalization | algebra |
---|---|---|---|
quantum spinning particle | Dirac operator | spectral triple | operator algebra |
quantum spinning string | Dirac-Ramond operator | 2d SCFT | vertex operator algebra |
That the 0-mode sector of a 2d SCFT – hence the quantum point particle limit of a quantum superstring dynamics – yields a spectral triple was maybe first highlighted in (Fröhlich-Gawędzki 93) by way of a series of concrete examples, such as the WZW model.
Here the role of the Dirac operator of the spectral triple is played by the Dirac-Ramond operator of the superstring, hence the operator whose index (in the large volume limit) is the Witten genus.
That hence the superstring quantum theory should be regarded as a kind of higher spectral triple was maybe first suggested in (Chamseddine 97), together with arguments that the associated spectral action indeed reproduces the action functional of the string’s target space effective supergravity theory. An exposition of this perspective is in (Fröhlich-Grandjean-Recknagel 97, section 7.2).
From Fröhlich 92, p. 11:
I still have hopes, perhaps romantic ones, that string theory, or something inspired by it, will come back to life again. I believe it is interesting to attempt to formulate string theory in an “invariant” way, quite like it is useful to formulate geometry in a coordinate-independent way. One might, for example, start with a family $\mathcal{F}$, of hyperfinite type $III_1$ von Neumann algebras – to be a little technical – indexed by intervals of the circle with non-empty complement (or of the super-circle). It may pay to formulate the starting point using the language of sheaves. $[...]$ This structure determines a braided monoidal C*-category with unit, …; briefly, a quantum theory. From a combination of such tensor categories (left and right movers) one would attempt to reconstruct (symmetries of) physical space-time. String amplitudes would correspond to arrows (intertwiners) of the tensor category. $[...]$ it would provide a general way of thinking about string theory that does not presuppose knowing the target space-time of the theory.
Later it was shown more formally (Roggenkamp-Wendland 03), reviewed in (Roggenkamp-Wendland 08), that there is a precise algebraically formalization of taking the “point particle limit” of a quantum string, by sending its vertex operator algebra to a spectral triple obtained by suitably retaining only worldsheet 0-modes.
In (Soibelman 11) this was used as a means to systematically study the large volume limit of effective string spacetimes (and hence aspects of the landscape of string theory vacua) by studying the spectral geometries (i.e. the Connes-style noncommutative geometries) of the spectral triples arising from the string’s point particle limit this way.
Now, since there is information lost in passing from a stringy “2-spectral triple” (a 2d SCFT) to its underlying point particle spectral triple, not all spectral triples are to be expected to have a lift to a 2-spectral triple (possibly corresponding to a UV-completion of the corresponding target space effective field theories).
In view of this, it is noteworthy that the spectral triple of the Connes-Lott-Chamseddine model shares a few key properties with the 2d SCFTs considered in string phenomenology:
The Connes-Lott-Chamseddine model is an encoding in a spectral triple of the standard model of particle physics coupled to gravity realized as a kind of spectral Kaluza-Klein compactification on a non-commutative fiber space down to ordinary 4d Minkowski spacetime (or possibly its Wick rotated Euclidean version). In order for this to work out, it turns out that the compactified non-commutative fiber space needs to have KO-dimension equal to $6$. (Here the fiber space is classically just a (“non-commutative”) point, but it appears as the singular collapsing limit of a space of finite dimension. This actual dimension is the KO-dimension.)
Hence the claim of the Connes-Lott-Chamseddine model is that if the standard model of particle physics is encoded as a singular limit of a Kaluza-Klein compactification modeled via a spectral triple then the dimensions of the KK-compactification are
with 4-dimensional base space and 6-dimensional fiber space, to a total of a 10-dimensional spacetime at high energy (after uncompactification of the fiber).
This, of course, is precisely the dimensionality of the target spacetime of perturbative string theory vacua for the critical superstring.
This point was highlighted in Connes 06, p. 8:
When one looks at the table (7.2) of Appendix 7 giving the KO-dimension of the finite space $[$ i.e. the noncommutative KK-compactification-fiber $F$ $]$ one then finds that its KO-dimension is now equal to 6 modulo 8 (!). As a result we see that the KO-dimension of the product space $M \times F$ $[$ i.e. of 4d spacetime $M$ with the noncommutative KK-compactification-fiber $F$$]$ is in fact equal to $10 \sim 2$ modulo 8. Of course the above 10 is very reminiscent of string theory, in which the finite space $F$ might bea good candidate for an “effective” compactification at least for low energies. But 10 is also 2 modulo 8 which might be related to the observations of Lauscher-Reuter 06 about gravity.
Algebraically, this arises from the fact that the BRST complex for the superstring worldsheet theory is consistent (has BRST differential squaring to 0) precisely if the corresponding 2d SCFT has conformal central charge 15, and each spacetime dimension contributes $1 \tfrac{1}{2}$ to this central charge (a contribution of 1 from each bosonic direction, and another $\tfrac{1}{2}$ for the corresponding fermionic contribution).
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 $\sigma$-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). This was further expanded on and used for the mathematical study of the large volume limit of string theory vacua in (Soibelman 11).
An early observation that the 0-mode sector of a 2d SCFT is a spectral triple, demonstrated in a series of concrete examples, is
The suggestion to understand, conversely, the string‘s worldvolume 2d SCFT as a higher spectral triple is due to
which claims to show that the corresponding spectral action reproduces the correct effective background action known in string theory. A more expository account of this perspective is in
A more formal derivation of how ordinary spectral triples arise as point particle limits of vertex operator algebras for 2d SCFTs then appears in
summarized in
A brief indication of some ideas of Yan Soibelman and Maxim Kontsevich on this matter is at
Further development of this and application to the study of the large volume limit of superstring vacua is in
based on
(discussing aspects of homological mirror symmetry).
Analogous detailed discussion based not on the vertex operator algebra description of local CFT but on the AQFT description via conformal nets is in
where 2d SCFTs are related essentially to local nets of spectral triples.
Exposition of these results is in
See also the references at geometric model for elliptic cohomology.
That the Connes-Lott models could be effective field theory-limits of perturbative string theory vacua is also mentioned in
Last revised on January 13, 2021 at 13:19:07. See the history of this page for a list of all contributions to it.