Ah, now a talk on a fundamental mathematical aspect of string theory.

Leonardo Rastelli is talking about *Distances between Conformal Field Theories*, following proposals by Maxim Kontsevich and Yan Soibelman as laid out and referenced notably in

- Yan Soibelman,
*Collapsing CFTs, spaces with non-negative Ricci curvature and nc geometry* (PSPM 83)

Recall that the basic premise of perturbative string theory is that spacetime together with the background gauge fields on it is encoded in terms of a 2-dimensional CFT that is, or behaves as if it is, the sigma-model that describes the quantum mechanics of single strings propagating in this spacetime, charged under these background fields.

Hence it is of interest to get an idea of “the space of all such 2dCFTs”.

In its entirety, very little is known about this space. One tiny corner that has been described in full detail by mathematical classification theorems is *rational 2d CFT* (this was achieved by *FRS formalism*).

Another tiny corner, consisting of sigma-models on target spaces which are Calabi-Yau fibrations with various assumptions on the background fields, attracted some attention a few years back, because it was a little less tiny than some people had hoped: this has come to be called the *landscape*.

To appreciate the general problem, it is useful to consider its analog for *particle phyiscs*: we may decide that it is a good idea to describe a spacetime with its background fields in terms of the quantum mechanics of a single *particle* propagating on this manifold, and charged under these fields. That quantum mechanics is given by three pieces of data: a Hilbert space of states, an algebra (of smooth functions on spacetime) densely embedded in it, and a certain operator on that space.

Such a triple of data is called a *spectral triple*. The study of target space geometries in terms of their spectral triples is known, somewhat unfortunately, a *Connes-style noncommutative Riemannian geometry* (or some variant of these words). A better term would be *spectral geometry*.

In any case, there is a rather well developed theory of such spectral geometry. The starting point of perturbative string theory is entirely analogous to this, just “lifted in dimension” by one, in a way that is a lot like categorification in various ways: it is like studying the moduli space of 2-spectral triples.

In Yan Soibelman’s article mentioned above the strategy is to turn this around and see what one can say about the 1-spectral geometry that arises from the 2-spectral stringy/CFT geometry in the limit that we assume the string to have vanishing extension.

Now back to Rastelli’s talk, finally: he points out how we are thus looking for analogs and generalizations of distance on moduli spaces of Riemannian metrics. He is scanning through various proposal of what one could define to be a sensible distance between CFTs, and what one knows about each proposal (not much in general).

The main proposal that he says he wants to propose now involves a concept he calls “conformal interfaces”, which is formulated in terms of boundary states for open string CFTs. I should better listen now and stop typing…

## Frenkel: Beyond topological field theory

Edward Frenkel gave a survey of his joint project with A. Losev and Nikita Nekrasov that they have been pursuing for serveral years already.

This starts with the observation is that there are several quantum field theories whose action functional is the sum of a “kinetic” and a “topological” piece, in such a way that the prefactors of those two pieces are usefully regarded as the real and the imaginary part of a single but complex coupling constant. The most famous example is probably Yang-Mills theory (see here) in which case this complex parameter notably controls (or is controled by) S-duality. Another famous example, actually closely related, which they amplify is the $(2|2)$-supersymmetric sigma-model, the superstring in a background field of gravity and B-field. Finally, to bring out the underlying formal structure most clearly, Frenkel and his coauthors observe that one can go down even one more step and still observe this phenomenon in the toy model of supersymmetric quantum mechanics. There is a useful review of their considerations in that simple case by Jacques Distler here.

The content of the program now is to consider the theory in a certain limit of that complex parameter, which is such as to make the possible physical configurations restrict to (“localize” to) only very special “instanton” configurations, which no longer form a huge space, but just a finite-dimensional manifold, over which the path integral is well defined. A famous examples of this is the A-model variant of the above 2-dimensional $\Sigma$-model, where this path integral famously computes the Gromov-Witten invariants of its target space.

The goal of the Frenkel-Losev-Nekrasov program is to go beyond well-known examples such as this one, considering corrections to the full “topological limit”. For instance they get Gromov-Witten invariants next not on base space itself, but on its jet space, and not a topological field theory but a “logarithmic conformal field theory”.

An exposition of the general program is given in

Notes on instantons in topological field theory and beyond(arXiv:hep-th/0702137)Since I see pretty much all the notes that I took in the talk today have their counterpart in that note, I’ll just point to that. A few more references and pointers are listed at the $n$Lab entry on

instanton.