The String Coffee Table

To save some time, as an introduction I simply quote the following summary from a lecture Soibelman has given last October at Vanderbilt University ($\to$).

He will explain the approach to Mirror Symmetry suggested in a joint project with Maxim Kontsevich. The aims is to explain the phenomenon of Mirror Symmetry in terms of homological algebra and non-commutative geometry. Discovered by physicists as a duality on a certain class of string theories, Mirror Symmetry turned out to be related to many deep questions of algebraic and symplectic geometry, algebra, number theory and differential equations. Non-commutative geometry provides an appropriate framework for study of what is called “D-branes” in the String Theory.

It combines physical idea of degenerating Conformal Field Theories with mathematical idea of Gromov-Hausdorff collapse of Calabi-Yau manifolds, as well as with unexpected relation to rigid analytic geometry. We suggest to view a given Conformal Field Theory as a kind of non-commutative space. Such non-commutative spaces can degenerate “at infinity”. Mirror symmetry can be explained in terms of the residual commutative geometry. On the algebraic side we will meet homotopy categories associated with compact symplectic manifolds. I am going to explain non-commutative formal geometry of those homotopy categories. There is another kind of non-commutative geometry of Mirror Symmetry. It is geometry of deformed Calabi-Yau manifolds (a kind of deformation quantization). I plan to discuss the way to construct such spaces starting with real manifolds equipped with an integral affine structure. This part of my lectures is also related to the so-called “tropical geometry”.

In his talk, Soibelman recalled some basics of Alain Connes’ spectral noncommutative metric geometry, in particular the concept of a spectral triple $(A, H, L)$, where $A$ is some algebra represented on the Hilbert space $H$ which carries an unbounded operator $L$ playing the role of the Laplace operator.

He emphasized how, using the GNS construction, we may think of the algebra $A$ as sitting inside $H$, such that the product $A \otimes A \overset{m}{\to} A$ gives rise to a product

on $H$.

This is constructed by the use of a formula that describes essentially something like a tree-level Feynman diagram with a single trivalent vertex. We start with an element of $A \subset H$ in each of the two incoming vertices, propagate them forward using the “euclidean propagator” $\exp(-t L)$, form the product at the vertex and propagate the result again along the outgoing edge. The sought after product is the limit of this procedure as $t$ tends to zero.

The point of this exercise is that 2D conformal field theories do give a realization of this setup - on loop space ($\to$).

The Hilbert space in question is that of states of the (closed) string

the role of the Laplace operator is plaed by the Virasoro generator $L_0 + \bar L_0$ and the operator product expansion provides us with a product on $H$ ($\to$)

Hence we do obtain something like a spectral triple for the geometry of loop space.

In order to find the effective geometry of target space, one can take a certain limit which sends $H \mapsto H_\mathrm{small}$ (nothing but what is known as the point particle limit of strings), which produces a (commutative for $\sigma$- models) algebra $A_0$.

The effective target space of the theory can then be identified with

Furthermore, one can start with the Segal axioms for CFT ($\to$), which tell us how to assign correlators to Riemann surfaces. Passing to the point particle limit these Riemann surfaces degenerate to graphs, and we are left with something that Kontsevich calls field theory on graphs.

As far as I understand this is nothing but what string theorists are essentially doing all along, but certainly Kontsevich gives it a more sophisticated, probably rigorous, form.

Soibelman spent the rest of his talk with discussing technical details of how to measure lengths in noncommtative target spaces. I will not reproduce that.