The String Coffee Table

Recall the standard scenario for topological strings:

Ordinary, conformally invariant, strings on a Calabi-Yau background $X$ feature an $N=2$ superconformal worldsheet symmetry. By redefining the supercharges of this theory in two different ways one obtains two different 2D topological field theories, the A-model and the B-model. Investigating the boundary conditions for open topological strings of this kind shows that they are in 1-1 correspondence with

- for the A-model: objects in the Fukaya category of Lagrangian submanifolds of $X$, enriched by ‘coisotropic branes’

- for the B-model: objects in the derived category of coherent sheaves over $X$.

Specifically, this means that a brane for the B-type topological string is defined by a complex

of coherent sheaves, hence

See this lightning summary for how this works in detail.

Now the crucial point. One can perform similar topological twists with $N=2$ QFTs in 2D which are not conformally invariant. One example for this are Landau-Ginzburg models (LG). This are certain 2D susy QFTs with a superpotential. Since everything under the sun is related, these LG models know one or two things about ordinary strings, too.

For one, there is a speculation that for certain choices of superpotential the LG model describes aspects of ordinary strings in RR backgrounds. Better yet, the version of mirror symmetry which applies to topologically twisted LG strings relates them to other QFTs, like sigma models on a Fano variety, which are conformal, even though only at the classical level. Generally, LG models lend themselves to calculations, and hence people calculate.

Given the deep (albeit partial) category-theoretic understanding of ordinary A- and B-topological strings and their mirror symmetry, the natural question to ask is how this picture changes when one passes to LG models.

Apparently before any physicist had any idea what the answer (or even the question) could be, the mathematician Kontsevich proposed a solution. He somehow figured out that B-branes in the LG model are described by twisted complexes of coherent sheaves, instead of ordinary complexes. In a ‘twisted complex’ the differential is not nilpotent, but squares to something proportional to the identity

The constant of proportionality here is precisely the superpotential of the LG model!

As far as I understand Maxim Kontsevich did not publish this insight, but told Dmitri Orlov about it. Orlov told Anton Kapustin and Yi Li. And these two finally figured out the physical derivation of Kontsevich’s insight:

A. Kapustin & Yi Li
D-Branes in Landau-Ginzburg Models and Algebraic Geometry
hep-th/0210296

Of course, physically, the answer is in the BRST charge. The ordinary derivation of the fact that a B-brane is derived by a complex of sheaves goes roughly like this:

There is the BRST charge $Q$ acting on open (topological) string states that stretch between manifold-like branes with fiber bundles over them. Turning on more general branes corresponds to inserting a boundary deformation of the theory, which means the BRST operator is deformed by a vertex operator $d$ as

It is required that $[Q,d] = 0$ which implies that in order for $Q_d^2 = 0$ to be true we need $d^2 = 0$. But $d$ decomposes as $d = \sum_n d_n$, which implies that the $d_n$ are maps forming a complex as above. (For more and more precise details follow the link I gave above.)

Now, for the LG model one finds that there are certain boundary terms one needs to introduce in order to preserve the supersymmetry for the open LG string. These terms imply that we no longer have $[Q,d] = 0$ for the deformation $d$. And hence $d^2 = 0$ no longer holds, which means that $d$ is a boundary condition defined by one of Kontsevich’s ‘twisted’ complexes.

In tractable models one translates the study of such twisted complexes to the study of pairs of matrices $E$ and $J$ such that $EJ + JE = W \mathrm{Id}$. That’s the reason why Wolfgang Lerche is talking about Matrix Factorization and open Topological Strings. In fact, there seems to be a category of matrix factorizations, and this category is equivalent to a category of coherent sheaves describing LG B-model branes.

And this is where the stuff begins which is really new and interesting…

And this is also where I am running out of time (and expertise, to be honest).

Robert said he might write a little more about this.