The String Coffee Table

Segal defined a 2D conformal field theory to be a certain functor from the category $\hat M$ of Riemannian surfaces to (topological) vector spaces.

When we pass from a conformal theory to a topological theory two things happen:

1) Riemannian surfaces are replaced by topological surfaces.

(That’s the obvious one.)

2) The space of states receives the structure of a complex, whose differential is the nilpotent operator $Q$, for instance that obtained by twisting a conformal supercharge.

(That’s a little more subtle.)

This means that both the source but also the target category need to be changed. Instead of vector spaces, the target should now be a the category of vector spaces equipped with the structure of a complex. Morphisms here are chain maps of complexes.

That’s again rather obvious. The non-obvious part, which is the crucial one for the definition of TCFT, is the re-definition of the source category.

The ‘correct’ way to do this has apparently been suggested independently by Segal and by Getzler, inspired possibly by remarks made by Kontsevich:

E. Getzler
Batalin-Vikovisky Algebras and Two-Dimensional Topological Field Theories
hep-th/9212043

G. Segal,
Stanford Lectures (1999)
(available here,
see in particula lecture 5)

One can apparently demonstrate (though I could’t sketch the way how to do this right now) that the path integral of a topological 2D theory obtained from twisting a conformal theory defines certain differential forms on moduli spaces. More precisely, when evaluated on surfaces with $m$ incoming and $n$ outgoing (closed) states the functional integral encodes a differential form $\omega(m,n)$ on the moduli space of Riemannian surfaces with values in the complex of homomorphisms from the complex of the ingoing states $E(m)$ to the complex of outgoing states $E(n)$, i.e.

One can then regard such differential forms as morphisms that replace the Riemannian surfaces of the CFT, retaining only their topological (cohomological) information.

Hence one constructs a new category, $C(M)$ whose objects are finite sets (the sets of incoming and/or outgoing states) and whose morphisms are differential forms $\omega$ on moduli space as above.

Denoting the category of chain complexes (over $\mathrm{Vect}$) as $E(\mathrm{Vect})$ we hence get the following

Definition: A 2D TCFT is a tensor functor

Being a functor, such a TCFT lives in a functor catgeory $\mathbf{TCFT}$. One of the contributions of Kevin Costello is a better understanding of the nature of this category $\mathbf{TCFT}$.

In

K. Costello
Topological Conformal Field Theories and Calabi-Yau Categories
math.QA/0412149

it is shown, among other things, that $\mathbf{TCFT}$ is equivalent to the catgeory of what are called (extended) Calabi-Yau $A_\infty$ categories.

A Calabi-Yau category is to a Frobenius algebra like a category is to a monoid: the composition operation in a category with a single object can be regarded as the multiplication operation in a monoid (e.g. in an algebra). Allowing for more than one object sort of generalizes the concept of an algebra to that of - a category.

Now, one way to characterize a Frobenius algebra is to say that it is an algebra $A$ with a non-degenerate ‘trace’ operation

(Here $k$ denotes the ground field we are working on). So if we think of $A$ as the space of morphisms of a category with a single object $\bullet$, we’d have a map

In this sense it is straightforward to say what a Frobenius structure on any (linear) category $C$ should be. We define it to be a set of maps

for all $x \in \mathrm{Obj}(C)$, satisfying some non-degeneracy property.

That defines a Calabi-Yau category. At this point, it would probably be more natural to call this a Frobenius category. The reason for invoking Calabi and Yau is of course due to the application this category finds in TCFT…

The theorem above in fact involved not ordinary CY-categories, but their $A_\infty$-version. An $A_\infty$-category is to an ordinary category like an $A_\infty$-algebra is to an ordinary algebra, where an $A_\infty$-algebra is an algebra in which associativity only holds up to specified homotopy which satisfies a condition up to higher homotopy and so on, ad infinitum. (For details see for instance section 6.1 in Kostello’s paper.)

Now, CY-categories appear naturally as categories of complexes of topological D-branes in Calabi-Yau manifolds $X$. These branes are described (I, II) by objects in $D^b(X)$ the bounded derived category of coherent sheaves on $X$. And $D^b(X)$ happens to be a Calabi-Yau category.

As Kevin Costello discusses in section 1.3 of his paper, one would hence expect that an open TCFT describes the topological $B$ model which gives rise to these branes. But this cannot be quite right, for instance because $D^b(X)$ is not $A_\infty$, while $\mathbf{TCFT}$ is. Some alternative interpretations are discussed, but remain conjectural.

There is much, much more content in Kevin Costello’s paper(s). But for now I’ll leave it at that.