January 10, 2006

TCFT, Part I

Posted by Urs Schreiber I should finally begin to learn a little bit about what goes under the curious name topological conformal field theory (TCFT). Kevin Costello had already pointed me to his work on TCFT last year, and, more recently, Aaron Bergman has emphasized the relevance of this work here.

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

(1)$\mathrm{CFT}:\stackrel{̂}{M}\to \mathrm{Vect}\phantom{\rule{thinmathspace}{0ex}}.$

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 $\omega \left(m,n\right)$ on the moduli space of Riemannian surfaces with values in the complex of homomorphisms from the complex of the ingoing states $E\left(m\right)$ to the complex of outgoing states $E\left(n\right)$, i.e.

(2)$\omega \left(m,n\right)\in {\Omega }^{•,•}\left(\stackrel{̂}{M}\left(m,n\right),\mathrm{Hom}\left({E}^{\left(m\right)},{E}^{\left(n\right)}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

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, $C\left(M\right)$ 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 $E\left(\mathrm{Vect}\right)$ we hence get the following

Definition: A 2D TCFT is a tensor functor

(3)$\mathrm{tcft}:C\left(M\right)\to E\left(\mathrm{Vect}\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

In

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

it is shown, among other things, that $\mathrm{TCFT}$ 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

(4)$\mathrm{tr}:A\to k\phantom{\rule{thinmathspace}{0ex}}.$

(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 $•$, we’d have a map

(5)$\mathrm{tr}:\mathrm{Hom}\left(•,•\right)\to k\phantom{\rule{thinmathspace}{0ex}}.$

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

(6)${\mathrm{tr}}_{x}:\mathrm{Hom}\left(x,x\right)\to k$

for all $x\in \mathrm{Obj}\left(C\right)$, 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 ${D}^{b}\left(X\right)$ the bounded derived category of coherent sheaves on $X$. And ${D}^{b}\left(X\right)$ 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 ${D}^{b}\left(X\right)$ is not ${A}_{\infty }$, while $\mathrm{TCFT}$ 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.

Posted at January 10, 2006 3:30 PM UTC

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Re: TCFT, Part I

One can apparently demonstrate (though I couldn’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.

Very briefly, in a TFT amost all calculations boil down to zero mode calculations. In effect, everything “localizes” on the space of bosonic zero modes. The Grassmann zero modes now get interpreted as sections of bundles over the space of bosonic zero modes, and the BRST operator acts as an exterior derivative or $\overline{\partial }$ (depending upon context) living on that bosonic zero mode moduli space, which you can see by looking at how it acts on the fields.

As a result, when one builds BRST-invariant operators in the theory, eg: ${\psi }_{1}...{\psi }_{n}{\lambda }_{1}...{\lambda }_{n}$ they’ll get interpreted as (possibly bundle-valued) differential forms on the moduli space. Typically $\psi ↔\mathrm{dz}$, etc.

This is an incredibly useful trick for, for ex, computing anything.

Ex: If you want to count BRST-closed states mod exact states, on the face of it it’s an ugly mess, but after you apply the dictionary above it becomes a trivial math problem.

It’s one of these things that, when you work through the details, it’s more or less clear.

Also, this intuition applies to TFT’s in any dimension, not just 2d, and not necessarily TCFT’s.

Posted by: Eric Sharpe on January 11, 2006 5:28 PM | Permalink | Reply to this

Re: TCFT, Part I

I know how Grassman zero modes tend to look like differential forms, but I still felt unsure about that particular statement which I cited. Probably it’s obvious once I really think about it.

Posted by: Urs on January 11, 2006 5:30 PM | Permalink | Reply to this

Re: TCFT, Part I

I know how Grassman zero modes tend to look like differential forms, but I still felt unsure about that particular statement which I cited. Probably it’s obvious once I really think about it.

That’s exactly it – once you work through the details, you’ll slap yourself. :-)

Actually there’s a good readable short account in Witten’s old paper from ‘91 or so called “Mirror manifolds and topological field theory.”

It’s an introduction to the A and B models in 2d, written for mathematicians, and it shows explicitly how this works.

After you’ve seen how the RR states in the A, B models are described in terms of differential forms, the next thing to look at is an old paper by Distler & Greene (Brian, the same one who gets on TV) called “Aspects of (2,0) compactifications.”

In the middle of that is an analysis of state-counting in heterotic compactifications, and you’ll see bundle-valued differential forms (sheaf cohomology) appearing there.

Posted by: Eric Sharpe on January 11, 2006 5:32 PM | Permalink | Reply to this
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