## May 29, 2006

#### Posted by urs

Here are some links I would like to record, without having the time to discuss them in detail.

1)

At the 83rd PSSL Jon Woolf has given a lecture series on derived categories ($\to$, $\to$), and the slides are available online.

2)

There is a paper

E. Lupercio & B. Uribe
Topological Quantum Field Theory, String and Orbifolds
hep-th/0605255

apparently originating in some lecture notes, which tries to survey some crucial facts of TQFT, with an eye on the authors speciality, which are orbifolds ($\to$) and orbifold string topology ($\to$).

The discussion is a little cursory, though some elementary concepts (like groupoids) are described in detail. The authors emphasize how Feynman’s path integral can be understood as being induced by a correspondence (a “span”), and indicate roughly how this point of view neatly leads to TQFT, Gromov-Witten invariants, Floer theory, and - in the last part of the paper - to orbifold string topology.

3)

Kevon Costello keeps developing his approach to topological string theory, called TCFT ($\to$).

His latest paper

K. Costello
Topological conformal field theories and gauge theories
math.QA/0605647

introduces a new qualitative step.

The starting point is a bundle of associative algebras with something like a pointwise trace on it, such that it becomes something like a bundle of Frobenius algebras.

I guess if we think of each of these algebras as a category with a single object, equipped with the given trace operation, such a bundle of algebras constitutes an example for a Calabi-Yau category, as Costello calls it ($\to$), apparently following Kontsevich.

But the crucial ingredient considered now is a nilpotent graded derivation $Q$ on that algebra bundle, constituting an elliptic complex. The result is called an elliptic space with a Calabi-Yau structure.

(Unfortunately, this defintion is given in place of an introduction.)

$Q$ is of course going to play the role of a supercharge or, rather, of a BRST operator. The point of the paper is to establish a relation between TCFT and various gauge theories, notably flavors of Chern-Simons and of Yang-Mills theory, all of which will be described in one way or another by an action formally having the familiar form

(1)$S\left(a\right)=\mathrm{Tr}\left(\frac{1}{2}aQa+\frac{1}{3}{a}^{3}\right)\phantom{\rule{thinmathspace}{0ex}}.$

In fact, ordinary Chern-Simons theory is easily obtained from the above setup by simply letting our algebra be that of differential forms with values in the endomorphisms of some vector bundle $E$ on some manifold $M$; and $Q$ the deRham differential.

The first nontrivial point of the exercise is the the example in section 2.3, where Kevin Costello constructs an elliptic space with “Calabi-Yau structure” such that the action functional of Chern-Simons form on it produces - not Chern-Simons theory - but Yang-Mills theory.

But the aim is to actually write down something like a TCFT version of cubic open string field theory. Therefore, next, the setup is lifted to the case where we are considering the moduli space of metrised ribbon graphs (which models aspects of the moduli space of Riemann surfaces) and algebras of differential forms over this. (Recall ($\to$) that this is supposed to be a substitute for some category of topological cobordisms, which the more naïve minds among us would expect to make an appearance here).

Serious work is required - and presented - in order to make sense of the path integral for the Chern-Simons-like action for this case. Apparently the claim is (p. 29, right above section 7), that the topological open (and closed?) TCFT string field theory described this way is identical (?) to Chern-Simons field theory.

Sorry if that’s not quite right, I haven’t had the time to read the paper in detail. Hopefully I will in the near future. Unfortunately neither an introduction, nor summary nor conclusion is provided, that might have helped in navigating the main message.

Posted at May 29, 2006 9:55 PM UTC

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