### Generalized Worldsheets?

#### Posted by Urs Schreiber

I am curently at the Amalfi coast, attending the annual IIASS conference ($\to $). The talks of most interest to me are yet to come, but here is a quick note on a talk we heard yesterday.

Fedele Lizzi reported on recent attemts

Fedele Lizzi, Sachindeo Vaidya, Patrizia Vitale
*Twisted Conformal Symmetry in Noncommutative Two-Dimensional Quantum Field Theory
*

hep-th/0601056

to define conformal field theory on a noncommutative *parameter space*, more precisely, to construct something that would deserve to be called a “Virasoro algebra living on the noncommutative Moyal plane”.

Essentially, what is done in the above paper is an application of the general prescription described in

Paolo Aschieri, Christian Blohmann, Marija Dimitrijevic, Frank Meyer, Peter Schupp, Julius Wess
*A Gravity Theory on Noncommutative Spaces*

hep-th/0504183.

The main point is that, when products of functions are deformed by the Moyal star, one can introduce a related twist on the Leibnitz rule for derivations on the original function algebra such that symmetries represented on the original algebra remain unbroken when sent to the deformed algebra. This is intended to allow one to have, for instance, ordinary Poincaré or conformal symmetry implemented on noncommutative spaces.

I must say I haven’t looked closely enough at these constructions yet to say anything of value about the technical details. (But Fedele Lizzi himself emphasized that their construction is little more than a first idea at the moment.) I am wondering, though, what the big picture is that is lurking in the background here, that, which does not depend, in particular, on restricting attention to the Moyal product.

I think the question is this:

Can one sensibly define a generalization of 2-dimensional conformal field theory on some sort of generalized Riemann surfaces, which are not ordinary manifolds?

(Since it lead to some discussion after the above mentioned talk, I should maybe emphasize that the question here is not about noncommutativity in target space. I hear that people have defined and investigated CFTs with the target being a quantum group, for instance. But this is not what should be the issue here, I think. These are still CFTs defined on the Riemann sphere, the complex torus, etc.

But is there anything known about how to generalize the concept of a conformal 2-dimensional field theory to something whose *parameter* spaces are, say, … general 2-dimensional *schemes*? Or, maybe better, (since we want something like a conformal structure on our generalized parameter space), where the parameter spaces are noncommutative Riemann surfaces defined following Connes’ NCG?

If anybody knows relevant references, plase drop me a note!

Of course one thing that immediately comes to mind are constructions like Matrix Strings or other worldsheet discretizations that make an appearance here and there. But I am not sure I have ever seen a definition of CFT in these context in a way that deserves this name.

If I were to make a guess myself, I would maybe note the following. It seems like one can capture 2-dimensional CFT by a notion of 2-transport ($\to $), i.e. by 2-functors which assign CFT propagators to Riemann surface elements. In the general formalism governing this construction, there is nothing which forces one to take the domain 2-category of these 2-functors to really be one whose 2-morphisms are surface elements. The general construction works for much more general 2-categories. Hence, from that point of view, it would be sort of straightforward to define a 2d-CFT on a generalized parameter space to be a certain 2-transport 2-functor on suitably generalized domain 2-categories. Maybe.

Posted at April 12, 2006 8:22 AM UTC
## Re: Generalized Worldsheets?

I am wondering, though, what the big picture is that is lurking in the background here, that, which does not depend, in particular, on restricting attention to the Moyal product.The Moyal product is, in a sense whose exact definition I have forgotten, the unique associative deformation of the commutative product in the function algebra. Locally, of course, but to solve global problems you must first get it right locally. Thus, unless you want to give up associativity (and then anything can happen, I guess), you are stuck with either the Moyal or the commutative product.

In general, if your algebra admits a realization as vector fields over an N-dimensional manifold, you get its Virasoro-like extensions by restriction from the non-central Virasoro extensions of the full algebra of vector fields in N dimensions. This is because such a realization is nothing but an embedding of your algebra into vect(N), just as a matrix representation is an embedding of a finite-dimensional algebra into gl(N). Billig has even constructed some of these extensions globally and on the group level, see math.GR/0302007.