April 12, 2006

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

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9 Comments & 1 Trackback

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.

Posted by: Thomas Larsson on April 12, 2006 9:41 AM | Permalink | Reply to this

Re: Generalized Worldsheets?

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.

I should have been more precise: I would like to see structure behind the most general case, where I make no assumptions on my algebra, except, probably that it is ${C}^{*}$.

Posted by: urs on April 12, 2006 2:05 PM | Permalink | Reply to this

Re: Generalized Worldsheets?

???

Associativity is probably the minimal assumption you can make. Anyway, before you can solve the global problem, you should first figure out the local solution.

People tried to construct things like Virasoro algebras over non-commutative geometries 15-20 years ago (I tried things like that, too). I have never heard of any striking results in this direction.

Posted by: Thomas Larsson on April 12, 2006 2:42 PM | Permalink | Reply to this

Re: Generalized Worldsheets?

You write:

???

Hm, not sure what is going on. Let’s try to get back on the same page:

Not every (associative, yes) ${C}^{*}$ algebra comes from a Moyal star product on some functions space, unless I am missing something. Agreed? (Otherwise, please clarify my confusion.)

Posted by: urs on April 12, 2006 2:53 PM | Permalink | Reply to this

Re: Generalized Worldsheets?

Any deformation of a function algebra is locally Moyal, I think. Locally, we can choose a Fourier basis, E_m = exp(im.x), satisfying

E_m * E_n = E_m+n

A deformation must be of the form

E_m * E_n = c(m,n) E_m+n,

where the c(m,n) satisfy a cocycle condition, whose unique solution is

c(m,n) = exp(ihw(m,n))

where h is Planck’s constant and w the symplectic metric. So in that sense Moyal is the unique deformation of the function algebra.

Posted by: Thomas Larsson on April 12, 2006 3:23 PM | Permalink | Reply to this

Re: Generalized Worldsheets?

Ok, I believe this. Now let me try to clarify your “???”:

Suppose I want to try to give a definition of something like a CFT which is defined not on Riemann surfaces, in particular not on manifolds, but on something more general. On a ring spectrum, say.

I could try to understand this by doing what you are referring to, and what Lizzi et al did: I could try to continuously deform away from the ordinary algebra of (continuous) functions on my Riemann surface.

This deformation, as you say, is governed by the Moyal star.

But, personally, I don’t want to be distracted by details of the Moyal star. Since I don’t see that anyone has the slightest idea on how to define multipoint higher genus correlators for something like a CFT on Moyal-deformed surfaces, I would rather like to first understand what general concept that could be in the first place. “What is the principle of a “CFT” defined on a general scheme?”, for instance, if any exists.

But possibly this is asking for too much, ok. So, since you mentioned 20 years old work on the idea of having a CFT on a Moyal plane, could you provide some pointers to references?

Posted by: urs on April 12, 2006 3:35 PM | Permalink | Reply to this

Re: Generalized Worldsheets?

But possibly this is asking for too much, ok. So, since you mentioned 20 years old work on the idea of having a CFT on a Moyal plane, could you provide some pointers to references?

Sorry, no, I have forgotten, and it was pre-arXiv anyway. A name that comes to mind is Cosmas Zachos.

In those days, things like Virasoro, Kac-Moody and Moyal algebras were quite new to most people (remember that very few did string theory before 1984), and people did naive experiments with them. My own (non-unique) idea to generalize the Virasoro algebra was simply to consider the algebra generated by

L_m = E_m d

where E_m satify Moyal and d is a derivation. I don’t think it has a central extension, though.

Posted by: Thomas Larsson on April 14, 2006 5:38 AM | Permalink | Reply to this

Re: Generalized Worldsheets?

Reminds of this old s.p.r. thread: NCG/SUSY/*-Product

Posted by: Eric on April 16, 2006 12:50 AM | Permalink | Reply to this

(R)CFT on more general 2-Categories

Eric wrote:

Reminds of this old s.p.r. thread: NCG/SUSY/*-Product

Right. That was also about the Moyal star. The Moyal star is neat, interesting, deserves our attention, can be used for lots of explicit constructions and what not - but nevertheless I would rather forget the Moyal star for a moment! :-)

Let me try to rephrase what I was hoping to get at.

What is 2-dimensional conformal field theory? For the special case of rational conformal field theory we have the following answer ($\to$):

A full rational conformal field theory (RCFT) (where “full RCFT” means a consistent assignment of $n$-point correlators to all kinds of Riemann surfaces) is specified by two pieces of data.

1) There is the complex-analytic content of the RCFT. This is specified by the local symmetries of the RCFT, encoded in the operator algebra of the Virasoro current and possibly other symmetry currents (like affine current algebras in WZW models).

This “chiral” component of the RCFT gives us, for every Riemann surface, a vector space of potential correlation functions (called “conformal blocks”), namely a vector space of all those would-be correlation functions that are consistent with the local symmetries of the RCFT (with the chiral Ward identities).

In order to get a full CFT, we need to pick from these potential correlation functions the actual correlation functions, such that our choice is consistent with the sewing and cutting of Riemann surfaces. This choice is additional information not contained in the Virasoro and other chiral operators.

2) Hence there is also topological content to an RCFT. This turns out to be encoded in a (special, symmetric) Frobenius algebra object $A$ in the representation category of the chiral algebra (namely the algebra of open string states for any one boundary condition in our RCFT). In practice this means, that the way to pick a consistent set of correlation functions from the spaces of conformal blocks obtained in 1) is to choose a dual triangulation for every Riemann surface and, roughly, interpret it as a “flow chart” for some algebraic computation using that algebra $A$. The result of that computation is a vector, and interpreted as a vector in the space of conformal blocks, this yields the correlation function.

There are many aspects of this that one could ponder. In the present context I am interested in the following observation:

The above construction has nicely abstracted away from any details of an RCFT to the crucial structure.

full RCFT = chiral data + internal Frobenius algebra

Given this alone, we can assign consistent $n$-point correlators to all Riemann surfaces.

We don’t even need to know that our chiral data is obtained from Virasoro currents, all we need is that it behaves like chiral data. Technically, all we need is a modular tensor category $C$ and and a certain algebra object in $C$.

My question is: Can we also abstract away from the nature of Riemann surfaces? I.e., can we use the data given by a symmetric special Frobenius algebra in a modular tensor category and assign vectors to something else than ordinary Riemann surfaces, such that some analog of the sewing constraints holds??

If that “something else” is a Moyal-deformed Riemann surface that’s fine. But since I don’t see how restricting attention to technical details of Moyal star NCG helps to adresss the general question, I would rather like to ignore this for a moment!

Here is an observation close to your heart, Eric. Let me sketch how one could imagine performing the generalization that I am talking about to a setup where Riemann surfaces are replaced by Diamond complexes.

As you may have seen, I am in the process ($\to$) of showing that the above mentioned “flow chart computation” on dual triangulations of Riemann surfaces is secretly the result of applying a locally trivialized 2-functor to our Riemann surface. This way of looking at things has the advantage that one sees what structure of the Riemann surface we really need: what we need is some 2-category whose composition of 2-morphisms behaves like gluing of little pieces of Riemann surface!

It is easy to construct such a 2-category ${P}_{2}$ for instance from any 2-dimensional diamond graph. (We have discussed similar - if not the same - things before.)

Let the objects of ${P}_{2}$ be the points of the diamond complex, let the 1-morphisms be the edges and those freely generated by composing these, and let the 2-morphisms be the diamonds in between four edges, and all those obtained by freely composing these. Finally, allow for a means to identify edges in order to obtain topologically nontrivially situations (as sketched in section 1.3 of these notes).

There is nothing much which can stop you from applying the whole construction of RCFTs as outline above to a situation where instead of 2-categories of faces in Riemann surfaces we use such 2-categories obtained from diamond complexes.

In fact, somebody should seriously think about this, the diamond complex setup might even allow to carry through the full program without any serious modification. After all, the light-cone structure implicit in the diamond structure defines a conformal structure. Furthermore, restriction to diamonds may not be a restriction at all as far as Riemann surfaces go. We know ($\to$) that every Riemann surface may be decomposed into a collection of conformal rectangles with appropriately identified boundaries.

In any case, from my point of view the interesting question is this:

How much can we generalize the 2-category of 2-paths in Riemann surfaces and still be able to define (R)CFTs on it?

I would not be surprised if it turns out that there is a 2-category whose 2-morphisms are to be regarded as Moyal-deformed disks and that, with due care, we can define an RCFT 2-transport on this 2-category. But fiddling around with the details of the Moyal star may not be the best way to see if and how this can work. Instead, I believe we would first need to understand which general properties of a 2-category we need in order to be able to define an (R)CFT on it.

P.S. Since I thought it might be worthwhile to move this from the comment section to a place where it is more visible, I have turned the above comments into a new entry ($\to$).

Posted by: urs on April 18, 2006 11:22 AM | Permalink | Reply to this
Read the post (R)CFT on more general 2-Categories
Weblog: The String Coffee Table
Excerpt: RCFTs can apparently be defined as 2-functors on 2-paths in Riemann surfaces. Can this be generalized to other domain 2-categories?
Tracked: April 18, 2006 11:42 AM

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