(R)CFT on more general 2-Categories

April 18, 2006

Posted by Urs Schreiber

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I am grateful for the comments to the recent entry “Generalized Worldsheets?”, but I feel they indicate that I did not manage to get my main point across. I now tried again, providing more details, in the latest comment to that entry ( $\to$ ), which, for that reason, I thought I should lift to blog top-level. It is reproduced below.

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What IS 2-dimensional conformal field theory, really?

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!

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 ( $\to$ ).

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.

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 outlined 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.

Posted at April 18, 2006 11:31 AM UTC

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Re: (R)CFT on more general 2-Categories

Hi Urs, interesting topic in my opinion, but I’m mostly confused about what it precisely means so I’ll scatter some comments and questions. They may be stupid, however, owing to my present confusion.

First though, let me just say that I find the idea interesting partly because it looks like an alternative to the deformation of RCFT that one usually considers, i.e. something like relaxing the conditions on the category C, and also consequently generalising the modular functor. I would expect in that case, though, that the domain category can still be thought of as some category of decorated surfaces. At least that’s the basic idea I have of deforming RCFT to the non-rational case.

Anyway, as I understand your idea, it would mean that one should generalise the modular functor such that the domain category might possibly be thought of as C-decorated somethings, where C is still modular, but the somethings are no longer surfaces. Do you agree with this?

What confuses me about this is that I have no clue what one would mean by “sewing” of things which are not geometrical. In the RCFT case it is possible to define a RCFT quite abstractly, in purely categorical language, such that, essentially, the only reference to the geometrical notion of sewing is in the axioms of a modular functor. I would certainly guess that this could easily be generalised in many ways which would have absolutely nothing to do with CFT if one does not have some good characterisation of sewing that can be applied to more general categories. So, this is my main confusion, how does one characterise sewing abstractly? Maybe this is in fact what you’re trying to say with your example, that it’s contained in the structure of some 2-category, or 2-functor?

You’ll have to forgive me for still not having grasped the 2-cat stuff you’re using, I know I should by now…
I hope I managed to communicate my confusion well enough, let me know otherwise, cause I’d be very greatful if you could clear some of my confusions regarding these matters

Cheers!
Jens

Posted by: Jens on April 18, 2006 2:09 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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Hi Jens, great to hear from you!

as I understand your idea, it would mean that one should generalise the modular functor such that the domain category might possibly be thought of as C-decorated somethings, where C is still modular, but the somethings are no longer surfaces. Do you agree with this?

Yes, exactly, that’s what I was thinking of! I don’t know if it can be done (though I hope it could), but this seems to be the issue which people like Fedele Lizzi should eventually address in the context of their work, which I mentioned ( $\to$ ).

So, this is my main confusion, how does one characterise sewing abstractly? Maybe this is in fact what you’re trying to say with your example, that it’s contained in the structure of some 2-category, or 2-functor?

That’s a very good point, I think. There are two things I can say in response to this:

1) First of all, indeed, as you indicate, the mere fact that we characterize our domain by a 2-category and our correlators by 2-functors does implement at least some notion of sewing. It’s nothing but (2-)functoriality, saying that it does not matter if I first compute correlators on a surface $\Sigma_1$ , then on a surface $\Sigma_2$ , then somehow compose these results, or if I first “sew” both surfaces to obtain $\Sigma_1\circ \Sigma_2$ and then compute correlators on that.

That’s just what functoriality is about, and it is, at least morally, the very reason why it makes sense to go ahead and define a local QFT to be some sort of fucntor.

2) On the other hand, I have not yet managed to understand if the mere fact that we can identify a 2-functor behind the scenes of FFRS (“Fjelstad-Fuchs-Runkel-Schweigert” :-) (as I think we can) tells us something about the main result in FFRS V. I am hoping it does, but somehow seeing this is still beyond me.

In fact, maybe you can help me understanding this better. Ingo once told me something about how it should be possible to regard an $A$ -ribbon “ending” on a boundary somehow in terms of a couple of $U_i$ -field insertions, namely those appearing in $A = \oplus_{i\in S} U_i$ . I can see how this must be true on heuristic grounds, but within FFRS formalism I am having trouble giving this idea a precise meaning.

Posted by: urs on April 18, 2006 2:51 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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One more comment:

So, as I understand it, your idea of generalising RCFT would mean generalising a modular functor to some non-geometric category. This would btw be the type of MF that includes sewing in the axioms. If that’s corectly understood, I just now came to think of something that I have no idea whether it a) has been done, b) is completely trivial, or c) would be interesting. The fact that the idea is so straightforward I guess points towards b). Anyway, the idea would be to formulate the notion of a topological modular functor (in the language of Bakalov & Kirillov in terms of continuous functions (or something similar taking into account $C$ -decorations). That would be something like a commutative $C^*$ modular functor. Have you encountered, or thought, something along these lines? I’ve never, that’s for sure, but unless a) or b) applies, it certainly looks very interesting to me. And it would (presumably) be a suitable avenue for non-commutative generalisations.

Posted by: jens on April 18, 2006 3:08 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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the idea would be to formulate the notion of a topological modular functor (in the language of Bakalov & Kirillov in terms of continuous functions (or something similar taking into account $C$ -decorations). That would be something like a commutative $C^*$ modular functor.

I believe this is what I had in mind when I mentioned $C^*$ -algebras ( $\to$ ). But let me make sure that we are talking about the same idea:

The idea would be to replace any mentioning of topological spaces $S$ in the definition of the 3DTFT that the modular functor gives rise to, by the corresponding $C^*$ algebra $A(S) = C^\infty(S)$ of continuous functions on these spaces. There must be a way to read off from a commutative $C^*$ -algebra the boundary of the corresponding topological space $A(S) \mapsto A(\partial S)$ and to determine an operation on two $C^*$ -algebras $A(S)\otimes A(S') \mapsto A(S \cup S')$ whenever the boundaries of $S$ and $S'$ match.

In short, there should be a category structure on commutative $C^*$ -algebras which implements the gluing of $d$ -dimensional topological spaces. The idea is to replace the category $d\mathbf{Cob}$ of $d$ -dimensional cobordisms by this algebraic category and define a modular functor in terms of that.

Once this is done, we would hope to simply be able to drop the reqirement that our $C^*$ -algebras be commutative. Voila, a modular functor on a “generalized space”.

That’s the idea that I had in mind. I don’t know if anyone has looked into this. The idea itself may be trivial, but I guess working out the consequences is hardly trivial and I bet it has not been done yet.

(But if anyone knows otherwise, please let us know.)

Posted by: urs on April 18, 2006 4:36 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

Yes, that’s the basic idea I had in mind, although when I wrote it I had more precisely the 2D modular functor in mind, and not 3D TFT. They are essentially the same, but I just thought it might be easier to generalise a setting where there is only one type of geometrical objects (extended surfaces) instead of two (ext. surfaces + cobordisms).
Also, I have a personal preference, which may very well be misguided, to stay in 2D unless it’s necessary to use the 3DTFT.

I furthermore agree that although the idea is straightforward (which I still think), it’s most likely non-trivial to work out the details, if at all possible ;-)

Re: A-ribbons and boundary field insertions. Am I right in asuming that you’re wondering about the precise relation between the formalisms of FFRS V and Ingo’s Streetfest procedings?
If so, then I can tell you how I understand it, although it will have to wait until I have more time…must first gather my thoughts on this again. If that’s not what you’re refering to, could you maybe be a little more specific?

Posted by: Jens on April 18, 2006 6:32 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

Am I right in asuming that you’re wondering about the precise relation between the formalisms of FFRS V and Ingo’s Streetfest procedings?

Yes, essentially. All comments you have will be highly appreciated.

Posted by: urs on April 18, 2006 7:11 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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Well, I hope the following can be to some help, if it’s not enough then just bombard me with specific questions…hopefully I’ll be able to shed some light on it.

In the original formalism (with small variations through parts I-V), a boundary field insertion is basically labeled by two left $A$ -modules, $M$ and $N$ , an object $U$ , and an element $\psi\in\text{Hom}_A(M\otimes U,N)$ . If we only allow for one left $A$ -module, namely $A$ itself (which is the situation in the Streetfest proceedings), then the intertwiner $\psi$ lies in the space $\text{Hom}_A(A\otimes U,A)\in\text{Hom}_A(A\otimes U,A$ .

Now, the reciprocity (see part I, and originally one of Ostrik’s papers) tells you that there is a canonical isomorphism $\text{Hom}_A(A\otimes U,A)\cong\text{Hom}(U,A)$ , so boundary field insertions in this case correspond to subobjects of $A$ . The isomorphism acts as $\phi\in\text{Hom}(U,A)\mapsto m\circ(id_A\otimes\phi)\in\text{Hom}_A(A\otimes U,A)$ , where $m$ is the multiplication of $A$ . So, what one does now is to disregard individual boundary insertions (all the information of those can be found in the algebra $A$ ), and simply replace every boundary insertion by the object $A$ and the intertwiner $m$ (really $m$ if the boundary is ingoing, $\Delta$ if it is outgoing). If you want to retain a correlator with a specific boundary field insertion given by an object $f:U\hookrightarrow A$ , you can basically take the connecting manifold, glue it to a cylinder over the double with straight ribbons and one coupon labeled by $f$ . This gives you the original construction, using that reciprocity tells you how the intertwiner looks.

Was that understandable?

Anyway, this only works for the boundary condition $A$ . To generalise this, I guess one should label boundary insertions with algebras and bimodules according to your post about RCFT and Quiver Reps. I haven’t thought too much about this though.

Posted by: Jens on April 19, 2006 12:40 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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Was that understandable?

Yes, very much so. Thanks!

I knew about the reciprocity theorem, but I had not realized that this implies that open $A-A$ string insertions are necessarily just subobjects of $A$ . Of course that’s precisely as one would expect.

Now, using this insight, doesn’t that alone pretty much solve the sewing problem for open worldsheets already?

(Probably not, what additional subtleties are there? I have to admit that FFRS V is the part which I have spent least time with, so far.)

Posted by: urs on April 19, 2006 1:40 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

Now, using this insight, doesn’t that alone pretty much solve the sewing problem for open worldsheets already?

I guess, in hindsight, yes. Still, I certainly did not have this picture clear while working on part V (although I’m sure it was more or less clear for my collaborators), and the formalism in part V is what translates most directly to the usual CFT-language. In any case, I think conceptually the results in part V are almost as clear, the bulk of the paper is just doing everything very carefully…

Posted by: Jens on April 19, 2006 2:31 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

Thanks.

Maybe another question: Intuitively it seems that closed string sewing should be a direct consequence of open string sewing. But apparently there is no way to just demonstrate open string sewing and then lean back and claim that this deals with closed string sewing automatically?

I will need to think more about this sewing stuff, and in particular have another close look at FFRS V before I can say/ask anything else of substance.

Posted by: urs on April 19, 2006 3:47 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

Maybe another question: Intuitively it seems that closed string sewing should be a direct consequence of open string sewing. But apparently there is no way to just demonstrate open string sewing and then lean back and claim that this deals with closed string sewing automatically?

That is also my intuition in fact. I still think that is true, but may require some better understanding of the bulk algebra for a RCFT. My ideas are rather vague, but if you allow vagueness, I expect that there is some more natural way to define the bulk algebra than the present, which might make it obvious why closed string sewing follows immediately from open string sewing.

Posted by: Jens on April 19, 2006 4:01 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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I expect that there is some more natural way to define the bulk algebra than the present

What do you mean by “bulk algebra” here? Do you mean the prescription for how to choose the $C$ -decoration of the bulk triangulation?

Posted by: urs on April 19, 2006 6:08 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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What do you mean by “bulk algebra” here? Do you mean the prescription for how to choose the C-decoration of the bulk triangulation?

Sorry, I didn’t explain what I meant by this. What I’m refering to is the (commutative Frobenius) algebra object in $\mathcal{C}\times\overline\mathcal{C}$ that is assigned to a closed string boundary. The algebra of closed string states so to speak. In math.CT/0512076 it’s defined to be $C_l(A\times\mathbf{1}\otimes T_{\mathcal{C}})$ for a ssFa $A$ in $\mathcal{C}$ , and with the algebra $T_{\mathcal{C}} = \oplus_{i,j}U_i\times\overline U_j$ as an object.

Posted by: Jens on April 19, 2006 8:46 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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Ah, I see, I should have guessed that this is what you were referring to.

I would think the most elegant way to define the bulk algebra is as the image of the projector given by the cylinder, i.e. by the $A$ -ribbon graph which decorates the standard cylinder. (I mean a straight $A$ ribbon running from boundary to boundary of the cylinder connected with another one running around its non-contractible cycle.)

This way of looking at things is for instance emphasized by Lauda&Pfeiffer ( $\to$ ).

Hm, on the other hand I recall I wanted to check what happens to this projector in the presence of a nontrivial twist (in the sense of twist in ribbon categories) and never did. Does it project on either the left or right center $C_\ell(A)$ , $C_r(A)$ as in section 5.5 of FRS I? I’d hope it does, but now I am not sure what happens to the twist and how left and right center are distinguished by the cylinder projector.

It’s good that we talk about this stuff. I wanted to work this projection stuff out a while ago and never got around to doing it. Now I should!

Posted by: urs on April 19, 2006 9:16 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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Quick reply before leaving for today: I agree that’s the elegant way of defining it, but as far as I can see one needs to work out a few things before it is even possible to define it this way. Maybe possible via 2-transport? Anyway, I’ve been trying to figure this out as well, and I think I have some idea at least, but I need to sit down and work it out thoroughly.

Hm, on the other hand I recall I wanted to check what happens to this projector in the presence of a nontrivial twist (in the sense of twist in ribbon categories) and never did. Does it project on either the left or right center C ℓ(A), C r(A) as in section 5.5 of FRS I? I’d hope it does, but now I am not sure what happens to the twist and how left and right center are distinguished by the cylinder projector.

Hmm, I’m not exactly sure which setting you are looking at, a genuinely braided category? That’s the difficult part right, because the algebra should no longer lie in $\mathcal{C}$ , but in $\mathcal{C}\times\overline\mathcal{C}$ . For a symmetric category I believe it does (haven’t checked in detailed, but have checked similar stuff), but there of course the left and right centers coincide.

Posted by: Jens on April 19, 2006 10:04 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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Hmm, I’m not exactly sure which setting you are looking at, a genuinely braided category?

Sorry, I should have been more precise. I was thinking about the move depicted for instance on page 3 of these slides (by Jeffrey Morton). This indicates how that cylinder I mentioned gives rise to the projector onto the center of an algebra internal to $\mathrm{Vect}$ .

Unless I am mixed up, when you try to perform this move not in $\mathrm{Vect}$ but in a ribbon category, you’ll see that you get the diagram on the very right, but with in addtion a ribbon twist inserted at one point (which is invisble in the case of $\mathrm{Vect}$ , of course).

That’s the difficult part right, because the algebra should no longer lie in […] but in […]

I see. I guess now I get your point on how it would be nice to get a better handle on this bulk algebra. :-)

I’ll try to think about it.

Posted by: urs on April 20, 2006 8:39 AM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

I was thinking about the move depicted for instance on page 3 of these slides (by Jeffrey Morton). This indicates how that cylinder I mentioned gives rise to the projector onto the center of an algebra internal to Vect.

Ahh, I see. Well, I agree that if you tried to interpret the same move in a ribbon category then you would get an additional twist. Still, I don’t think that’s what one would like to do.

Posted by: Jens on April 20, 2006 4:34 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

I don’t think that’s what one would like to do.

Ok, I need to think about this when I am less tired. It just seems that in part this cylinder projector is forced upon us, since it necessarily is at work when you have an open string coming in and a closed coming out (the “zipper” move in Lauda/Pfeiffer language, which there precisely defines the map to the closed string algebra).

Posted by: urs on April 20, 2006 6:51 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

Well, as far as I can see it’s not clear how to translate this from TFT to CFT. Would like to discuss this further, but that’ll have to wait.

Posted by: Jens on April 20, 2006 7:08 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

This way of looking at things is for instance emphasized by Lauda+Pfeiffer.

I should mention, BTW, that I saw a copy (from a distance) of an actual manuscript by Moore and Segal today. I don’t know how complete it was, but perhaps their paper may be forthcoming.

I’m not exactly sure what you’re referring to here (I’m just dropping into the comment thread), but I learned today (from Dan Freed) that the image of the map from the closed string algebra to the center of the open string algebra can be thought of as living in the endomorphisms of the identity functor. This is related to Costello’s work because the Hochschild (co?)homology is just the derived version of this.

Posted by: Aaron Bergman on April 20, 2006 12:24 AM | Permalink | Reply to this

bulk algebra

the image of the map from the closed string algebra to the center of the open string algebra can be thought of as living in the endomorphisms of the identity functor. This is related to Costello’s work because the Hochschild (co?)homology is just the derived version of this.

Sounds very interesting. Right now I am not sure how to decode this, though. The identity functor on which category are you refering to? That of Frob algebras?

Posted by: urs on April 20, 2006 11:37 AM | Permalink | Reply to this

Re: bulk algebra

The category where the objects are boundary conditions and the Homs are the vector spaces of string states.

Posted by: Aaron Bergman on April 20, 2006 3:13 PM | Permalink | Reply to this

Re: bulk algebra

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Ok, so endomorphisms of the identity are given by assignments

(1)

a \mapsto u_a

where $a$ is a D-brane and $u_a$ an $a$ - $a$ string state, making these naturality squares commute:

(2)

\array{ a & \overset{u_a}{\rightarrow} & a \\ v_a\;\;\downarrow && \downarrow \;\; v_a \\ a &\overset{u_a}{\rightarrow}& a }

(and similarly for diagrams where we have a different branes $a$ and $b$ in the top and bottom line), where composition of arrows is the operator product of the corresponding vertex operators.

Hm, this clearly does specify the correct condition for the topological string. On the other hand, for the problem that Jens is refering to (related to a nontrivial braiding in the category for the CFT case) this seems not to be the right answer. (?)

Posted by: urs on April 20, 2006 6:26 PM | Permalink | Reply to this

Re: bulk algebra

Hm, this clearly does specify the correct condition for the topological string. On the other hand, for the problem that Jens is refering to (related to a nontrivial braiding in the category for the CFT case) this seems not to be the right answer. (?)

Well, mainly because, in a (non-topological) CFT, I don’t wan’t to assign an object in the “chiral” category to a closed string. I don’t see how that could even give something consistent, given that the geometric category is symmetric, or do you not agree?

Posted by: Jens on April 20, 2006 7:18 PM | Permalink | Reply to this

Re: bulk algebra

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Yes, I perfectly agree. Seems like we are stuck at a point where should stop chatting and just work some things out. :-) Unfortunately, right now I am forced to be busy with less intersting things…

Posted by: urs on April 20, 2006 8:25 PM | Permalink | Reply to this

Re: bulk algebra

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thought of as living in the endomorphisms of the identity functor.

Robin Houston mentions something closely related over on his bosker blog.

Posted by: urs on June 10, 2006 5:34 PM | Permalink | Reply to this

derived

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This is related to Costello’s work because the Hochschild (co?)homology is just the derived version of this.

If you could point me directly to the point in his papers related to this particular statement that would be a real treat.

I just heard a talk by Daniel Huybrechts on stability conditions for derived categories $D^b(X)$ of coherent sheaves on $X=$ K3 surfaces (based on the work by Bridgeland).

This reminded me, that the collection of “physical” B-branes, i.e. the stable BPS branes, is actually still a subcategory of $D^b(X)$ , i.e. still a derived category.

This again means that if we could correctly analyze the brane content of the physical theory in the first place, this, too, should involve derived categories.

Now, as I went on about in a recent entry ( $\to$ ), using FRS we completely understand the abstract nature of the “landscape” of rational CFT string backgrounds. It is something like the collection of bimodule subcategories internal to modular tensor categories.

As I tried to point out, we can think of this as a collection of lax “quiver representations” with values in the modular category. But this suggests a natural category structure on this collection, which, in fact, should be abelian.

Hence it is sort of tempting to consider the derived category of that category of RCFT backgrounds.

While interesting, this leads me into completely unknown territory. Maybe its misguided altogether.

Posted by: urs on April 20, 2006 8:43 PM | Permalink | Reply to this

Re: derived

If you could point me directly to the point in his papers related to this particular statement that would be a real treat.

I learned it from David Ben-Zvi, so I’m not sure of a reference. Maybe something by Caldararu might talk about it?

Posted by: Aaron Bergman on April 20, 2006 8:53 PM | Permalink | Reply to this

Re: derived

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I can talk a little more now about Hochschild cohomology. It is true (only on the level of the dg-category, I think), that $HH^n$ is equal to natural transformations from $Id \rightarrow [n]$ . If you want to phrase this in terms of the usual derived category of sheaves, say, you get something like $HH^n = Ext^n_{X\timesX}(O_\Delta,O_\Delta)$ where $\Delta$ is the diagonal. It follows from a generalization of a theorem of Hochschild-Kostant-Rosenberg that, for the derived category of coherent sheaves, we have $HH^n(D(X)) = \bigoplus_{i+j=n} H^i(X,\wedge^j TX)$ which you’ll recognize as the right answer from the B-model.

For the relation of HH to the dg-category, there is this paper by Toen. Caldararu has a nice discussion of the definition just in terms of sheaves in this paper. There’s more on the HKR isomorphism (and the relation to K-theory) here.

Posted by: Aaron Bergman on June 10, 2006 5:59 PM | Permalink | Reply to this

Re: derived

I can talk a little more now about Hochschild cohomology.

Great, thanks a lot for the information.

I’ll take a look at this as soon as time permits.

Posted by: urs on June 12, 2006 9:22 AM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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Hi Urs,

When I read your paragraph

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??

I think you know the first thing that popped into my mind. Then I found out two paragraphs later that you were intentionally setting me up :)

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 ( $\to$ ).

It is about time! :)

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.

If you were walk down this path, I would be excited to sit on the sidelines and comment as the journey evolves. I can almost guarantee you that formulating things using diamond complexes will lead to significant simplification and conceptual transparency. That is the pattern we saw emerge from our adventures and I completely expect it to continue the deeper you go.

And “Hi” Jens. You will soon discover (if it isn’t completely obvious already) how clueless I am about all this stuff. I am an engineer afterall. Not even that. I now work in finance *gasp* :)

Anyway, I just thought I would make a totally naive comment regarding something you said.

So, as I understand it, your idea of generalising RCFT would mean generalising a modular functor to some non-geometric category.

I’m guessing that the term “geometric category” probably has some really specific mathematical definition (one that would give me a nose bleed if I heard it). However, in the case of a diamond complex, I just thought I would point out that some things that might not seem “geometric” in the tradition sense of Riemannian manifolds may deserve the right to be thought of as geometric. A diamond complex is kind of like a “discrete” or “finitary” or “combinatorial” manifold although there is no notion of a underlying continuum (Hausdorf space) in sight.

You are probably too busy and/or not interested enough, but I would be really thrilled to hear if you had any opinion about the paper Urs and I wrote on discrete differential geometry on diamond complexes ( $\to$ )(which I changed the name at the last minute to “causal graph” in an apparent failed attempt to avoid the possible misconception that what we did was limited to cubic lattices and also to draw attention to the parallels with Sorkin’s poset stuff).

Personally, and I think I can speak for Urs to a certain extent, I (we?) think that there is something fundamental in all that discrete differential geometry stuff. One of the fascinating aspects of it that I have made use of in mathematical finance is that discrete calculus on a diamond graph leads naturally to a discrete version of stochastic calculus.

The relationship between noncommutative geometry and stochastic calculus was first observed (to the best of my knowledge) by Dimakis & Mueller-Hoissen. I think I can safely claim to be the first person to ever apply NCG to mathematical finance :)

Noncommutative Geometry and Stochastic Calculus: Applications in Mathematical Finance

The discrete version on a diamond complex (binary tree) came later

Financial Modelling using Discrete Stochastic Calculus

Anyway, I am pretty sure that noone who reads this will find any of those interesting :) The real point is just to demonstrate that there is some neat (and deep?) mathematics involved with diamond complexes.

One other thing I thought was neat (fundamental?) is that there is some curious relationship between white noise and Wiener processes (besides the obvious), which Urs and I discussed a little bit here

White Noise and Wiener Processes

Anyway, I know nothing about RCFTs, but I am just really happy to see Urs thinking about constructing them within the framework of discrete differential geometry on diamond complexes and I look forward to trying to follow along.

Best regards,
Eric

Posted by: Eric on April 19, 2006 6:46 AM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

And “Hi” Jens. You will soon discover (if it isn’t completely obvious already) how clueless I am about all this stuff. I am an engineer afterall. Not even that. I now work in finance *gasp* :)

Hi Eric! :) I’m quite sure you’re not completely clueless, seeing that you wrote a paper with Urs and all.

I’m guessing that the term “geometric category” probably has some really specific mathematical definition (one that would give me a nose bleed if I heard it). However, in the case of a diamond complex, I just thought I would point out that some things that might not seem “geometric” in the tradition sense of Riemannian manifolds may deserve the right to be thought of as geometric. A diamond complex is kind of like a “discrete” or “finitary” or “combinatorial” manifold although there is no notion of a underlying continuum (Hausdorf space) in sight.

Nope, at least not the way I was using the term. I simply meant some category where the objects are in some loose sense geometric, like some type of manifolds for instance, or (as is relevant for RCFT) extended surfaces.

I guess (as is maybe clear from the discussion above) that something like what you describe, things that are not obviously geometric but for some reason deserve to be called geometric (or maybe, things that share certain properties with geometric structures), is needed to define a sensible generalisation of RCFT along the lines Urs has been talking about here. Personally I don’t really have a clue as to what precise properties one should require. It looks probable to me that, as Urs comments above, the relevant properties are abstracted in terms of “2-structures”, but I’ve no idea about any details. Maybe I should better shut up and try to understand Urs’ work on understanding FRS as a locally trivialised 2-functor first.

You are probably too busy and/or not interested enough, but I would be really thrilled to hear if you had any opinion about the paper Urs and I wrote on discrete differential geometry on diamond complexes (→)

Well, I have to admit that I haven’t read it…although there may be motivation for me to do so. Also, I’m completely clueless regarding those ideas (i.e. discrete diff. geometry), and any comments from me would most likely be worthless I’m afraid.

Best,
Jens

Posted by: Jens on April 19, 2006 2:20 PM | Permalink | Reply to this

Re: (R)CFT on more general 2-Categories

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As Jens says, for now the term “geometric $n$ -category” is not a definition for a special kind of $n$ -category, but rather indicates that we happen to be given some $n$ -category and want to think of its $p$ -morphisms as some sort of $p+x$ -dimensional spaces. (At least that’s how I am using it currently.)

Similarly for $n$ -transport. For me, $n$ -transport is just any $n$ -functor on a geometric $n$ -category. Hence it just any old $n$ -functor, but with the difference that I happen to interpret its domain in a special way.

(If you like, compare my remarks on precisely this issue in the very begining of my notes $\to$ ).

It’s exactly this fact which suggests that we consider domain $n$ -categories for $n$ -transport whose $p$ -morphisms are not at all like ordinary $p+x$ -dimensional “spaces”.

When showing that locally trivialized $n$ -transport gives rise to lots of structures which are encountered in gerbes and QFT, the only property of the domain $n$ -category that one really needs to require is that there is some sort of a site structure around, which allows to say what it means to restrict the $n$ -functor to “local patches” of its domain $n$ -category.

Posted by: urs on April 19, 2006 3:11 PM | Permalink | Reply to this

Algebraic Cobordisms

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Motivated by further private email exchange, I have sent the following message to sci.math.research:

For any integer $n$ , we define the category $\mathrm{nCob}$ to be that whose objects are diffeomorphism classes of ( $n-1$ ) dimensional manifolds, and whose morphisms are diffeo classes of $n$ -dimensional manifolds cobording their source and target objects. Composition is gluing of manifolds at their boundaries.

I would like to understand the construction of this category in terms not of the manifolds themselves, but in terms of their algebras of smooth functions.

There should be a nice way to read off the algebra of functions supported on the boundary from the algebra of functions on the entire manifold with boundary; and to encode the gluing of manifolds in terms of a certain gluing operation on their algebras of functions.

Of course the motivation behind all this is that I would like to understand if there is something like a noncommutative version of $\mathrm{nCob}$ .

It feels like these questions should have basic, well-known answers, but I am probably not familiar enough with the relevant literature in order to know.

I’d be grateful for any comments and pointers to relevant literature.

Posted by: urs on April 21, 2006 3:28 PM | Permalink | Reply to this

Re: Algebraic Cobordisms

Would it be misguided to try to construct something like this on diamonds? *returns to lurking*

Posted by: Eric on April 21, 2006 6:51 PM | Permalink | Reply to this

Re: Algebraic Cobordisms

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Would it be misguided to try to construct something like this on diamonds?

Right, in that case we can easily see what we need to do.

So say our algebra is that of functions on finite sets and say it comes to us enhanced to a full differential graded algebra, i.e. as the grade 0-component.

Then this is the same as encoding a directed graph structure with the set of vertices being our finite set. We can read off the “boundary” from that graph.

To formulate that more algebraically, let’s assume our graph looks well behaved 2-dimensional, in that our dg algebra has well-behaved elements up to degree 2 and is empty in higher degree. We know this means that at least locally the graph is of diamond form (square lattice, two incoming, two outgoing edges at every bulk vertex).

We can now easily capture the boundary vertices algebraically. They correspond to those functions (grade 0 elements) whose differential lives in a smaller space than those of bulk points (since the differential is the weighted sum of the outgoing and incoming edges, some of which are not present at the boundary).

Hm, right, so perhaps what we need is a catgeory not just of algebras, but of dg-algebras.

Posted by: urs on April 22, 2006 5:16 PM | Permalink | Reply to this

Algebraic Cobordisms

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Turns out there is indeed a theory of algebraic cobordisms.

Posted by: urs on June 8, 2006 4:37 PM | Permalink | Reply to this

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April 18, 2006