## April 7, 2005

### New preprint: From Loop Groups to 2-Groups (and the String Group)

#### Posted by Urs Schreiber

I am happy to be able to announce a new preprint:

J. Baez, A. Crans, U. Schreiber & D. Stevenson

From Loop Groups to 2-Groups

Abstract:

We describe an interesting relation between Lie 2-algebras, the Kac–Moody central extensions of loop groups, and the group $\mathrm{String}\left(n\right)$. A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2-group is a categorified version of a Lie group. If $G$ is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras ${𝔤}_{k}$ each having $𝔤$ as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on $G$. There appears to be no Lie 2-group having ${𝔤}_{k}$ as its Lie 2-algebra, except when $k=0$. Here, however, we construct for integral $k$ an infinite-dimensional Lie 2-group ${𝒫}_{k}G$ whose Lie 2-algebra is equivalent to ${𝔤}_{k}$. The objects of ${𝒫}_{k}G$ are based paths in $G$, while the automorphisms of any object form the level-$k$ Kac–Moody central extension of the loop group $\Omega G$. This 2-group is closely related to the $k$th power of the canonical gerbe over $G$. Its nerve gives a topological group $\mid {𝒫}_{k}G\mid$ that is an extension of $G$ by $K\left(ℤ,2\right)$. When $k=±1$, $\mid {𝒫}_{k}G\mid$ can also be obtained by killing the third homotopy group of $G$. Thus, when $G=\mathrm{Spin}\left(n\right)$, $\mid {𝒫}_{k}G\mid$ is none other than $\mathrm{String}\left(n\right)$.

[Update: I am aware of that problem with the incorrectly-displayed TeX code above. I am hoping to find the solution to that problem soon.]

There are two central theorems here:

1) The weak Lie 2-algebras ${𝔤}_{k}$ defined in HDA VI are equivalent to infinite-dimensional strict Fréchet Lie 2-algebras ${𝒫}_{k}𝔤$. These are related to the Kac-Moody central extension ${\Omega }_{k}𝔤$ of the loop algebra $\Omega 𝔤$ and come from infinite-dimensional Fréchet Lie 2-groups ${𝒫}_{k}G$.

2) The so-called ‘nerve’ $\mid {𝒫}_{k}G\mid$ of ${𝒫}_{k}G$ is, for $G=\mathrm{Spin}\left(n\right)$, the topological group $\mathrm{String}\left(n\right)$.

The conclusion of the paper is the following:

We have seen that the Lie 2-algebra ${𝔤}_{k}$ is equivalent to an infinite-dimensional Lie 2-algebra ${𝒫}_{k}𝔤$, and that when $k$ is an integer, ${𝒫}_{k}𝔤$ comes from an infinite-dimensional Lie 2-group ${𝒫}_{k}G$. Just as the Lie 2-algebra ${𝔤}_{k}$ is built from the simple Lie algebra $𝔤$ and a shifted version of $𝔲\left(1\right)$:

(1)$0⟶\mathrm{b}𝔲\left(1\right)⟶{g}_{k}⟶𝔤⟶0\phantom{\rule{thinmathspace}{0ex}},$

the Lie 2-group ${𝒫}_{k}G$ is built from $G$ and another Lie 2-group:

(2)$1⟶{ℒ}_{k}G⟶{𝒫}_{k}G⟶G⟶1$

whose geometric realization is a shifted version of $\mathrm{U}\left(1\right)$:

(3)$1⟶B\mathrm{U}\left(1\right)⟶\mid {𝒫}_{k}G\mid ⟶G⟶1\phantom{\rule{thinmathspace}{0ex}}.$

None of these exact sequences split; in every case an interesting cocycle plays a role in defining the middle term. In the first case, the Jacobiator of ${𝔤}_{k}$ is $k\nu :{\Lambda }^{3}𝔤\to ℝ$. In the second case, composition of morphisms is defined using multiplication in the level-$k$ Kac–Moody central extension of $𝒪G$, which relies on the Kac–Moody cocycle $k\omega :{\Lambda }^{2}𝒪𝔤\to R$. In the third case, $\mid {𝒫}_{k}G\mid$ is the total space of a twisted $B\mathrm{U}\left(1\right)$-bundle over $G$ whose Dixmier–Douady class is $k\left[\nu /2\pi \right]\in {H}^{3}\left(G\right)$. Of course, all these cocycles are different manifestations of the fact that every simply-connected compact simple Lie algebra has ${H}^{3}\left(G\right)=ℤ$.

We conclude with some remarks of a more speculative nature. There is a theory of ‘2-bundles’ in which a Lie 2-group plays the role of structure group [3, 4]. Connections on 2-bundles describe parallel transport of 1-dimensional extended objects, e.g. strings. Given the importance of the Kac–Moody extensions of loop groups in string theory, it is natural to guess that connections on 2-bundles with structure group ${𝒫}_{k}G$ will play a role in this theory.

The case when $G=\mathrm{Spin}\left(n\right)$ and $k=1$ is particularly interesting, since then $\mid {𝒫}_{k}G\mid =\mathrm{String}\left(n\right)$. In this case we suspect that $2$-bundles on a spin manifold $M$ with structure $2$-group $𝒫G$ can be thought as substitutes for principal $\mathrm{String}\left(n\right)$-bundles on $M$. It is interesting to think about ‘string structures’ [16] on $M$ from this perspective: given a principal $G$-bundle $P$ on $M$ (thought of as a $2$-bundle with only identity morphisms) one can consider the obstruction problem of trying to lift the structure $2$-group from $G$ to ${𝒫}_{k}G$. There should be a single topological obstruction in ${H}^{4}\left(M;ℤ\right)$ to finding a lift, namely the characteristic class ${p}_{1}/2$. When this characteristic class vanishes, every principal $G$-bundle on $M$ should have a lift to a $2$-bundle $𝒫$ on $M$ with structure $2$-group ${𝒫}_{k}G$. It is tempting to conjecture that the geometry of these $2$-bundles is closely related to the enriched elliptic objects of Stolz and Teichner [20].

Posted at April 7, 2005 9:28 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/547

### Re: New preprint: From Loop Groups to 2-Groups (and the String Group)

I guess it is time to learn about these things. Maybe I can start with an embarrassingly elementary question: what is this String group good for? I understand (to some extent) the reason why a physicist wants to kill the first homotopy group of SO(n) — it is because the physicist wants to be able to define the parallel transport of fermion fields. When it is possible, lifting from SO(n) to Spin(n) involves (roughly) fixing a Z_2-valued ambiguity for every 1-cycle of spacetime, which one describes as the choice of periodic or antiperiodic boundary conditions for the fermions. Is there an analogous “problem” that one encounters in string theory and gets solved by lifting to String(n)? Is there a Z-valued ambiguity for something associated with 3-cycles of spacetime?

Posted by: Andy Neitzke on April 8, 2005 6:46 AM | Permalink | Reply to this

### Re: New preprint: From Loop Groups to 2-Groups (and the String Group)

It’s all about elliptic cohomology. If I remember it, the obstruction to lifting to String(n) is p_1 / 2. This is just the sigma model anomaly. Thus, we need the lifting in order to define string fields. Killingback wrote about this way back when, along with some other people.

Just like we can think of a spin structure on spacetime as an orientation of loop space, I believe we can think of a string structure as a spin structure on loop space, cf, various Witten stuff.

Since an element of elliptic cohomology really ought to be a global twisting of string field, in the same way a vector bundle is a twisting of an ordinary function, I keep on hoping Urs will write down a sigma model action for a non-abelian string. But if he doesn’t do it soon, I might be forced to learn this stuff, too :)

But in the meantime, my exceptional collections are keeping me somewhat busy.

Posted by: Aaron Bergman on April 8, 2005 8:37 AM | Permalink | Reply to this

### Re: New preprint: From Loop Groups to 2-Groups (and the String Group)

But how precisely?? I know that a 2-connection in a 2-bundle with structure 2-group being that group ${𝒫}_{k}\mathrm{Spin}\left(n\right)$ looks a lot like Stolz&Teichner’s enriched elliptic objects.

Both are 2-functors from 2-categories of surface elements, both crucially involve the group $\mathrm{String}\left(n\right)$ and both assign a circle worth of morphisms to a surface element. If they are not ‘the same’ they must be closely related. But how exactly? Can you see it?

If I remember it, the obstruction to lifting to String(n) is ${p}_{1}/2$.

Indeed. The string class in ${H}^{3}\left(L\left(X\right),ℤ\right)$ of the free loop space is nothing but ${p}_{1}/2$ in ${H}^{4}\left(X,ℤ\right)$ integrated over the loop, roughly.

A nice discussion is given by Danny Stevenson and Michael Murray in math.DG/0106179.

This is just the sigma model anomaly.

That of the super sigma-model, only, right?

BTW, I can see why this must be how it is, but would get into trouble if I tried to prove it. Can you sketch a proof?

Thus, we need the lifting in order to define string fields.

I see, that’s a good way to put it which directly motivates the stuff you say further below.

The paper I mentioned above builds on these Killingback results and kind of extends them in a way. In particular there the relation between $L\left(G\right)$ bundles over $M$ and $G$-bundles over ${S}^{1}×M$ gets into the picture.

This is related to M-theory somehow. But again, I have only a murky picture of what should be a crisp statement.

So for instance there is a shift in the quantization condition for the 4-form field-strength ${G}_{4}$, which is nothing but ${p}_{1}/2$. What does that mean in this context?

Just like we can think of a spin structure on spacetime as an orientation of loop space, I believe we can think of a string structure as a spin structure on loop space, cf, various Witten stuff.

So this tells us that we can have Dirac operators on loop space.

Since an element of elliptic cohomology really ought to be a global twisting of string field, in the same way a vector bundle is a twisting of an ordinary function,

I am glad you say so. This is part of what I tried getting at repeatedly in our discussion on sps. Certainly I did not express myself well.

Namely is it the string field itself which ought to be thought of as a function and hence as a local section of some ‘bundle’? Following the reasoning reviewed by Lazaroiu in hep-th/0305095 it seems that we should more generally generalize the string field to an object in ${D}_{infy}^{b}\left(X\right)$. (Where it becomes a morphism in a complex.)

If we regard that object as a function which is really a local section we arrive at the line 2-bundles that I talk about here.

As I discuss, the 2-connections in these line 2-bundles again look very much alike Stolz&Teichner’s enriched elliptic objects.

I keep on hoping Urs will write down a sigma model action for a non-abelian string.

Hey, I already did! Didn’t I?? :-)

I did it for strict structure 2-groups and 2-bundles whose base 2-space is really a 1-space. I have ideas in stock how to do it for weak structure 2-groups, too, and how that involves ${n}^{3}$-scaling behaviour.

That was the whole point of this entry which is also discussed in this set of slides and will be discussed at length soon in this document.

Or do you mean these constructions are not what we should be looking for?

But if he doesn’t do it soon, I might be forced to learn this stuff, too :)

If you ever feel like collaborating, just let me know. :-)

But in the meantime, my exceptional collections are keeping me somewhat busy.

I still feel that the freedom in choosing exceptional collections of D-branes is related to the gauge freedom we have in the line 2-bundles mentioned above. I might be wrong, of course. If I am wrong about this, I am hoping that you will eventually convince me of it.

Posted by: Urs on April 8, 2005 12:43 PM | Permalink | Reply to this

### Re: New preprint: From Loop Groups to 2-Groups (and the String Group)

If I knew how exactly this was all related to elliptic cohomology, I’d have already written the paper :). I just have semi-informed speculation.

That of the super sigma-model, only, right?

You need fermions to get the sigma-model anomaly, I think. Certainly you need something chiral.

BTW, I can see why this must be how it is, but would get into trouble if I tried to prove it. Can you sketch a proof?

A proof that p_1/2 is the sigma-model anomaly or that the string structure is the same as the sigma model anomaly? Assuming the latter, besides the Killingback, there’s a paper by Witten in a conference proceedings, I think, that might be helpful. I think it’s “Global Anomalies in String Theory”, but I could be misremembering.

So for instance there is a shift in the quantization condition for the 4-form field-strength G 4 , which is nothing but p 1 /2 . What does that mean in this context?

The shift in the M-theory 4-form is lambda/2, where lambda is p_1/2.

Just like we can think of a spin structure on spacetime as an orientation of loop space, I believe we can think of a string structure as a spin structure on loop space, cf, various Witten stuff.

So this tells us that we can have Dirac operators on loop space.

And the index of this gives us the elliptic genus.

Namely is it the string field itself which ought to be thought of as a function and hence as a local section of some ‘bundle’?

I’m scared of the word “local” there. “Bundle”, too.

Following the reasoning reviewed by Lazaroiu in hep-th/0305095 it seems that we should more generally generalize the string field to an object in D infy b(X). (Where it becomes a morphism in a complex.)

That would be an open string, not a closed string, I’d think. I don’t think the derived category, which basically tells us what boundary conditions are, is going to tell us all that much about closed string states.

I keep on hoping Urs will write down a sigma model action for a non-abelian string.
Hey, I already did! Didn’t I?? :-)

I did it for strict structure 2-groups and 2-bundles whose base 2-space is really a 1-space. I have ideas in stock how to do it for weak structure 2-groups, too, and how that involves n 3 -scaling behaviour.

That was the whole point of this entry which is also discussed in this set of slides and will be discussed at length soon in this document.

While there are many impressive and rather frightening diagrams there, I didn’t see an action written down anywhere. Should I look more closely?

I still feel that the freedom in choosing exceptional collections of D-branes is related to the gauge freedom we have in the line 2-bundles mentioned above. I might be wrong, of course. If I am wrong about this, I am hoping that you will eventually convince me of it.

As I said, it seems to me that all this 2-stuff is related to closed strings, while derived categories and the like are related to open strings. These aren’t completely disparate, but I can’t say I see how what you’re getting at, to the extent that I understand it, would make sense.

Posted by: Aaron Bergman on April 8, 2005 5:49 PM | Permalink | Reply to this

### Re: New preprint: From Loop Groups to 2-Groups (and the String Group)

The shift in the M-theory 4-form is $\lambda /2$, where lambda is ${p}_{1}/2$.

Ok, so the shift is ${p}_{1}/4$. But it vanishes in any case when loop space is string class.

Namely is it the string field itself which ought to be thought of as a function and hence as a local section of some ‘bundle’?

I’m scared of the word ‘local’ there. ‘Bundle’, too.

Of course the quotation marks on ‘bundle’ were supposed to make us think of 2-bundle. Too bad that you don’t see the line 2-bundle here which I am seeing. Possibly I am hallucinating.

While there are many impressive and rather frightening diagrams there, I didn’t see an action written down anywhere. Should I look more closely?

It suffices to read the following couple of summarizing sentences:

We have a theorem that a nonabelian surface holonomy for strict 2-groups is a 2-functor $\mathrm{hol}$ from the 2-groupoid of bigons ($\sim$ surface elements up to thin homotopy) to the strict 2-group which locally (i.e. on elements ${U}_{i}$ of a good cover) assigns to a bigon

(1)${\gamma }_{1}\stackrel{\Sigma }{\to }{\gamma }_{2}$

(where ${\gamma }_{i}$ are paths with coinciding endpoints and $\Sigma$ is a surface with these paths being its boundary)

the 2-group element

(2)${W}_{A}\left({\gamma }_{1}\right)\stackrel{{𝒲}_{\left(A,B\right)}\left(\Sigma \right)}{\to }{W}_{A}\left({\gamma }_{2}\right)$

where ${W}_{A}\left({\gamma }_{i}\right)$ is the ordinary holonomy of a local connection 1-form $A\in {\Omega }^{1}\left({U}_{i},\mathrm{Lie}\left(G\right)\right)$ and ${𝒲}_{\left(A,B\right)}\left(\Sigma \right)$ is the holonomy along a path in the path space ${P}_{s}^{t}\left({U}_{i}\right)$ over ${U}_{i}$ of the path-space 1-form

(3)${\Omega }^{1}\left({P}_{s}^{t}\left({U}_{i}\right),\mathrm{Lie}\left(H\right)\right)\ni {𝒜}_{\left(A,B\right)}={\oint }_{A}\left(B\right)$

obtained from the 2-form $B\in {\Omega }^{2}\left({U}_{i},\mathrm{Lie}\left(H\right)\right)$ by pulling it back to the path and integrating it over the path while parallel transporting it (with $A$ and an action

(4)$\alpha :G\to \mathrm{Aut}\left(H\right)$

given by the nature of our 2-group) to the source of ${\gamma }_{i}$.

This is described in gory detail in hep-th/0412325.

In general, the surface whose surface holonomy you are computing won’t fit into a single ${U}_{i}$. So you triangulate your surface such that each face ${\Sigma }_{i}$ fits into a single ${U}_{i}$ and compute all the $𝒲\left({\Sigma }_{i}\right)$ as above. Then you need to come up with a way to glue all of them together to get a single $𝒲\left(\Sigma \right)$ in a way that is well defined and in particular invariant under the 2-gauge transformations in your 2-bundle.

There is a single frightening diagram which tells you how to do that. It’s number thirteen or so, right below the sentence

Hence category-LEGO logic tells us that the definition of global 2-holonomy must be as follows:

I thought these diagrams were cute. They made me think of playing with LEGO. There are hexagons, squares and triangles in the game and there is only a single way how they fit together at their edges. The hard part, which is maybe a little frightening, is to show that what must ‘obviously’ be correct actually is correct, i.e. gauge invariant. (Note however that I have not tried yet to prove invariance of the choice of triangulation. I expect the proof of that to be completely analogous to that for abelian gerbe holonomy.)

Using these diagrams our functor $\mathrm{hol}$ is actually globally defined. And hence given any surface (bigon) $\Sigma$ with source and target edge identified, $\mathrm{hol}\left(\Sigma \right)$ is its 2-holonomy (for strict 2-groups here).

There are obvious ways how to ‘trace’ over an automorphic strict 2-group element, just like you would for ordinary holonomy. This is discussed by Girelli&Pfeiffer, for instance. Call that operation $\mathrm{Tr}$.

This means that a $\sigma$-model for nonabelian strings charged under a strict 2-group is given by the exponentiated action

(5)$\mathrm{exp}\left(S\left(\Sigma \right)\right)=\mathrm{exp}\left({S}^{\mathrm{kin}}\left(\Sigma \right)\right)\cdot \mathrm{Tr}\left({\mathrm{hol}}_{\left\{{A}_{i},{B}_{i}\right\}}\left(\Sigma \right)\right)$

with the above $\mathrm{hol}$.

For instance for that 2-group ${𝒫}_{k}\mathrm{Spin}\left(n\right)$ which is related to $\mathrm{String}\left(n\right)$ the 2-group element associated to any closed surface is of the form

(6)$\mathrm{hol}\left(\Sigma \right)=p\stackrel{\left(1,c\right)}{\to }p$

with $p\in {P}_{0}\mathrm{Spin}\left(n\right)$ an element of the based path group over $\mathrm{Spin}\left(n\right)$ and $\left(1,c\right)\in \stackrel{̂}{{\Omega }_{k}\mathrm{Spin}\left(n\right)}$ an element in the center of the Kac-Moody centrally extended loop group over $\mathrm{Spin}\left(n\right)$ with $c\in U\left(1\right)$.

In this case the trace can simply chosen to be

(7)$\mathrm{Tr}\left(p\stackrel{\left(1,c\right)}{\to }p\right)=c\phantom{\rule{thinmathspace}{0ex}}.$

As I said, it seems to me that all this 2-stuff is related to closed strings, while derived categories and the like are related to open strings.

Depends. It’s the open strings which most naturally become morphisms, since they have source and target.

Posted by: Urs on April 8, 2005 6:59 PM | Permalink | Reply to this

### Re: New preprint: From Loop Groups to 2-Groups (and the String Group)

Sorry for the bad formatting and the weird typos in the original version of my above message. I have corrected that now. If you wondered about the original message please have another look.

The reason for the mess I posted was that I am currently on a CFT conference in Bonn and was using somebody’s personal computer at University of Bonn in the evening. Before I could properly finish the above message they were kicking me out of that room (it was late, I guess…) and I hastily submitted the comment in its awkward form.

Posted by: Urs on April 9, 2005 12:33 PM | Permalink | Reply to this

### Re: New preprint: From Loop Groups to 2-Groups (and the String Group)

what is this String group good for?

The String group enters crucially into the construction of enriched elliptic objects in Stolz&Teichner’s work. This is physically relevant since these enriched elliptic objects are something like a rigorous version of a superstring.

Now, Stolz&Teichner’s article is sufficiently complex that it is not at all easy for me to point out why exactly the String group appears there the way it does. I could mumble a couple of things about Eilenberg-MacLane spaces and so on, but in the end it would not really condense to the brief answer that should exist.

But I believe the general heuristic argument is, as you indicate yourself, that $\mathrm{String}\left(n\right)$ is to Dirac operators on loop space (= worldsheet supercharges) much like $\mathrm{Spin}\left(n\right)$ is to ordinary Dirac operators.

Regarding ‘rigorous’ one should mention the following two points, though:

1) An enriched elliptic object is like a superstring where the bosonic modes are treated classically while only the worldsheet fermions are quantized. This is due to the fact that a mathematician does not know how to make rigorous sense of the bosonic part of the path integral.

2) Stolz&Teichner’s work is famously still unpublished und unfinished. As far as I have heard one of the reasons for that is that it is not clear how to make the horizontal composition of ‘bigons’ (pointed worldhsheet elements as they appear in 2-categorical language) which carry a conformal structure precise, since at the connceting point we don’t have a smooth manifold anymore and hence technical problems with conformal structure on the composite piece of worldsheet appear.

That much is for sure. Now, there is the still speculative aspect (as far as I am aware at least) that Aaron keeps mentioning:

Like the obstruction to a spin structure measures the failure of the existence of spinor fields, the obstruction to a string structure should measures the failure of the existence of string fields for the superstring.

Fields are not really functions but are sections of bundles. Hence the string field should really be a ‘section’ of something and only locally look like the naive field that one usually thinks of. That something plausibly has a ‘structure group’ related to $\mathrm{String}\left(n\right)$.

As you can see in the recent entries here at the SCT and in my long discussion with Aaron and others on sci.physics.strings, I am currently believing that there is a notion of ‘line 2-bundle’ and that work on derived categoric description of D-branes as well as this stuff on elliptic cohomology indicates that the 2-sections of such line 2-bundles are what generalized string fields (= objects of derived categories describing D-brane configurations) really are. I might be wrong, of course, this is currenly my pet idea. See my reply to Aaron’s comment for more.

Finally, concerning your question about $ℤ$-valued ambiguities I can unfortunately only say that these expectations sound very reasonable, but that I don’t know. Maybe somebody does. If not, it should be possible to figure this out, I hope.

Posted by: Urs on April 8, 2005 1:28 PM | Permalink | Reply to this

### Re: New preprint: From Loop Groups to 2-Groups (and the String Group)

Hi,

I remember the good old days of reading dialogs on SPR between Oz and the Wizard. In this place, I feel like Oz, so I hope a naive question now and then from the peanut gallery won’t distract too much.

I won’t even pretend to understand what Urs is working on these days, but I do understand the desire to search for Dirac operators. Although, instead of Dirac operators, I was looking for Hodge stars, (thanks to Urs) it became clear to me (at least in elementary diff geo) that the two were related since you can construct Dirac operators from exterior derivatives and their adjoints. Is a similar Dirac-Kaehler-like thing happening in some freaky scary space that I have no chance of ever understanding?

(Back to lurking…)

Eric

Posted by: Eric on April 8, 2005 2:39 PM | Permalink | Reply to this

### 2-Dirac operators

Great question.

As explained in

C. Lazaroiu: Generalized Complexes and String Field Theory

and reviewed in

C. Lazaroiu: D-Brane Categories

from boundary string field theory we get a dG-category of string fields. The objects are D-branes, the morphism are string fields describing states of string stretching between given two D-branes. There is some extra structure describing string interactions and of course there is also the BRST operator which acts as a graded Leibnitz nilpotent operator on the algebra of the string fields.

So if you adopt the point of view of 2-NCG and see the algebra of string fields (which are morphisms!) as a categorification of an ordinary algebra (at this point not quite in the sense of HDA2 but see below) then the BRST operator $d$ here is sort of a 2-deRham operator.

If there is what is called cohomological splitting we get an inner product with respect to which there is an adjoint ${d}^{†}$ of the BRST operator and the massless fluctuation around the vacuum of the string field action are described by those string fields $\varphi$ which satisfy

(1)$d\varphi =0={d}^{†}\varphi$

(cf. pp. 10-11 of the above hep-th/0305095. )

But now I would like to ask/remark the following:

The above dG-category is not the last word, since it knows only about manifold-like D-branes and knows no brane/anti-brane annihilation. We really need to consider complexes in this category which leaves us with the ${A}_{\infty }$-enhancement ${D}_{\infty }^{b}\left(X\right)$ of ${D}^{b}\left(X\right)$.

This is also nice, I think, in the sense that objects in these derived category more cleanly fit into the idea of 2-NCG. In that context we should think of these objects, locally at least, as elements of a 2-Hilbert space on which a 2-Dirac operator should be represented.

So the question here is: Does the $d$ from above extend to a nilpotent graded functor

(2)$d:{D}_{\infty }^{b}\left(X\right)\to {D}_{\infty }^{b}\left(X\right)$

on the ${A}_{\infty }$ enhanced derived category of coherent sheaves on target space??

It sure acts naturally on the objects in that category, but does it extend to a functor?

If yes, then so should ${d}^{†}$.

Now, as I have emphasized before, objects in ${D}_{\infty }^{b}\left(X\right)$ look rather like local 2-sections than like elements of a 2-algebra. The 2-algebra acting on them is rather that containing the autoequivalences of our D-brane category. If twe represent the latter equivalently by ${D}_{\infty }^{b}\left(\mathrm{kQ}-\mathrm{Mod}\right)$ for $\mathrm{kQ}$ the path algebra of some quiver $Q$, then this 2-algebra is something like ${D}_{\infty }^{b}\left(\mathrm{kQ}-\mathrm{Mod}-\mathrm{kQ}\right)$.

Does $d$ extend to a functor on this guy, too?

I don’t know. But if it would we’d be incredibly close to having a 2-spectral triple

(3)$\left(𝒜,ℋ,{𝒟}^{i}\right)$

with

(4)$𝒜\sim {D}_{\infty }^{b}\left(\mathrm{kQ}-\mathrm{Mod}-\mathrm{kQ}\right)$

the 2-algebra,

(5)$ℋ\sim \left\{\left(Gammma{D}_{\infty }^{b}\left(\mathrm{kQ}-\mathrm{Mod}{\right)}^{n}\right\}$

a 2-Hilbert space of suitable sections of 2-vector bundles (modules of $𝒜$)

and

(6)${𝒟}^{±}=d±{d}^{†}$

two 2-Dirac operators.

Nils Baas has promised a paper on 2-Dirac operators this spring. I am looking forward to comparing this with these observations.

Posted by: Urs on April 8, 2005 5:21 PM | Permalink | Reply to this
Read the post Seminar on 2-Vector Bundles and Elliptic Cohomology, I
Weblog: The String Coffee Table
Excerpt: Review of the 2-vector approach towards elliptic cohomology. Part I.
Tracked: February 2, 2006 7:03 PM
Read the post Remarks on String(n)-Connections
Weblog: The String Coffee Table
Excerpt: Is Stolz/Teichner's String-connection a 2-connection in an associated 2-bundle?
Tracked: March 11, 2006 3:13 PM
Read the post Stevenson, Henriques on String(n)
Weblog: The String Coffee Table
Excerpt: Stevenson and Henriques have articles on the String group.
Tracked: March 23, 2006 9:27 AM
Read the post Categorified Gauge Theory in Chicago
Weblog: The String Coffee Table
Excerpt: Conference with emphasis on categorified gauge theory, covering gerbes and the String group.
Tracked: April 4, 2006 10:12 AM
Read the post Jurco on Gerbes and Stringy Applications
Weblog: The String Coffee Table
Excerpt: Jurco reviews some facts concerning nonabelian gerbes in string theory.
Tracked: April 19, 2006 11:38 AM
Read the post Talk in Bonn on the String 2-Group
Weblog: The String Coffee Table
Excerpt: Talk on the String 2-Group in Bonn.
Tracked: April 26, 2006 8:21 PM
Read the post 10D SuGra 2-Connection
Weblog: The n-Category Café
Excerpt: On the Lie 2-algebra governing 10-dimensional supergravity.
Tracked: August 28, 2006 3:51 PM
Read the post On n-Transport: 2-Vector Transport and Line Bundle Gerbes
Weblog: The n-Category Café
Excerpt: Associated 2-transport, 2-representations and bundle gerbes with connection.
Tracked: September 7, 2006 2:01 PM
Read the post Puzzle Pieces falling into Place
Weblog: The n-Category Café
Excerpt: On the 3-group which should be underlying Chern-Simons theory.
Tracked: September 28, 2006 3:29 PM
Read the post Kosmann-Schwarzbach & Weinstein on Lie Algebroid Classes
Weblog: The n-Category Café
Excerpt: Kosmann-Schwarzbach on classes of Lie algebroids generalizing the character of a Lie algebra.
Tracked: October 10, 2006 1:29 PM
Read the post WZW as Transition 1-Gerbe of Chern-Simons 2-Gerbe
Weblog: The n-Category Café
Excerpt: How the WZW 1-gerbe arises as the transition 1-gerbe of the Chern-Simons 2-gerbe.
Tracked: October 29, 2006 4:55 PM
Read the post A 3-Category of twisted Bimodules
Weblog: The n-Category Café
Excerpt: A 3-category of twisted bimodules.
Tracked: November 3, 2006 2:09 PM
Read the post Dijkgraaf-Witten Theory and its Structure 3-Group
Weblog: The n-Category Café
Excerpt: The idea of Dijkgraaf-Witten theory and its reformulation in terms of parallel volume transport with respect to a structure 3-group.
Tracked: November 6, 2006 8:25 PM
Read the post Chern-Simons Lie-3-Algebra inside derivations of String Lie-2-Algebra
Weblog: The n-Category Café
Excerpt: The Chern-Simons Lie 3-algebra sits inside that of inner derivations of the string Lie 2-algebra.
Tracked: November 7, 2006 8:55 PM
Read the post 2-Monoid of Observables on String-G
Weblog: The n-Category Café
Excerpt: Rep(L G) from 2-sections.
Tracked: November 24, 2006 5:34 PM
Weblog: The n-Category Café
Excerpt: A conference on bundles and gerbes, another one on topology, and comments on associated 2-vector bundles and String connections.
Tracked: April 19, 2007 8:56 PM
Read the post Zoo of Lie n-Algebras
Weblog: The n-Category Café
Excerpt: A menagerie of examples of Lie n-algebras and of connections taking values in these, including the String 2-connection and the Chern-Simons 3-connection.
Tracked: May 10, 2007 6:21 PM
Read the post Connections on String-2-Bundles
Weblog: The n-Category Café
Excerpt: On connections on String 2-bundles.
Tracked: June 3, 2007 4:04 PM
Read the post On BV Quantization. Part I.
Weblog: The n-Category Café
Excerpt: On BV-formalism applied to Chern-Simons theory and its apparent relation to 3-functorial extentended QFT.
Tracked: August 17, 2007 10:42 PM
Read the post Lie n-Algebra Cohomology
Weblog: The n-Category Café
Excerpt: On characteristic classes of n-bundles.
Tracked: September 7, 2007 6:00 PM
Read the post Obstructions, tangent categories and Lie n-tegration
Weblog: The n-Category Café
Excerpt: Thoughts on n-bundle theory in terms of Lie n-algebras.
Tracked: September 24, 2007 10:00 PM
Read the post Cohomology of the String Lie 2-Algebra
Weblog: The n-Category Café
Excerpt: On the Lie 2-algebra cohomlogy of the String Lie 2-algebra and its relation to twisted K-theory.
Tracked: October 2, 2007 10:45 PM
Read the post On Lie N-tegration and Rational Homotopy Theory
Weblog: The n-Category Café
Excerpt: On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan's old article on this issue in particular.
Tracked: October 20, 2007 4:34 PM
Read the post Chern-Simons States from L-infinity Bundles, III: States over the Circle
Weblog: The n-Category Café
Excerpt: On computing the states of Chern-Simons theory over the circle from the L-infinity algebraic model of the Chern-Simons 3-bundle over BG.
Tracked: February 4, 2008 6:08 PM
Read the post Construction of Cocycles for Chern-Simons 3-Bundles
Weblog: The n-Category Café
Excerpt: On how to interpret the geometric construction by Brylinksi and McLaughlin of Cech cocycles classified by Pontrjagin classes as obstructions to lifts of G-bundles to String(G)-2-bundles.
Tracked: February 12, 2008 1:03 PM
Read the post Teleman on Topological Construction of Chern-Simons Theory
Weblog: The n-Category Café
Excerpt: A talk by Constant Teleman on extended Chern-Simons QFT and what to assign to the point.
Tracked: June 17, 2008 6:55 PM
Read the post Higher Structures in Math and Physics in Lausanne
Weblog: The n-Category Café
Excerpt: A workshop in Lausanne on Higher Structures in Mathematics and Physics
Tracked: November 2, 2008 5:19 PM

Post a New Comment