Re: Self-dual strings and M5-brane anomalies

October 22, 2004

Re: Self-dual strings and M5-brane anomalies

Posted by Urs Schreiber

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[This is a followup to Luboš’s blog entry Self-dual strings and M5-brane anomalies. Let’ see if it is possible to have an inter-blog discussion.]

When I read Berman’s & Harvey’s hep-th/0408198 a while ago I learned a bit more about how difficult it really is to understand the situation with $Q_5 \gt 1$ coincident 5-branes.

Important for me at that point was the reference to

X. Bekaert & M. Henneaux & A. Sevrin: Chiral forms and their deformation (2000)

which demonstrates that it is impossible to have a local deformation of an abelian theory of self-dual 2-forms to a non-abelian one.

This result is perhaps reminiscent of the insight of

F. Girelli & H. Pfeiffer: Higher gauge theory - differential versus integral formulation (2004)

that the ‘naive’ (depending on what you still consider naive) local extension of the YM Lagrangian to nonabelian 2-forms constituting a local connection for a strict structure 2-group is really equivalent to ordinary gauge theory again and does not yield the expected extension.

The same problem from another point of view is that one can write down the dynamical fields that are expected to describe the coincident 5-branes by deriving data describing a nonabelian gerbe from M5-brane anomaly cancellation as in

P. Aschieri & B. Jurčo: Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory (2004) ,

but then it is not known (yet) how to construct gauge invariant quantities, holonomies and ‘Lagrangian-like’ objects from that data.

Though there are some first hints, I believe. I think I can show that the nonabelian gerbes (without connection) considered by Aschieri & Jurčo are equivalent to the data describing 2-transitions in 2-bundles (without connection), which were very recently introduced in

T. Bartels: Categorified gauge theory: 2-Bundles (2004) .

That’s good, because the 2-categoric context of 2-bundles should allows us to simply categorify the objects that we are looking for in the ordinary incarnation to get their nonabelian 2-form version.

In particular, it should be possible to have a 2-morphisms from the 2-groupoid of 2-paths in the base space of a 2-bundle to the structure 2-group and thus get a holonomy of non-abelian 2-forms. It remains to be seen if the non-abelian cocycle data introduced that way is still equivalent to Aschieri&Jurč’s nonabelian gerbes with connection, but the success in the case without connection suggests that this must be true.

The good thing is that if we pick the base 2-space of our 2-bundle to be $B$ with point space

(1)

B^1 = \mathcal{M}

being ‘spacetime’ (i.e. the 5-brane worldvolume) and the arrow space

(2)

B^2 = \Omega\mathcal{M}

the space of based loops over $\mathcal{M}$ (i.e. the configuration space of closed strings in the 5-brane) then such a 2-connection gives rise to a connection on the loop space $B^2$ (as the entire 2-bundle gives rise to an ordinary bundle over $\Omega$ \mathcal{M}) and we know some things about how to get holonomies from such nonabelian path-space connections and how they can in principle be more general than the local strict 2-group holonomies considered by Girelli&Pfeiffer (though it remains to be better understood exactly how they are more general).

In fact, this leads me to the cubic scaling of degrees of freedom that Lubš talked about in his post.

So if I understand correctly the usual asymptotic scaling with $n^2$ , where $n$ is the dimension of the Cartan sub-algebra of the gauge group) is simply an indication of the fact that fields in the adjoint rep and in particular the gauge bosons themselves have to make up $n \times n$ matrices and thus appear in bunches of $n^2$ .

So if objects in theories on $n$ $Q_5$ branes scale faster than $n^2$ this might indicate that the gauge connection in these theories requires for its specification more than a $\mathrm{Lie}(\mathrm{SU}(n))$ -valued differential form. (What Luboš addresses as fields carrying ‘three indices’).

But that’s exactly what is the case in general for the 2-connections and the connections on loop space! Bekaert,Henneaux&Sevrin in their paper mention the famous old result by Teitelboim (Phys. Lett. 167B (1986) 63) that no ‘straightforward’ (as they say) non-abelian extension of the 2-form field exists, which is based on the assumption that the holonomy of a 2-form gauge field $B$ along a string worldsheet $\Sigma$ is

(3)

W_B[\Sigma] = \exp\left(i \int_\Sigma B\right) \,,

even in the nonabelian case.

But the boundary state deformation considerations that I gave in hep-th/0407122 as well as the relation to the 2-group connections shows that (what Alvarez et. al had, in a special case, considered before and what was also expressed by Hofmann) that this fails because parallel transport of the non-abelian $B$ to a reference fiber has to be included, generalizing the above to

(4)

W_B[\Sigma] = \mathrm{P}\exp\left( i \int_\Sigma W_A^{-1} B W_A \right) \,.

This form of a path space connection evades Teitelboim’s no-go theorem if the 2-form $B$ and the 1-form $A$ satisfy a certain condition, dubbed r-flatness by Alvarez, which can be shown to be equivalent to the exchange law arising in a sesqui-connection (a 2-connection into a sesqui-group).

There is some discussion necessary concerning the uniqueness of this form, but in any case it shows that the gauge fields expected to describe this non-abelian parallel transport of strings involves more than one $\mathrm{Lie}(\mathrm{SU}(n))$ -valued differential form and hence more than $n^2$ ‘fields of data, namely (at least) two such forms. This holds locally and indeed the general results on nonabelian gerbes say that there must be even more forms involved.

Now this does not yet prove the $n^3$ scaling, but this is kind of suggestive to be part of the explanation why the scaling is $\gt n^2$ .

Posted at October 22, 2004 7:43 PM UTC

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Re: Re: Self-dual strings and M5-brane anomalies

Hi, Urs,

I guess you have read this literature a lot more thoroughly than I ever did, but the last time I looked at it, I remember seeing an N^3 popping out in an unexpected place: if you look at Hofman’s paper on nonabelian 2-forms, as part of the data of his connection on loop space, he uses a field with 3 indices that gets interpreted as determining an algebra structure on the N-dimensional “Chan-Paton” vector space. Roughly, this field looks like it describes the way that the fivebranes are knitted together by condensation of “pant” diagrams, which would allow strings on the M5-brane worldvolume to change their Chan-Paton indices as they interact with one another. Is it related to what you are discussing above?

Posted by: Andy Neitzke on October 24, 2004 2:15 AM | Permalink | Reply to this

Re: Re: Self-dual strings and M5-brane anomalies

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if you look at Hofman’s paper on nonabelian 2-forms, as part of the data of his connection on loop space, he uses a field with 3 indices that gets interpreted as determining an algebra structure on the N-dimensional “Chan-Paton” vector space

Are you talking about the field $M$ in equation (21) of hep-th/0207017? (As far as I am aware Christiaan Hofman has never published the announced ‘Nonabelian Wilson surfaces’. (?))

That equation says among other things that a connection for a nonabelian 2-form theory contains more data than (locally) fits into a single Lie-agebra valued 1-form, so yes, that’s part of what I was talking about.

But I have to admit that I hadn’t considerd the possibility that the $n^3$ degrees of freedom might be precisely correspond to those encoded in that field $M$ which encodes the algebra product, because usually this $M$ is thought to be fixed. Hofman himself fixes it later on, so that for instance it no longer appears in his table 2.

But it is intriguing to locate the $n^3$ scaling here.

On the other hand, Hofman does not discuss any restriction on the connection coming from the requirement that one can construct a surface holonomy from it. That might in principle reduce the available degrees of freedom. But then, how many degrees of freedom are there really in $\mathrm{Hom}(\mathfr{g}^{\otimes 2},\mathfr{g})$ ?

Posted by: Urs Schreiber on October 24, 2004 5:45 PM | Permalink | Reply to this

Re: Re: Self-dual strings and M5-brane anomalies

Exactly, that’s the M I was talking about, in

Hofman, Christiaan. “Nonabelian 2-forms,” hep-th/0207017.

At the level of numerology, that N^3 seems to jump right out. Well, finding physics in it would be quite another matter…

Just for my own curiosity, now I am looking at my old notes and trying to translate the vague ideas I had into the things I have seen you saying recently. Does the condition that one can construct a surface holonomy have something to do with the vanishing of the “generalized curvature,” constructed from A,B,M by adding the curvature of A to the adjoint action of B with respect to the algebra product?

Posted by: Andy Neitzke on October 25, 2004 7:30 AM | Permalink | Reply to this

Re: Re: Self-dual strings and M5-brane anomalies

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finding physics in it would be quite another matter…

Yes, and the first question to be answered in order to address this question would probably be what the space of states that Hofman’s operators are supposed to act on really is.

I have a vague idea that the space of multi-integrals that Hofman uses and that goes back to work by Chen, Getzler and others probably, might even be the correct space of states for tensionless strings, which would make sense in the context of 5-branes. I have made some comments on this recently here.

One problem is that

a) there seems to be no complete understanding and/or agreement of how to represent the constraints of tensionless string on some Hilbert space in the first place

c) it is not clear what such a worldsheet description would buy us. As Wolfgang Lerche pointed out, it might be completely inadequate for describing anything on the 5-branes. But others have expressed the idea that it might in fact capture at least some physics, like Hofman does in his concluding remarks.

If we had a physical world-sheet like interpretation of these multi-integrals it seems conceivable that the physics of the $M$ -operator on these integrals could be identified.

To me it seems like a hint that when you generalize the boundary states that describe nonabelian 1-forms in the obvious way to inlcude a nonabelian 2-form you automatically get from the deformed (possibly tensionless) string’s supercharges nonabelian connections on loop space that have a well-defined surface holonomy.

Does the condition that one can construct a surface holonomy have something to do with the vanishing of the ‘generalized curvature’, constructed from $A$ , $B$ , $M$ by adding the curvature of $A$ to the adjoint action of $B$ with respect to the algebra product?

In principle yes, but concerning Christiaan Hofman’s paper there is a catch.

I am not sure how to even in principle define any notion of surface holonomy from the Hochschild complex operators that Christaan Hofman talks about. Maybe that’s related to the fact that he never seems to have published the announced paper on Nonabelian Wilson Surfaces? I should have asked him about this last time I talked to him.

The problem is the direct generalization of the ‘shuffle product’ to the nonabelian case. In the abelian case we have

(1)

\oint (B) \wedge \oint (\omega_1, \dots, \omega_n) = \sum \pm \oint (\omega_1, \dots, \omega_k,B,\omega_{k+1},\dots, \omega_n)

and Hofman takes this to remain true in the non-abelian case, which motivates his (15). But this turns $\oint (B) \wedge$ from an operator of exterior multiplication by some loop space 1-form into a more sophisticated object, namely some sort of derivation on multi-integrals. That’s because the ordinary exterior multiplication with the nonabelian $\oint (B)$ would yield

(2)

\oint(B) \wedge \oint (\omega_1,\dots, \omega_n) = \sum \pm \oint (\omega_1^{a_1}, \dots, \omega_k^{a_k},B^{a},\omega^{a_{k+1}}_{k+1},\dots, \omega_n^{a_n}) \otimes t_a t_{a_1} t_{a_2}\cdots t_{a_n}

with the scalar part of $B$ shuffled but the $\mathfr{g}$ -valued part in the original order.

Similar comments apply to his $M$ operator. From my perspective $M$ is really a new kind of exterior derivative, as I have discussed here, rather than a generalized ‘1-form’ or something.

Without a good physical understanding of what all this is supposed to describe we can of course define what we like and see if it is interesting. But I don’t know how to get surface holonomies from Hofman’s connections.

What I do know is that given any covariant exterior derivative $d + A$ , for $A$ some $\mathfr{g}$ -valued 1-form we can define a holonomy $W_A = \mathrm{P}\exp\left(i \int A\right)$ . I could imagine that this can be generalized, but then there seem to appear a couple of details that require further discussion.

On the other hand, from various other points of view like

- the boundary state deformation technique

- the considerations by Alvarez et al.

- the theory of 2-groups and 2-connections (i.e. the 2-bundle context)

it seems more natural to have a connection on loop space locally described by an honest 1-form $\oint_A (B)$ . This can straightforwardly be integrated to a surface holonomy and the result equals the 2-group holonomy of a 2-connection (i.e. of a 2-functor from the 2-groupoid of bigons in target space into a given 2-group) cooked up from $A$ and $B$ . To me this is a strong indication that this is the right thing to do. 2-connections should be the right sort of connections on 2-bundles (though this is still under investigation) and 2-bundles can be shown to be equivalent to nonabelian gerbes (at least I think I can). There is much to be better understood here, though.

Now, to finally answer your question: The surface holonomy of $\oint_A (B)$ is well defined if and only if it is what Alvarez called r-flat, namely invariant under reparameterizations of any given surface, obviously. One can show that r-flatness is equivalent to the 2-category ‘exchange law’ in 2-connections. So far, all known interesting nonabelian solutions to r-flatness are even flat. That’s where the flatness condition that you mentioned comes from.

It is an open problem to understand if there are non-flat r-flat connections on loop space and if (what is the case so far for all known examples) there are r-flat connections that are not equivalent to those satisfying $dt(B) + F_A = 0$ .

Of course, also in this context, which is not exactly that discussed by Christiaan Hofman, we can consider modifying the exterior derivative on loop space by using the $\sim n^3$ degrees of freedom contained in $M$ . Maybe this is even the key to getting more interesting r-flat connections on loop space. But I’d have to think about it.

Posted by: Urs Schreiber on October 25, 2004 1:40 PM | Permalink | Reply to this

Re: Re: Self-dual strings and M5-brane anomalies

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I wrote:

I am not sure how to even in principle define any notion of surface holonomy from the Hochschild complex operators

Maybe I should expand on what the difficulties are that I encounter.

They are similar to the difficulties that one also encounters when trying to define the action of the loop space exterior coderivative $d^*$ on those multi-integrals: In general the action of that operator on a multi-integral yields something that has no (at least no obvious) representation as a linear combination of such multi-integrals.

Not every $p$ -form on loop space that one can write down is manifestly a linear combination of these multi-pull-back-integrals!

(Maybe one can prove a theorem that any $p$ -form on loop space can be written as a linear combination (or a limit of such) of multi-pull-back integrals of forms on target space? But I don’t know. I could imagine that something like this might be true, but it is far from obvious and probably quite subtle.)

This becomes a problem for the following reason:

For finite dimensional ordinary geometry, given a covariant exterior derivative $\nabla = d_A = d + A$ and a path $\gamma$ with tangent $t$ we can define the gauge covariant holonomy $W$ as the solution of

(1)

\iota_t (d+A)W = 0

(2)

\Leftrightarrow \mathcal{L}_t W = -\{\iota_t, A\}W \,,

where $\mathcal{L}$ is the Lie derivative. and $\{\iota_t, A\}$ is the contraction of $A$ with $t$ .

For this prescription to work $A$ can be more general than simple exterior multiplication with a 1-form together with an action on some internal degrees of freedom. So that’s what we need.

But when this prescription is generalized to loop space one encounters the problem that the inner product $\iota_t$ of an arbitrary loop space vector $t$ with a multi-pull-back integral does not in general yield another such multi-pull-back integral-type form. So if $A$ is something that is defined only on these multi-integrals, we won’t be able to integrate the above for $W$ . That’s the problem.

The above presciption does work on loop space however for those loop space vectors $t = t^{(\mu,\sigma)} \frac{\delta}{\delta X^{(\mu,\sigma)}}$ that come from target space vectors $v$ , i.e. for

(3)

t^{(\mu,\sigma)} = v^\mu(X(\sigma)) \,,

where $X : (0,1) \to \mathcal{M}$ is the loop.

It is clear that this is a severe restriction, just consider any loops with self-intersections.

But maybe that’s sufficient for some purposes. Hm…

Posted by: Urs Schreiber on October 25, 2004 7:00 PM | Permalink | Reply to this

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