Gerbes, strings, and Nambu brackets

November 1, 2005

Posted by Urs Schreiber

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John Baez asks me to forward the following question to the Coffee Table. It’s an intriguing question, related (to my mind at least) to the question how ‘categorified quantum theory’ and string/loop physics are related. But it’s much more concrete than that.

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[begin of forwarded message]

One of the puzzling things about the process of categorifying line bundles to get gerbes is how it relates to geometric quantization.

Line bundles are classified by integral closed 2-forms. Gerbes are classified by integral closed 3-forms.

Integral closed 2-forms show up everywhere in classical mechanics. The symplectic structure on the phase space of a classical system is a closed 2-form, and it’s integral if it satisfies the ‘Bohr-Sommerfeld quantization condition’. The corresponding line bundle becomes important when we actually go ahead and quantize the system! Wavefunctions are sections of this bundle.

What’s the analogous story for integral closed 3-forms, if any???

My only guess is that ‘Nambu brackets’ are involved. I don’t know much about these. I just got some papers in the mail about them:

Thomas Curtright, Cosmas Zachos Quantizing Dirac and Nambu Brackets AIP Proc. 672 (2003) 165-182 hep-th/0303088

T. L. Curtright, C. K. Zachos Branes, Strings, and Odd Quantum Nambu Brackets hep-th/0312048

Cosmas Zachos, Thomas Curtright Branes, Quantum Nambu Brackets, and the Hydrogen Atom Czech. J. Phys. 54 (2004) 1393-1398 math-ph/0408012

I haven’t gotten around to digging up earlier papers, which might be easier to understand. But anyway…

A closed 2-form $\omega$ lets you define the Poisson bracket $\{f,g\}$ of two functions $f$ and $g$ . Similarly, a closed 3-form lets you define the bracket $\{f,g,h\}$ of three functions. I’m guessing that this is an example of a ‘Nambu bracket’, and that Nambu went on to define even higher brackets.

Somehow these higher brackets are related to strings and branes. Despite the above papers, I’m not sure how! But thanks to my work with Urs Schreiber on higher gauge theory and string theory, I can’t help but guess some analogy like this:

The electromagnetic field $F$ is a closed 2-form, which should be added to the usual symplectic structure on the cotangent bundle of spacetime to obtain the Poisson brackets describing the dynamics of a charged particle. (This is true and well-known.)

The Kalb-Ramond field $G$ is a closed 3-form, which should be added to the usual 3-form on something or other to obtain the Nambu brackets describing the dynamics of a charged string. (This is just a guess!)

Does anyone know anything about this???

As if that weren’t mysterious enough already, there’s also the question of this analogy:

(1)

\array{ \text{Poisson brackets}\; \{f,g\} & \text{Nambu brackets}\; \{f,g,h\} \\ \text{symplectic structure} & \text{nondegenerate closed 3-form} \\ \text{K&auml;hler structure} & \text{???} }

In geometric quantization we pick a Kähler structure having our symplectic structure as its imaginary part, so we can talk about holomorphic sections of the corresponding line bundle - these are the allowed wavefunctions.

What’s the analogous thing for a nondegenerate closed 3-form, if anything???

[end of forwarded message]

Posted at November 1, 2005 1:30 PM UTC

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12 Comments & 1 Trackback

Mechanics on Loop Space

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I would love to know the full answer to this question.

I bet, though, that one aspect of a partial answer would go like this:

Given the 3-form on target space $M$ (let me call it $H \in \Omega^3(M)$ ), we obtain a (Chen-) 2-form $\int \mathrm{ev}^* H$ on loop space in the usual way [1].

Hence if we can deal with infinite-dimensional mechanics, we now can do symplectic geometry and classical mechanics on loop space given appropriate 3-forms on target space.

I expect that, in as far as free string dynamics can be described using symplectic geometry on loop space, the 2-form $\int \mathrm{ev}^* H$ for $H$ the Kalb-Ramond field strength will play precisley the role of the electromagnetic field strength $F$ in ordinary dynamics. This is certainly true for all kinds of expressions that one finds from string dynamics. And it is technically nothing but what we found for 2-bundles with 2-connection. There the local connection 1-form

(1)

\mathcal{A}(\gamma) = \int_\gamma W_\gamma(\mathrm{ev}^* B)

on loop space has the loop space 2-form curvature

(2)

\mathcal{F}(\gamma) = \int W_\gamma(\mathrm{ev}^* H)

coming from the target space 3-form $H = \mathbf{d}_A B$ .

In fact, this is also how people construct a line bundle on loop space given an abelian gerbe on target space. (I know you know this, but let me emphasize it in the context of this question). They take the 3-form representing the class of the gerbe and cook up a 2-form on loop space from it, which there classifies a line bundle.

All this addresses the second and maybe the third line of the little table in your question. I have no clear idea how Nambu brackets fit into this particular picture.

—–

[1]

I’ll recall for those readers unfamiliar with this procedure how $\int \mathrm{ev}^* H$ is defined.

$\mathrm{ev}$ is the ‘evaluation map’

(3)

\array{ \mathrm{ev} &:& LM \times [0,1] &\to& M\\ && (\gamma,\sigma) &\mapsto& \gamma(\sigma) }

which sends a loop and a parameter value to the position of that loop at that parameter value in target space.

$\int \mathrm{ev}^* H$ denotes the 2-form on loop space obtained by pulling back $H$ to $LM\times [0,1]$ along $\mathrm{ev}$ and then integrating one ‘leg’ over the given loop at which we want to know the value of the 2-form. In local coordinates this is simply

(4)

(\int \mathrm{ev}^* H)(\gamma) = \int_0^1 d\sigma\; H(\gamma(\sigma))_{\mu\nu\rho} \frac{d\gamma^\rho}{d\sigma}(\sigma) \, \mathbf{d}\gamma^\mu(\sigma) \wedge \mathbf{d}\gamma^\nu(\sigma) \,,

where $\mathbf{d}$ is the exterior differential on loop space. This differntial (and the way it appears above) is made precise using the theory of loop space geometry by K.-T. Chen, otherwise known as ‘diffeology’.

Posted by: Urs Schreiber on November 1, 2005 2:44 PM | Permalink | Reply to this

Re: Mechanics on Loop Space

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Thanks for your comments, Urs. I made a little progress on this puzzle today talking with Danny Stevenson. Some positive progress (thinking of new ideas) and some negative progress (realizing my old ideas were stupid). Right now I’ll just describe the negative progress.

A nondegenerate closed 2-form $\omega$ gives an isomorphism between the tangent bundle and the cotangent bundle, and thus gives a map, say #, from 1-forms to vector fields. This lets us define Poisson brackets as follows: $\{f,g\} = \omega(#df,#dg)$ Alas, a nondegenerate closed 3-form $\nu$ does not give us such an isomorphism. It thus does not let us define Nambu brackets as follows: $\{f,g,h\} = \nu(#df,#dg,#dh)$ unless we can get our hands on the isomorphism # in some other way.

For Nambu brackets what we really need is trivector field $N$ satisfying some equation; then we can define $\{f,g,h\} = N(df,dg,dh)$ This is just like how Poisson brackets work. For Poisson brackets we start with a bivector field $\Pi$ satisfying some equation, and define $\{f,g\} = \Pi(df,dg)$ This nondegenerate bivector field is called the Poisson tensor. One way to get it is starting from a symplectic structure, as above. But, there are Poisson tensors that don’t arise this way.

So, it’s a bit trickier than I thought, but I still think there has to be some 3-form analogue of symplectic geometry, and geometrically quantizing this must be related to gerbes the way geometric quantization of symplectic manifolds is related to line bundles.

More later - including maybe a comment on Kea’s comment, once I remember what I once knew about “generalized geometry”.

Best, jb

Posted by: john baez on November 2, 2005 6:06 AM | Permalink | Reply to this

Nambu brackets from 3-forms

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I woke up this morning at about 6 am - early for me - and started thinking about math.

I made some more negative progress on Nambu brackets. But, this time it was “double negative” progress: realizing that I was being stupid when I thought my previous ideas were stupid. So, I guess this is positive progress.

(Both negative and positive progress are forms of progress, but positive progress is more fun.)

True: if you give me a nondegenerate closed 3-form $\nu$ it doesn’t give an isomorphism # from 1-forms to vector fields, so we can’t define Nambu brackets by $\{f,g,h\} = \nu(#df,#dg,#dh)$

But, there’s something smarter that might actually work! A 3-form gives a map sending vector fields to 2-forms: $u \mapsto \nu(u,-,-) .$ Suppose we can find a one-sided inverse to this map; call it #. Then we can define a ternary bracket by $\{f,g,h\} = \nu(#(df \wedge dg),h)$ And, maybe if $\nu$ and # satisfy some nice properties, these ternary brackets will satisfy nice properties too!

The above formula nicely fits Nambu’s notion that in his theory, it takes two Hamiltonians to define time evolution. Two functions f and g give a 2-form $df \wedge dg$ which gives a vector field $#(df \wedge dg)$ .

Note that I say “one-sided inverse” because an honest two-sided inverse would be too much to hope for… except when the phase space is 3-dimensional, making the space of vectors have the same dimension as the space of 2-forms. This is a way in which the idea I’m talking about now is doomed to be less beautiful than symplectic geometry.

In fact, I took a look at Nambu’s original paper yesterday (you can see it online if you have the journal subscription), and I think he considered the 3-dimensional case! More generally, I think Nambu was interested in n-ary brackets on an n-dimensional phase space.

But, in applications to string theory the relevant phase spaces are infinite-dimensional, so we might be able to demand an isomorphism between vectors and 2-forms! I actually think this could work….

You probably think I’m nuts, because the whole idea of defining time evolution by two Hamiltonians sounds crazy. However, I think it may not be so crazy: I think it may be compatible with more familiar approaches to physics. At least I hope so. I’m not trying to do something bizarre; I’m just trying to see how 3-forms can do for physics something like what 2-forms already do.

There’s a lot more to say, but….

Posted by: john baez on November 2, 2005 3:11 PM | Permalink | Reply to this

Re: Nambu brackets from 3-forms

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You probably think I’m nuts, because the whole idea of defining time evolution by two Hamiltonians sounds crazy.

Quite the opposite, actually. Yesterday I was toying around with a similar approach, since there actually are naturally two Hamiltonian-like gadgets for a string.

Namely

(1)

L

and

(2)

\bar L

:-)

More precisely, we don’t only have the Hamiltonian constraint

(3)

H \sim L + \bar L

which generates worldvolume $\tau$ -evolution (as for the particle)

but also the reparameterization constraint

(4)

P \sim L - \bar L

which generates worldvolume $\sigma$ -translations.

I’ll try to think about how this observation might merge with what you mentioned.

Posted by: Urs Schreiber on November 2, 2005 3:54 PM | Permalink | Reply to this

Re: Nambu brackets from 3-forms

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Wait, something is wrong here:

Suppose we can find a one-sided inverse to this map; call it $#$ . Then we can define a ternary bracket by

(1) $\{f,g,h\} = v(#(df\wedge dg),h)$

Namely $#(df\wedge dg)$ is a vector, while $h$ is still a function. This is not what you can feed into $v$ .

There are obvious things one could try to try to fix this…

Posted by: Urs Schreiber on November 2, 2005 4:04 PM | Permalink | Reply to this

Re: Gerbes, strings, and Nambu brackets

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Well, guessing here without thinking, surely Hitchin’s Generalised Geometry is important. In fact I found this Hitchin talk from July 2005.

Posted by: Kea on November 2, 2005 3:07 AM | Permalink | Reply to this

Re: Gerbes, strings, and Nambu brackets

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surely Hitchin’s Generalised Geometry is important

Generalized geometry is all about talking about an abelian gerbe in terms of the (2-)algebroid

(1)

TM \oplus T^*M \overset{\rho}{\to} TM \,.

(which, I think, should be the Atiyah-2-algebroid of the associated 2-bundle, but anyway).

One idea that comes to mind is that there is a 2-term $L_\infty$ -like structure at work in the background. This happens to come with a trinary bracket which is indeed related to the curvature 3-form. This is a possible candidate for something related with Nambu brackets.

Posted by: Urs on November 2, 2005 10:07 AM | Permalink | Reply to this

Twisted Poisson brackets

Hi Urs,

about your last coffee table entry. I do not know whether this is exatly what John has in mind, but you can twist a Poisson bracket by a closed three form (not necessarily nondegenerate). I think it appeared first time as a twisted Poisson sigma model by Klimcik and Strobl (math.SG/0104189) and by Severa and Weinstein (math.SG/0107133) (I do not know in which order) in the context very close
to Hitchin’ s generalized geometry. Then Severa and independetly me (with Paolo, Peter Schupp and Igor Bakovic, hep-th/0206101) have described the quantization of these twisted Poisson brackets. You can interpret it as a stack of algebras (or a noncommutative gerb, really gerb in the sense as Chaterjee uses it in his theses. It is just one example of what you could call a nonabelian gerbe). If you think this is relevant will be glad to discuss more.

Regards,
Brano

Posted by: Branislav Jurco on November 2, 2005 11:32 AM | Permalink | Reply to this

Re: Twisted Poisson brackets

Thanks for the references, Brano! The paper by Klimcik and Strobl is indeed relevant, in that it uses a 2-form on the target space of a 2d nonlinear sigma model to modify the Poisson brackets on the classical phase space of this model. It also does the same using a 3-form on the target space, by assuming that the 2-manifold in question is the boundary of a 3-manifold. (This is a kind of Wess-Zumino-Witten trick.)

All this fits my philosophy nicely, except that I want to a novel sort of “phase space” for this kind of theory in which the symplectic structure is replaced by a closed 3-form.

Posted by: john baez on November 2, 2005 4:24 PM | Permalink | Reply to this

Re: Twisted Poisson brackets

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“All this fits my philosophy nicely, except that I want a novel sort of phase space …..”

Do you mean phase space, or just coadjoint orbit or something like that? There seems to be a lot of work using gerbes in the context of geometric quantization by people like Brylinski. In one of his gerbe papers he mentions (and references) the connection to group-valued moment maps , which sounds like an interesting generalization of the usual Hamiltonian picture.

Just a thought.

Posted by: Kea on November 3, 2005 4:55 AM | Permalink | Reply to this

‘Covariant Hamiltonian Formalism’

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I had a vague recollection from long long ago (it seems) that C. Rovelli once thought about related things. A little searching yielded his article

C. Rovelli, Covariant Hamiltonian Formalism for Field Theory: Hamilton-Jacobi equation on the space $\mathcal{G}$ , gr-qc/0207043

He thinks in terms of “3-branes” there, instead of “1-branes” (“strings”), but the conceptual problem is the same.

So in particular he proposes to replace the 1-form

(1)

\theta_1 = \sum_i p_i \, dq^i

from ordinary mechanics by the 4-form

(2)

\theta_4 = p_{ijkl} dq^i \wedge dq^j \wedge dq^k \wedge dq^l \,.

Like our symplectic form is (locally) the differential of $p_i dq^i$

(3)

\omega_2 = d\theta_1 = dp_i \wedge dq^i

(everything expressed here locally) he then uses the 5-form

(4)

\omega_5 = d\theta_4

(see pp. 12-13)

as a generalization of that in order to write down ‘covariant Hamiltonian formalism’ for “3-branes”. Of course what he really considers is GR in 3+1 dimensions.

Posted by: Urs Schreiber on November 2, 2005 5:12 PM | Permalink | Reply to this

Read the post Classical, Canonical, Stringy
Weblog: The String Coffee Table
Excerpt: A stringy 3-form generalization of 2-form symplectic canonical mechanics.
Tracked: November 2, 2005 8:12 PM

Re: Gerbes, strings, and Nambu brackets

Have a look at

http://arxiv.org/abs/hep-th/0202173

Posted by: Mike on November 20, 2005 5:54 PM | Permalink | Reply to this

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November 1, 2005