### Gerbes, strings, and Nambu brackets

#### Posted by urs

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.

**[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:

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]**

## Mechanics on Loop Space

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

on loop space has the loop space 2-form curvature

coming from the target space 3-form $H={d}_{A}B$.

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’

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 $\mathrm{LM}\times [\mathrm{0,1}]$ 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

where $d$ 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’.