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 lets you define the Poisson bracket of two functions and . Similarly, a closed 3-form lets you define the bracket 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 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 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)
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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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 (let me call it ), we obtain a (Chen-) 2-form 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 for the Kalb-Ramond field strength will play precisley the role of the electromagnetic field strength 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)
on loop space has the loop space 2-form curvature
(2)
coming from the target space 3-form .
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 is defined.
is the ‘evaluation map’
(3)
which sends a loop and a parameter value to the position of that loop at that parameter value in target space.
denotes the 2-form on loop space obtained by pulling back to along 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)
where 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’.
Re: Gerbes, strings, and Nambu brackets
Well, guessing here without thinking, surely Hitchin’s Generalised Geometry is important. In fact I found this
Hitchin talk from July 2005.
Re: Gerbes, strings, and Nambu brackets
surely Hitchin’s Generalised Geometry is important
Generalized geometry is all about talking about an abelian gerbe in terms of the (2-)algebroid
(1)
(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 -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.
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
‘Covariant Hamiltonian Formalism’
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 ,
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)
from ordinary mechanics by the 4-form
(2)
Like our symplectic form is (locally) the differential of
(3)
(everything expressed here locally) he then uses the 5-form
(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.
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
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 (let me call it ), we obtain a (Chen-) 2-form 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 for the Kalb-Ramond field strength will play precisley the role of the electromagnetic field strength 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 .
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 is defined.
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.
denotes the 2-form on loop space obtained by pulling back to along 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 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’.