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February 16, 2007

QFT of Charged n-Particle: T-Duality

Posted by Urs Schreiber

Last time I described how the idea of pull-push propagation in quantum mechanics should look like when we refine the formalism to quantization on a category, or even to quantization on an nn-category, i.e. when we systematically replace spaces by categories and regard, for instance, a string not just as an interval [0,1] [0,1] \subset \mathbb{R} but as a poset par={ab} \mathrm{par} = \{a \to b\} \, propagating not just on a target space X X but on the corresponding category of 2-paths tar=P 2(X). \mathrm{tar} = P_2(X) \,.

In particular, I drew a pasting diagram that descibed the pull-push of a section,

(1) ev *1 conf×par e phas ev *tra \array{ & \nearrow\searrow^{\mathrm{ev}^*1} \\ \mathrm{conf}\times\mathrm{par} & \;\Downarrow e& \mathrm{phas} \\ & \searrow \nearrow_{\mathrm{ev}^*\mathrm{tra}} }

of an nn-bundle with connection tra:tarphas \mathrm{tra} : \mathrm{tar} \to \mathrm{phas} through a suitable correspondence.

(2) hist×worldvol out * in * conf×par conf×par ev ev tar tra e phas , \array{ && \mathrm{hist}\times \mathrm{worldvol} \\ & \multiscripts{^{\mathrm{out}^*}}{\swarrow}{} && \searrow^{\mathrm{in}^*} \\ \mathrm{conf}\times \mathrm{par} && \Leftarrow && \mathrm{conf}\times \mathrm{par} \\ & {}_{\mathrm{ev}} \searrow && \swarrow_{\mathrm{ev}} & \downarrow \\ && \mathrm{tar} && \downarrow \\ && \mathrm{tra}\downarrow\;\; & \stackrel{e}{\Leftarrow}& \downarrow \\ && \mathrm{phas} & \leftarrow } \,,

supposed to describe the quantum evolution of the state ψe\psi \simeq e corresponding to that section over the worldvolume worldvol\mathrm{worldvol}.

I claim that this is the natural operation of a worldvolume on a state. And I claim that it is once again crucial that we have understood a section as a transformation (1) between transport functors. Notice that, by passing to the components of (2), the 2-morphisms filling this diagram– which are forced upon us by the transformation nature of sections – turn the bare correspondence hist out * in * conf conf \array{ && \mathrm{hist} \\ & \multiscripts{^{\mathrm{out}^*}}{\swarrow}{} && \searrow^{\mathrm{in}^*} \\ \mathrm{conf} && && \mathrm{conf} } into a correspondence with an (n1)(n-1)-bundle on the correspondence space P hist out * in * conf conf. \array{ && \mathrm{P} \\ && \downarrow \\ && \mathrm{hist} \\ & \multiscripts{^{\mathrm{out}^*}}{\swarrow}{} && \searrow^{\mathrm{in}^*} \\ \mathrm{conf} && && \mathrm{conf} } \,. Moreover, by the rules for composition of transformations of functors, the pull-push through this correspondence automatically and naturally incorporates the action of that bundle on the section pulled up the correspondence space.

Such a transformation is known to categorify ordinary linear operations, as recalled in Fourier-Mukai, T-Duality and other linear 2-Maps.

In particular, (topological) T-duality for 2-particles is an example for such a transformation, as described in Mathai on T-Duality, II: T-dual K-classes by Fourier-Mukai.

1) First a word on T-duality:

In the context of quantum theory, we conceive the geometry of effective target space, i.e. of the space that our quantum nn-particle propagates in, entirely in terms of its observable effects on that nn-particle.

We probe space by letting our quantum nn-particle roam in it.

At least for n=1n=1 (but conceivably also for higher nn) this is exactly the idea of Alain Connes’ notion of (“spectral”) geometry:

Geometry is whatever is encoded in the spectrum of a suitale Laplace-like operator (the Hamiltonian of our particle) or Dirac-like operator (the supercharge of our superparticle).

Taking this seriously, we are lead to find kinds of geometries that have, in particular, notions of automorphisms that go beyond the diffeomorphisms of ordinary manifolds.

One of the most famous of these “generalized diffeomorphisms” is T-duality. This arises, quite generally, for “2-particles” – also known as strings.

The idea is, roughly, that when probing a space of the form X×T, X \times \mathbf{T}\,, where T\mathbf{T} is the 1-torus, by letting a 2-particle roam in it, we usually cannot distinguish it from X×T^, X \times \hat \mathbf{T} \,, where T^\hat \mathbf{T} is the dual torus.

The reason is, roughly, that the spectrum of the Hamiltonian of the 2-particle contains contributions from

a) the ordinary momentum of the 2-particle

but also

b) from the linear extension of the 2-particle.

As we pass from a 2-particle on a torus to that on the dual torus, these two contributions interchange their role, but such that the total spectrum (of the loop space Laplace operator) remains unaffected!

So the notion of geometry as induced by the quantum theory of the 2-particle has a notion of automorphism not known to 1-particles. These isomorphisms are called T-dualities. For the obvious reason (“T”=torus).

This is, at least, the standard way to descrtibe T-duality in string theory. However, the basic mechanism is much more general than any mentioning of “momentum” and “linear extension” might suggest: in the context of what is known as Topological T-Duality people study the effect of T-duality in particular on twisted K-theory, without ever considering the dynamics of any 2-particle.

We have seen that K-theory, which classifies Chan-Paton bundles on D-branes (and twisted K-theory, which classifies twisted Chan-Paton bundles known as gerbe modules) is a purely kinematical aspect of the 2-particle.

So: topological T-duality is the kinematical aspect of T-duality.

Now a word on T-Folds:

In as far as T-duality is a “generalized diffeomorphism” of a quantum geometry, it makes sense to regard local such diffeomorphisms.

Like an ordinary manifold is something that locally looks like n\mathbb{R}^n such that the transitions are diffeomorphisms of n\mathbb{R}^n, we can imagine defining a T-fold to be something that locally looks like a manifold, with the transitions being given by T-dualities.

I am not aware that this idea has been adopted seriously by mathematicians and developed rigorously (but maybe it has: I’d be grateful for references!), but string theorists do have thought about this. See for instance

C. M. Hull
A Geometry for Non-Geometric String Backgrounds

Finally: the example

In order to obtain an easily tractable but interesting example of pull-push evolution of 2-particles, I would like to consider a 2-particle on a target space 2-category tar\mathrm{tar} which models a very simple T-fold structure.

The idea is to model the T-fold transitions in the T-fold in the same spirit that group action transitions are modeled in an orbifold: we simply include additional morphisms between points that are related by the respective transformation. (For how orbifolds are described by groupoids this way see Moerdijk on Orbifolds, II.)

So given a torus T\mathbf{T}, and its dual torus T^\hat \mathbf{T} (both really just being S 1S^1), consider the 2-category P 2 T(S 1×S 1) P_2^T(S^1 \times S^1) which is defined to be that generated by 2-paths in T\mathbf{T} P 2(T) P_2(\mathbf{T}) and 2-paths in T^\hat \mathbf{T} P 2(T^) P_2(\mathbf{\hat T}) as well as 1-morphisms that are pairs of points T×T^ \mathbf{T} \times \hat \mathbf{T} and T^×T. \hat \mathbf{T} \times \mathbf{T} \,.

More generally, we might consider T-duality on a target space of the form X×TX \times \mathbf{T}. Then we would suitably define tar=P 2 T(X×S 1×S 1) \mathrm{tar} = P_2^T(X \times S^1 \times S^1) in the more or less obvious way.

The 2-particle on that target space will couple to a Kalb-Ramond background field, which can be regarded as nothing but a 2-functor tra:P 2 T(X×S 1×S 1)Σ(1dVect ). \mathrm{tra} : P_2^T(X \times S^1 \times S^1) \to \Sigma(1d\mathrm{Vect}_\mathbb{C}) \,.

On the generators P 2(X×T)P_2(X\times \mathbf{T}) and P 2(X×T^)P_2(X\times \hat \mathbf{T}) we take this to be any bundle gerbe with connection that we like.

But, since we are now on a T-fold, we will also have to define how tra\mathrm{tra} acts on the new T-duality morphisms in T×T^\mathbf{T} \times \hat \mathbf{T} .

Since tra\mathrm{tra} takes values in Σ(1dVect )\Sigma(1d\mathrm{Vect}_\mathbb{C}), this will be nothing but a line bundle P T×T^. \array{ P \\ \downarrow \\ \mathbf{T} \times \hat \mathbf{T} } \,.

Take this to be the Poincaré line bundle.

Finally, consider the open 2-particle of the form par=(ab) \mathrm{par} = (a\to b) and consider a simple cylindrical worldvolume of that wolldvol=(a a b b). \mathrm{wolldvol} = \left( \array{ a &\to& a' \\ \downarrow &\Rightarrow& \downarrow \\ b &\to& b' } \right) \,. Take the space of histories hist \mathrm{hist} to be maps from this to all T-duality morphisms.

Then the pull-push evolution on our 2-particle along this worldvolume should give a T-duality transformation on the Chan-Paton bundles over the endpoints aa and bb.

And I claim it does. Feeding this data into the pull-push defined by (2) reproduces, over the 2-particle’s endpoints, precisely the topological T-duality transformation.

I wanted to describe this in more detail. But now I did run out of time. Have to run now. To be continued (maybe in the comment section.)

Posted at February 16, 2007 12:15 PM UTC

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Re: QFT of Charged n-Particle: T-Duality

I have a little more time now. Will begin describing some of the details.

There a a couple of simple technical issues that are best seen by writing out everything on a piece of paper. For instance, I frequently use the isomorphism Hom(conf×par,phas)Hom(conf,Hom(par,phas)) \mathrm{Hom}(\mathrm{conf}\times \mathrm{par},\mathrm{phas}) \simeq \mathrm{Hom}(\mathrm{conf} ,\mathrm{Hom}(\mathrm{par},\mathrm{phas})) (which in particular means that the product in expressions like conf×par\mathrm{conf}\times \mathrm{par} is always taken to be the tensor product adjoint to the Hom. So in the 2-categorical setup that the present example lives in this means that ×\times is the Gray tensor product.)

The main points to see what is going on are the following:

First of all, while for a line bundle gerbe our parallel transport 2-functor sends morphisms to 1-dimensional vector spaces tra:tarΣ(1dVect ) \mathrm{tra} : \mathrm{tar} \to \Sigma(1d\mathrm{Vect}_\mathbb{C}) we really want to regard this as taking values inside 2-vector spaces tra:tarΣ(1dVect )Bim2Vect . \mathrm{tra} : \mathrm{tar} \to \Sigma(1d\mathrm{Vect}_\mathbb{C}) \hookrightarrow \mathrm{Bim} \hookrightarrow 2\mathrm{Vect}_\mathbb{C} \,. (For a detailed discussion of this inclusion see this.)

That’s important, because the choice of codomain, even though tra\mathrm{tra} only hits Σ(1dVect)\Sigma(1d\mathrm{Vect}), affects where the transformations of this 2-functor take values in – and hence where the states of our 2-particle take values in.

So take the codomain to be Bim\mathrm{Bim}, as always.

Then the first crucial thing to be aware of is that a transformation

e:ev *1ev *tra e : \mathrm{ev}^*1 \to \mathrm{ev}^*\mathrm{tra} is, over each endpoint of our 2-particle, (ab)(a \to b) a (twisted) vector bundle on XX.

I once gave a detailed discussion of the mechanism behind that in Flat Sections and Twisted Groupoid Reps.

The main point is that such a transformation comes, over each path xγy x \stackrel{\gamma}{\to} y in target space, from a 2-morphism in Bim\mathrm{Bim} (the front or back face of the naturality tin can diagram characterizing pseudonatural transformations of 2-functors) of the form Id e a(x) e(γ) e b(y) tra(γ) . \array{ \mathbb{C} &\stackrel{\mathrm{Id}}{\to}& \mathbb{C} \\ e_a(x)\downarrow \;\; &\;\;\Downarrow e(\gamma)& \;\;\downarrow e_b(y) \\ \mathbb{C} &\stackrel{ \mathrm{tra}(\gamma) }{\to}& \mathbb{C} } \,.

So e ae_a is a \mathbb{C}-bimodule bundle over XX, as is e be_b: but that just means these are vector bundles over XX.

This are the “Chan-Paton” vector bundles that the endpoints of the string couples to.

(This is really just another incarnation of the fact that morphisms of bundle gerbes, aka “stable morphisms”, are given by vector bundles.)

With this in hand, the remaining main point is to write out the combined transformation

(1) hist×worldvol out * in * conf×par conf×par ev ev tar tra e phas , \array{ && \mathrm{hist}\times \mathrm{worldvol} \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf}\times \mathrm{par} && \Leftarrow && \mathrm{conf}\times \mathrm{par} \\ & {}_{\mathrm{ev}} \searrow && \swarrow_{\mathrm{ev}} & \downarrow \\ && \mathrm{tar} && \downarrow \\ && \mathrm{tra}\downarrow\;\; & \stackrel{e}{\Leftarrow}& \downarrow \\ && \mathrm{phas} & \leftarrow } \,,

out in components, evaluated in the context that I specified at the end of the above entry.

The composition of transformations involved looks, for each T-fold-transition morphism xx^ x \to \hat x in components (over the endpoints aa and bb of the 2-particle, respectivey) like this: e(x) tra(xx^) . \array{ \mathbb{C} &\stackrel{e(x)}{\to}& \mathbb{C} &\stackrel{\mathrm{tra}(x\to\hat x)}{\to}& \mathbb{C} \\ } \,.

But this is exactly what says that we pull back the Chan-Paton bundle on T\mathbf{T} to the correspondence space T×T^\mathbf{T}\times \hat \mathbf{T}, and then tensor it with the Poincaré line bundle on that correspondence space (before pushing everything down again).

To see this, carefully write out what (1) says in components, taking care of the rules for horizontal and vertical composition of transformations.

Posted by: urs on February 16, 2007 3:47 PM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: Extended Worldvolumes
Weblog: The n-Category Café
Excerpt: Passing from locally to globally refined extended QFTs by means of the adjointness property of the Gray tensor product of the n-particle with the timeline.
Tracked: August 2, 2007 7:49 PM
Read the post Groupoidification from sigma-Models?
Weblog: The n-Category Café
Excerpt: On constructing sigma-models from differential nonabelian cocycles and how this gives rise to methods of groupoidification.
Tracked: December 30, 2008 4:27 PM

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