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January 24, 2007

The Globular Extended QFT of the Charged n-Particle: Definition

Posted by Urs Schreiber

After thinking about it for a while (A B C D E F G H I J) it seems that I am finally at a point where I can venture to state a comprehensive formal definition of the structure whose working title was the charged quantum nn-particle.

The following definition is taken from the beginning of

The Globular Extended QFT of the String propagating on the Classifying Space of a strict 2-Group

which develops one of simplest interesting examples in more detail (to be discussed in a followup post).

The two definitions, discussed in detail below, roughly go like this:

Definition 1. A charged nn-particle is a setup (parγconftartraphas) \left( \mathrm{par} \stackrel{\gamma \in \mathrm{conf}}{\to} \mathrm{tar} \stackrel{\mathrm{tra}}{\to} \mathrm{phas} \right) internal to nCatn\mathrm{Cat}.

Definition 2. The quantization of a charged nn-particle is the nn-functor on par\mathrm{par} obtained by pull-pushing tra\mathrm{tra} through the correspondence conf×par ev tar par. \array{ & & \mathrm{conf}\times \mathrm{par} \\ & \multiscripts{^{\mathrm{ev}}}{\swarrow}{}\;\; && \searrow \\ \mathrm{tar} &&&& \mathrm{par} } \,.

There is a mystery that demands to be understood:

Mystery: The theory of gerbes with connection in terms of local data exhibits a lot of structural resemblance to state sum models of 2-dimensional quantum field theory, topological as well as conformal.

Why is that?

Does this point to a deeper pattern that we might want to understand?

Can we maybe understand these involved state sum models from first principles?

After a little bit of reflection, I think the pattern is this:

  • nn-Bundles with connection are naturally conceived in terms of parallel transport nn-functors.
  • Coupling these nn-connections to an nn-particle amounts to transgressing these nn-functors to a suitable configuration space.
  • Quantizing these charged nn-particles amounts to pushing the transgressed nn-functors forward to a point.

From this point of view, evolution in the quantum field theory of the charged nn-particle is an nn-functor that is inherently obtained from the parallel transport nn-functor that expresses the background field that the particle propagates in.

Both, the original parallel transport nn-functor as well as the resulting quantum propagation nn-functor may be locally trivialized. For the former this yields the local description of gerbe holonomy. For the latter this yields the state sum description of QFT.

This situation may be visualized by the following cube, a more detailed description of which is given here.

The essence of the quantization step, going horizontally from left to right in the above cube, is, as I now believe, well captured by the following two definitions.

Definition 1. A charged nn-particle (parγconftartraphas) \left( \mathrm{par} \stackrel{\gamma \in \mathrm{conf}}{\to} \mathrm{tar} \stackrel{\mathrm{tra}}{\to} \mathrm{phas} \right) is

  • an (n1)(n-1)-category par\mathrm{par}, called parameter space and thought of as modelling the shape and internal structure of the nn-particle
  • an nn-category, tar\mathrm{tar}, called target space and thought of as modelling the space that the nn-particle propagates in
  • an nn-category phas=nVect\mathrm{phas} = n\mathrm{Vect}, being the nn-category of some notion of nn-vector spaces
  • an nn-functor tra:tarphas\mathrm{tra} : \mathrm{tar} \to \mathrm{phas}, thought of as encoding the parallel transport in an nn-bundle with connection on target space
  • a choice of sub-nn-category conf[par,tar]\mathrm{conf} \subset [\mathrm{par},\mathrm{tar}], thought of as encoding the configuration space of the nn-particle.

Of course this data wants to be interpreted internal to a suitable context. Depending on the strictness or weakness of notion of nn-category one uses, and depending on which additional structures – usually smooth local trivializability – one imposes on tra\mathrm{tra}, we are dealing with a strict or weak, continuous or smooth nn-particle, etc.

The main point is now

Definition 2. Given a charged nn-particle (parγconftartraphas), \left( \mathrm{par} \stackrel{\gamma \in \mathrm{conf}}{\to} \mathrm{tar} \stackrel{\mathrm{tra}}{\to} \mathrm{phas} \right) \,, its globular extended quantum field theory is the nn-functor q(tra):parphas q(\mathrm{tra}) : \mathrm{par} \to \mathrm{phas} which is the image of tra\mathrm{tra} under the morphism qq indicated in the following diagram. tar ev conf×par p par q: [tar,phas] ev * [conf×par,phas] p *¯ [par,phas] tra ev *tra q(tra). \array{ &\mathrm{tar} &\stackrel{\mathrm{ev}}{\leftarrow}& \mathrm{conf} \times \mathrm{par} &\stackrel{p}{\to}& \mathrm{par} \\ q : & [\mathrm{tar},\mathrm{phas}] &\stackrel{\mathrm{ev}^*}{\to}& [\mathrm{conf}\times \mathrm{par},\mathrm{phas}] &\stackrel{\bar {p^*}}{\to}& [\mathrm{par},\mathrm{phas}] \\ & \mathrm{tra} &\mapsto& \mathrm{ev}^* \mathrm{tra} &\mapsto& q(\mathrm{tra}) } \,.

Here conf×parevtar\mathrm{conf} \times \mathrm{par} \stackrel{\mathrm{ev}}{\to} \mathrm{tar} is the obvious evaluation map coming from restricting [par,tar]×parevtar[\mathrm{par},\mathrm{tar}] \times \mathrm{par} \stackrel{\mathrm{ev}}{\to} \mathrm{tar} along the inclusion conf[par,tar]\mathrm{conf} \hookrightarrow [\mathrm{par},\mathrm{tar}].

Precomposition with this pp defines a pull-back nn-functor p *:[par,phas][conf×par,phas] p^* : [\mathrm{par},\mathrm{phas}] \to [\mathrm{conf}\times\mathrm{par},\mathrm{phas}] and p *¯:[conf×par,phas][par,phas] \bar{p^*} : [\mathrm{conf}\times\mathrm{par},\mathrm{phas}] \to [\mathrm{par},\mathrm{phas}] denotes the adjoint of this nn-functor, a push-forward.

This quantization procedure sends a transport functor on target space to a transport functor on parameter space.

The meaning of the above definition is indicated by the following table.

This nn-functor q(tra)q(\mathrm{tra}) on parameter space is an extended QFT in that refines the original definition of a QFT as a 1-representation of a cobordism 1-category nQFT:nCob SVect n-QFT : n\mathrm{Cob}_S \to \mathrm{Vect} to an nn-functorial representation on nn-vector spaces, which assigns data at all dimensional levels of its domain.

I call the quantized transport q(tra)q(\mathrm{tra}) a globular extended QFT in order to distinguish it from a bunch of other proposals to define extended QFTs as representations of extended cobordisms categories. In such extended cobordisms categories, objects correspond to collections of points on parameter space (possibly empty), morphisms correspond to collections of arcs in parameter space (possibly empty), and so on.

On the other hand, in a globular extended QFT, an nn-morphism in the domain is precisely a single nn-arc in parameter space.

This greatly streamlines the handling of all local aspects of parameter space. It does however come at the cost that no topological nontrivial cobordisms can be handled globally without performing traces, as briefly indicated in section 1.3 of Transport Theory.

Caveat. Without further qualification, the above reproduces only the kinematics of what one ordinarily considers as quantization. Dynamics should follow the same pattern, but is more subtle.

On the other hand, whereas ordinary quantization produces a mere 1-functor QFT:nCob SVect, QFT : n\mathrm{Cob}_S \to \mathrm{Vect} \,, the above, being an extended QFT, produces an nn-functor q(tra):parnVect q(\mathrm{tra}) : \mathrm{par} \to n\mathrm{Vect} which contains quite a bit of information, even at the kinematical level. In particular, higher phenomena such as D-branes, string fusion, closed bulk insertions and various other higher structures – that may not even have names yet – are captured.

But more important is this:

there is a phenomenon (a kind of holography) which says, formulated as a principle, roughly that

dynamics of the quantum nn-particle \leftrightarrow kinematics of the quantum (n+1)(n+1)-particle

This is one of the main motivations for the entire setup discussed here. I shall come back to that later.

Quantization and Transgression.

We can understand the above definitions in a broader context. Notice that, given a charged nn-particle (parγconftartraphas) \left( \mathrm{par} \stackrel{\gamma \in \mathrm{conf}}{\to} \mathrm{tar} \stackrel{\mathrm{tra}}{\to} \mathrm{phas} \right) we can form the diagram conf p 1 tar ev conf×par p 2 par \array{ &&&& \mathrm{conf} \\ &&& \multiscripts{^{p_1}}{\nearrow}{}\; \\ \mathrm{tar} &\stackrel{\mathrm{ev}}{\leftarrow}& \mathrm{conf}\times \mathrm{par} \\ &&& \multiscripts{_{p_2}}{\searrow}{}\; \\ &&&& \mathrm{par} } and use this to push-pull transport functors on target space either to parameter space – this is the quantization we discussed – or to configuration space. The latter procedure is known as transgression.

Posted at January 24, 2007 5:03 PM UTC

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1 Comment & 30 Trackbacks

Read the post Globular Extended QFT of the Charged n-Particle: String on BG
Weblog: The n-Category Café
Excerpt: The string on the classifying space of a strict 2-group.
Tracked: January 26, 2007 2:52 PM

Re: The Globular Extended QFT of the Charged n-Particle: Definition

Just a very preliminary comment, provoked by
the definition of quantization of a charged
n-particle via a correspondence. That’s
very similar to one way of generating higher
homotopies, cf. inverting quasiisos (aka quisms).
And that in turn is relevant to the (some kind of ) equivalence of n-groupoids and n-homotopy types.

Some of the talks at the IHP Higher
Structures meeting are now posted at
http://www.math.psu.edu/ping/IHP07/slides/
and hopefully more will be added soon.
Breen’s talk was particularly relevant.

Posted by: jim stasheff on January 30, 2007 5:16 PM | Permalink | Reply to this
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