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August 3, 2007

QFT of Charged n-Particle: Towards 2-Functorial CFT

Posted by Urs Schreiber

[ Update: We now have a more developed write-up:

Jens Fjelstad and U. S.
Rational CFT is parallel transport
(pdf) ]


With Jens Fjelstad I am working on understanding 2-dimensional conformal field theory as an extended quantum field theory, that is: as a propagation 2-functor of a charged 2-particle.

Various aspects of this I mention every now and then. An overview of some of the existing work had accompanied my Fields Institute Talk: On 2-dimensional QFT: from Arrows to Disks.

Here is a description of some main aspects of the project.

Starting point – the FRS prescription for (rational) 2d CFT.

The most detailed understanding of (rational) 2-dimensional conformal field theory – including its relation to 3-dimensional topological field theory – currently available comes from the FRS formalism (statement, literature, more recent developments).

The local behaviour of a 2-dimensional conformal field theory, including the very dependence on the conformal structure, is encoded in the representation theory of a chiral vertex operator algebra. The FRS prescription gives a solution to the problem of constructing, essentially, a quantum propagation functor on arbitrary conformal cobordisms whose restriction to conformal annuli close to the identity reproduces the information encoded in the given chiral algebra (as in part 4 of this talk). The functoriality of this functor is what is known as a solution to the sewing constraints.

[This needs to be discussed in more detail.]

While it certainly has an internal elegance to it – being to a large degree a sophisticated version of what are known as state sum models, such as they are used for topological strings – the prescription which accomplishes this is somewhat baroque.

It consists of a set of rules for how to assign to any worldsheet a ribbon diagram running inside a 3-manifold cobounding the orientation bundle of the worldsheet, and colored by certain objects and morphisms in the modular tensor category of representations of the chiral algebra.

This 3-manifold with ribbon diagram inside is then fed into the Reshitikhin-Turaev prescription – by itself a not entirely straightforward mechanism – which constructs 3-dimensional topological field theories from a given modular tensor category.

This way one eventually obtains a vector in some vector space, which may finally be mapped by some isomorphism into the space of correlation functions of the CFT one wishes to construct.

[This needs to be discussed in more detail.]

It can be proven that the collection of correlators spit out by this cooking recipe indeed defines a full (rational) 2-dimensional CFT based on the specified chiral algebra.

While quite remarkable, this success leaves one left with the undeniable feeling that somewhere below all this machinery there must lie hidden a more fundamental mechanism which controls all this.


The goal: a 2-Functorial description of 2-dimensional CFT

Here we would like to collect evidence for the following statement:

1st Statement. The FRS formalism for 2-dimensional quantum field theory arises as the local data of a local trivialization of a propagation 2-functor going from a 2-categorical refinement of the category of surfaces to the 2-category of 2-vector spaces.

Quantization, Categorification and Local Trivialization form the edges of a cube of operations on n-functors

Moreover

2nd Statement. This propagation 2-functor is the component map of a transformation of propagation 3-functors of the corresponding 3-dimensional topological field theory.

To round this up, there is at least a partial understanding of the connection of nn-functorial quantum propagation to the classical theories it comes from:

3rd Statement. In the concrete special case that the 3-dimensional TFT is Dijkgraaf-Witten theory, the propagation 3-functor is indeed the quantization, realized as a certain canonical colimit (playing the role of the path integral) of a 3-functor on the given finite group with values in 3-vector spaces.

To collect the necessary motivation for drawing the bigger nn-categorical picture of quantum theory which we see emerge here, we shall start by demonstrating in detail how the FRS prescription arises from a 3-functorial construction for the special case where we restrict attention to just worldsheets of the form of a disk. (Hence we here just go From Arrows to Disks.)

But within that restricted setup, we quite generally discuss arbitrary boundary conditions as well as bulk and boundary insertions. In fact, the point is that all these structures appear by themselves from just the abstract mechanism of nn-functorial QFT.


A first clue: bulk field insertions and transformations of 3-functors

A key clue, simple to understand and yet very helpful to get an impression for what is going on, is the structure of the bulk field insertions in the FRS formalism.

Suppose we are looking, locally, at a piece of worldsheet where “a closed string state comes in from infinity” – a bulk field insertion. Whatever that really means, following the FRS prescription we are to describe the situation by a diagram of this kind:

Here the plane indicates the piece of worldsheet surface which we are looking at. The rectangle in the center is the place where the bulk field is inserted.

Every such bulk field is labeled by two representations of the chiral algebra (the “left- and right-moving part”). These appear in the diagram as the two ribbons running vertically. These are to be thought of as being labeled by the (identity morphism on) the simple objects U U and V V of the modular tensor category which these representations correspond to.

The horizontal ribbon line running inside the surface is, in general, what is called a defect line. While these are important for understanding the general situation (they encode for instance dualities like T-duality or t’Hooft operators, such as appear in the 2d QFT description of geometric Langlands), let us assume for the moment that in our example this represents what is called a trivial defect, in which case this line is essentially an artefact of our description of the situation, but not itself of intrinsic meaning. Still, the FRS formalism requires that wherever we want to include a bulk field insertion such a line has to be present. Moreover, it needs to be labeled, so we are told by the FRS recipe, by a Frobenius algebra object A A inside the given representation category (satisfying a couple of extra properties, called “specialness”, “symmetry” and possibly, if the formalism is to be applicable also to unoriented worldsheets, “Jandl“ness).

Given these labellings, one has to furthermore consider the objects UA U \otimes A and AV A \otimes V in the modular tensor category as AA-bimodules with the obvious action of AA on itself, where we act on UAU \otimes A from the left by braiding AA under UU while we act on AVA \otimes V by braiding AA over VV. Then, finally, the coupon in the center of the diagram is to be labeled by a morphsim ρ:UAAV \rho : U \otimes A \to A \otimes V in the modular tensor category which is a homomorphism of internal AA-bimodules.

This is the FRS labelling prescription for a bulk field insertion. Similar prescriptions exist for boundary conditions and boundary field insertions.

The main motivation for considering precisely this prescription is: it works. It just so happnes that when combined with the rest of the FRS formalism, this can be shown to result in some construction which correctly describes CFT correlators with bulk insertions.

But a hint as to why this works is possibly contained in the following statement, which we now turn to:

Clue #1. The above ribbon diagram is precisely the Poincaré-dual string diagram corresponding to a cylindrical 3-morphism in the 3-category ΣBim(C)\Sigma \mathrm{Bim}(C).

This 3-category ΣBim(C)\Sigma \mathrm{Bim}(C) is the 3-category naturally associated to any braided monoidal category:

- it has a single object

- it has one 1-morphism for each algebra object internal to CC

- it has one 2-morphism from AA to BB for every AA-BB-bimodule internal to CC

- it has one 3-morphism for every bimodule homomorphism internal to CC.

Composition along the single object is the tensor product on Bim(C)\mathrm{Bim}(C) which is inherited from the fact that CC is braided. Composition along 1-morphisms is the ordinary tensor product of bimodules and compositon along 3-morphisms is just the composition of bimodule homomorphisms.

Even though ΣBim(C)\Sigma \mathrm{Bim}(C) is a natural thing to consider, it may still appear like quite a mouthful to swallow. However, it turns out that ΣBim(C)\Sigma \mathrm{Bim}(C) is actually to be thought of as nothing but the 3-category of endomorphisms of the canonical 1-dimensional 3-vector space: this is the single object of ΣBim(C)\Sigma \mathrm{Bim}(C), while the 1-morphisms in there are the “linear operators” on it, etc.

The reader not on speaking terms with internal bimodules is therefore strongly encouraged to think, in the following, of ΣBim(C)\Sigma \mathrm{Bim}(C) as the home of the space of states over a point of a 3-particle described by a 3-dimensional TQFT– (i.e. a membrane-like particle, as described in QFT of Charged nn-Particle: Extended Worldvolumes ).

In fact, for our purposes we should slightly oversimplify and think of our 3-category as being 3Vect:=ΣBim(C), 3\mathrm{Vect} := \Sigma \mathrm{Bim}(C) \,, the category of 3-vector spaces. This is only slight abuse of terminology but very useful for understanding the general picture by comparison with that of ordinary (not nn-categorically refined ) quantum theory.

Then, what we should be considering are cylinders in 3Vect3\mathrm{Vect}. For us, a cylinder in an nn-category is the kind of nn-morphism which a transformation of nn-functors assigns to an (n1)(n-1)-morphism of the given domain nn-category.

For the first few nn, such cylinders look like this:

(This is taken from Extended QFT and Cohomology II: Sections, States, Twists and Holography)

Here we have n=3n=3 and hence “true 3-dimensional” cylinders.

Notice that when such cylinders really arise as components of transformations η:F 1F 2 \eta : F_1 \to F_2 of nn-functors, their top and bottom are fixed to be the images of some 2-morphism under the nn-functors FF and GG, respectively. This is because they need to form a naturality “square” at level nn. For n=3n=3 this looks like

(More details on 3-functors and their morphisms and higher morphisms are given in Picturing Morphisms of 3-Functors.)

It will turn out to be sufficient, for the present purpose, to consider a simple special case of 3-functors with values in 3-vector spaces: we can assume that they are restricted by the fact that they send every 1-morphisms to the identity.

Such cylinders in 3Vect3\mathrm{Vect} hence are labelled by

- an algebra AA on the left vertical seam

- an algebra BB on the right vertical seam

- an AA-BB bimodule on the front face

- another AA-BB bimodule on the rear face

- an II-II bimodule UU on the top

- and an II-II bimodule VV on the bottom.

Here II denotes the tensor unit in CC, equipped with the trivial algebra structure on it. Therefore UU and VV are nothing but mere objects in CC!

The collection of all cylinders of this form yields an interesting 2-category in its own right. It has been described in more detail in

The 1-Dimensional 3-Vector Space
Bulk Fields and Induced Bimodules
A 3-Category of twisted Bimodules.

With the concept of a cylinder in 3Vect3\mathrm{Vect} thus established, our clue#1 is now easily checked:

the Poincaré-dual string diagram

which discribes the cylindrical 3-morphism in 3Vect3\mathrm{Vect} is precisely the type of diagram which the FRS prescription tells us to associate to a bulk field insertion:

We take this as a first indication that

The 2-dimensional theory really arises, when regarded as an extended propagation 2-functor, as a transformation of a 3-functor.


A major clue: the full disk diagram

The simplest self-contained setup which is built around the concept of a single bulk insertion, like those discussed above, is that where the bulk insertion sits on a disk-shaped worldsheet.

This requires specifying some kind of boundary condition on the rim of the disk. The FRS prescription tells us to model this boundary condition by choosing an object NN in the modular tensor category with the structure of an internal AA-module on it, for AA the algebra object which already appeared above, labelling the ribbon that runs inside the worldsheet. Then we are to consider the ribbon graph which has this object NN running in a circle, with the AA-line inside it, and attached with its end to the module line, where at the points these ribbons meet they are glued by those morphisms in CC which encode the action of AA on NN.

More interestingly, one can consider the situation where two “open string states come in from infinity”, i.e. where we have two boundary insertions. Whatever that really mean, in the FRS formalism this is modelled by

- choosing two (possibly different) AA-module objects NN and NN' in CC, encoding the boundary conditions on the two ends of these open strings

- choosing for the two boundary field insertions morphisms ϕ 1:NAN \phi_1 : N \otimes A \to N' and ϕ 2:NAN, \phi_2 : N' \otimes A \to N \,, respectively.

Then instead of a single NN-line running around the disk we are to have NN and NN' run over a semi-circle each. And where these lines meet, we are to glue them with these morphisms ϕ 1\phi_1 and ϕ 2\phi_2.

So, in summary, the ribbon graph which the FRS prescription tells us to associate to a disk-shaped worldsheet with one bulk and two boundary field insertions looks like this:

Remarkably, for understanding the nn-categorical mechanism at work here in the background, it is quite helpful at this point to bring to mind the physical interpretation of this situation:

when considering the disk with two boundary insertions ϕ 1\phi_1 and ϕ 2\phi_2 we are really describing the situation where a quantum string (the “2-particle”, a linearly extended kind of particle) with quantum state |ϕ 1 |\phi_1 \rangle propagates along its way, thereby suffering some transformations of kinds – here essentially it merges or “emits” a closed string, transforming it, schematically, into a state U|ϕ 1. U |\phi_1 \rangle \,. Finally, we probe for the “probability amplitude” ϕ 2|U|ϕ 1 \langle \phi_2 | U |\phi_1\rangle that as a result of this propagation and interaction we see a string in the state ϕ 2\phi_2 running away to infinity.

A little reflection shows that this pairing of states is actually modeled after the Hom-functor:

- let Q:parnVect Q : \mathrm{par} \to n\mathrm{Vect} be our extended QFT nn-functor restricted to the (n1)(n-1)-categorical parameter space par\mathrm{par} of the nn-particle which we are describing (as discussed in more detail in QFT of Charged nn-Particle: Extended Worldvolumes)

- let I:parnVect I : \mathrm{par} \to n\mathrm{Vect} be the tensor unit of all such nn-functors.

Then

- a state is a morphism ϕ:IQ \phi : I \to Q

- a costate is a morphism ϕ *:QI \phi^* : Q \to I

- a (free) propagator is a morphism U:QQ. U : Q \to Q \,.

Hence a pairing of states is acually an (n1)(n-1)-functor Hom(I ϕ 1 Q,I ϕ 2 * Q):End(Q)End(I). \mathrm{Hom} \left( \array{ I \\ \;\;\downarrow^{\phi_1} \\ Q } \; , \; \array{ I \\ \;\;\uparrow^{\phi_2^*} \\ Q } \right) \;\; : \;\; \mathrm{End}(Q) \to \mathrm{End}(I) \,.

Remarkably, it turns out that the nn-categorical formalism knows all about the necessity of boundary conditions for extended objects, i. e. for nn-particles with n>1n \gt 1:

for n=1n=1, the objects of End(I)\mathrm{End}(I) are just complex numbers. But for n>1n \gt 1, one sees that the complex numbers sit inside End(I)\mathrm{End}(I), but that End(I)\mathrm{End}(I) is considerably larger.

Therefore a correlator is obtained from a pairing of states of the (n>1)(n \gt 1)-particle only after we choose an ii-trivialization of Hom(ϕ 1,ϕ 2 *):End(Q)End(I)\mathrm{Hom}(\phi_1,\phi_2^*) : \mathrm{End}(Q) \to \mathrm{End}(I), for i:Σ (n1)End(I) i : \Sigma^{(n-1)} \mathbb{C} \hookrightarrow \mathrm{End}(I) the canonical inclusion of the ground field into the endomorphisms of the trivial extended nn-dimensional QFT, hence a morphism End(Q) = End(Q) Cor(ϕ 1,ϕ 2) b Hom(ϕ 1,ϕ 2) Σ n1 i End(I). \array{ \mathrm{End}(Q) &\stackrel{=}{\to}& \mathrm{End}(Q) \\ {}^{\mathrm{Cor}(\phi_1,\phi_2)}\downarrow\;\;\; & \Downarrow^{b} & \;\;\;\downarrow {}^{\mathrm{Hom}(\phi_1,\phi_2)} \\ \Sigma^{n-1} \mathbb{C} &\stackrel{i}{\hookrightarrow}& \mathrm{End}(I) } \,.

It turns out that the choice of morphism bb here is what encodes the boundary conditions.

Since these boundary conditions are famously known, for n=2n=2, as “D-branes”, and since the morphism bb in that case, being a transformation of 2-functors, is given by 2-commuting diagrams known as tin can diagrams, the slogan is that we obtain D-branes from tin cans. This was developed in

D-Branes from Tin Cans: Arrow Theory of Disks
D-Branes from Tin Cans, Part II
D-Branes from Tin Cans, Part III: Homs of Homs

Of relevance for our purpose here is what this implies for the pairing of two states of the 3-particle:

the relation between the pairing of two states and the corresponding transformation which encodes the correlator is given by a diagram of the form

.

where on the left the two cylinders appearing are to be thought of as the two 3-states of our 3-particle, the cylinder on the right is the one encoding the actual value of their correlator, while the wedge-like pieces are the modification of transformations of 3-functors coming from the morphism called bb above.

(A little more on such manipulations with 3-functors can be found in Picturing Morphisms of 3-Functors.)

Solving this for the cylinder on the right shows that this is given by an expression of the following form

.

This indicates that the 3-morphism in 3Vect3\mathrm{Vect} which our extended QFT assigns to the disk looks like this:

As this indicates, the Poincaré dual string diagram of the 3-morphisms which describes the pairing of states of the membrane reproduces the FRS decoration prescription for a correlator of the 2-particle.

More precisely, to turn this into a well defined statement, one first needs to locally trivialize the 2-functor with respect to the canonical inclusion ΣCBim(C) \Sigma C \hookrightarrow \mathrm{Bim}(C) of the modular tensor category in the category of bimodules internal to that. Requiring the existence of such a local trivialization precisely encodes the demand that

The algebras appearing here have to be special Frobenius algebras.

Special Frobenius algebras come from special ambidextrous adjunctions, and these are what allow the local trivialization of 2-functors (see this for more).

Many of the structures appearing in the FRS formalism can actually be compared to similar structures appearing in the description of surface transport in 2-bundles with connection, by following the first edge of the cube. The reason is, we claim, that in both cases we are dealing with nn-functors and their local trivialization: in one case these nn-functors describe “classical propagation”, namely parallel transport in an nn-bundle with connection, in the other case they encode quantum propagation.

This close similarity becomes much more pronounced once one realizes that the local descent data for gerbes (2-bundles) is also controlled by Frobenius algebras, or rather by “Frobenius monoidoids”, only that these happen to have an invertible product operation, which makes the Frobenius property become hard to notice. (A remark on this can be found at Frobenius Algebroids with Invertible Products.)

More details on local trivialization and local description of 2-functors with values in (induced) bimodules is given in FRS Formalism from 2-Transport.

All the 2-morphisms discussed there have to be regarded as 2-dimensional cross-sections of the cylindrical 3-morphisms which we discussed so far. This then explains why they involve bimodules induced by tensoring with objects and acting by over- or underbraiding as used there.

By switching to this 2-dimensional notation, working out the local trivialization, and then finally passing from globular to string diagrams produces from our 3-morphism above finally the following planar diagram

This is the FRS disk diagram. Compare to section 4.3 of

Fuchs, Runkel, Schweigert TFT construction of RCFT correlators IV: Structure constants and correlation functions.

Posted at August 3, 2007 2:21 PM UTC

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8 Comments & 8 Trackbacks

Re: QFT of Charged n-Particle: Towards 2-Functorial CFT

Here is the diagram which illustrates the horizontal composition of those cylinders (in ΣBim\Sigma \mathrm{Bim} with top and bottom rims restricted to be identities), which translates in the CFT to having two disorder lines (one taking the CFT from its AA-phase to its BB-phase, the next one taking it to its CC-phase) whith two bulk disorder field insertions sitting on them:

Posted by: Urs Schreiber on August 6, 2007 2:27 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Towards 2-Functorial CFT

Here is maybe a nice example of how nn-categorical language unifies some of the structure one sees in this CFT context:

According to FRS, the most general boundary field insertion is a morphism of the form

Here UU is a rep of the chiral algebra, hence an “open string state” coming in. Its right boundary condition (“D-brane”) is labeled by the AA-module NN' (where AA is the algebra of open string states sitting on that right D-brane) and its left boundary condition by the BB-module NN (where BB is the algebra of string states on the left D-brane).

On top of that, there may be a defect line MM emanating from this boundary insertion (usually some duality operation, like T-duality or the kind of dualities Witten and Kapustin consider), respresented by an AA-BB- bimodule MM.

The black ribbon here denotes a BB-line which induces the tensor product over BB on these objects in the modular tensor category CC.

That means, we may really think of this as obtained from a string diagram in Bim(C)\mathrm{Bim}(C) of the form

Here 11 denotes the tensor unit object in CC equipped with the trivial algebra structure on it.

Now, maybe remarkably, one can see that all this data encoding a boundary field insertion in CFT is nothing but — a pseudonatural transformation (of 2-functors).

This becomes manifest as we pass from the string-diagram notation to the Poincaré-dual globular notation for 2-categories:

Hence the “boundary field insertion”, a “string 2-state”, namely a string state together with all the information about the D-branes its endpoints sit on (for instance “Chan-Paton bundles on D-branes”), is nothing but a morphism |ψ:IQ |\psi\rangle : I \to Q from a 2-functor II with values in the modular tensor category itself, regarded as sitting in ΣCBim(C) \Sigma C \hookrightarrow \mathrm{Bim}(C) into the 2-functor QQ, with values in Bim(C)\mathrm{Bim}(C), which assigns to the string it “space of 2-states”.

Posted by: Urs Schreiber on August 6, 2007 11:39 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Towards 2-Functorial CFT

great pictures urs!

Posted by: picture_watcher on August 7, 2007 12:50 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Towards 2-Functorial CFT

great pictures urs!

Thanks!

The colored pictures were all done by Jens Fjelstad using InkScape.

The b&w ones I did, as always, using xypic.

There are many more pictures to be drawn. If there is anyone out there interested in helping (with drawing but possibly also with thinking!), please drop me a note! Of course all fame will be shared equally. :-)

(By the way: the original inkscape .SVG output of our pictures looks even better, with gradient scaling neatly indicating the 3-dimensional nature of these cylinders. But I did’t manage to include the SVG code here. Even though it should work.)

Posted by: Urs Schreiber on August 7, 2007 10:41 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Towards 2-Functorial CFT

The diagrams of “the pairing of two states of 3-particle” - between boldface
“D-branes of tin cans, part III: homs of homs”
and
“Picturing morphisms of 3 functors” -
are very reminiscent of [virtual] helical ribbons.

Isn’t this to be expected in mechanics and ballistics?

Similar helices are found in nucleic and amino acids.

Perhaps these helices provide information of some type, probably different in different scales?

Posted by: Doug on August 7, 2007 12:59 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Towards 2-Functorial CFT

Here are more details on how the disk correlator looks like as a pasting diagram in a 2-category, and how the corresponding Poincaré-dual “string diagram” (or “tangle diagram”) arises.

The pairing of two states |ϕ 1:IQ |\phi_1\rangle : I \to Q and |ϕ 2:IQ |\phi_2\rangle : I \to Q via the Hom-functor, together with a choice of a suitable trivialization of the Heisenberg-state functor Hom(I ϕ 1 Q,I ϕ 2 Q):End(Q)End(I), \mathrm{Hom} \left( \array{ I \\ \;\;\downarrow^{\phi_1} \\ Q } \; , \; \array{ I \\ \;\; \uparrow^{\phi_2^\dagger} \\ Q } \right) \;\; : \;\; \mathrm{End}(Q) \to \mathrm{End}(I) \,, which encodes the boundary conditions, yields, as indicated above, for the disk correlator a pasting diagram of the sort

This and the following is described in more detail in

On correlators of the 2-particle

Here N AN_A and N BN_B are two modules which encode the D-brane that the left and right, respectively, end of the 2-particle in the state ϕ 1\phi_1 and ϕ 2\phi_2 sits on.

This has to be regarded as a 2-dimensional cross section through the cylinders in ΣBim(C)\Sigma \mathrm{Bim}(C) mentioned before. Hence the bimodule homomorphism ρ\rho in the center should really be thought of as being a morphism of induced bimodules, with an object UU in CC coming in from above the drawing plane and another object VV running off below the drawing plane, describing a bulk field insertion. For simplicity this 3-dimensional aspect is suppressed in the following.

So the correlator is a pasting diagram in Bim(C)\mathrm{Bim}(C). But we’d rather want it to be a pasting diagram in ΣC\Sigma C itself!

To do so, we notice the canonical inclusion ΣCBim(C) \Sigma C \hookrightarrow \mathrm{Bim}(C) and perform essentially a local trivialization of the above surface correlator with respect to this inclusion.

The result of this operation is that after an identical rewriting further component 2-cells appear, built from the morphisms of the special ambidextrous adjunctions that the special Frobenius algebras AA and BB are built from:

Notice that this is exactly equal to the smaller pasting diagram before. What happened is that a puffed-up version of the identity 2-cells on KK and on KK' has been inserted.

But after this insertion of two identities, we may rebracket, in a sense, and merge the 2-cells of the incoming and outgoing state, respectively, with half of these identities.

As a result, the disk pasting diagram turns into one in ΣC\Sigma C:

Passing equivalently to the dual tangle diagram then yields

The two green coupons represent the boundary field insertions. (This are the morphisms discussed before here.)

The yellow coupon in the center is again the bulk field insertion. As I said, the ribbons running off perpendicular to the drawing plane, as well as the corresponding labels, are suppressed.

Where the bulk field insertion sits, two defect lines meet, drawn in red, which interpolate between the AA- and the BB-phase of the disk.

The outer red line merges with one side of the adjunctions to produce the gray rim, which is the one-sided module that describes the two D-branes on the boundary of the disk-shaped worldsheet.

Where two parts of an adjunction meet we get entirely gray bands. These are the algebra lines for the algebra AA and BB-respectively.

Posted by: Urs Schreiber on August 7, 2007 5:30 PM | Permalink | Reply to this
Read the post The Canonical 1-Particle, Part II
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Excerpt: Taking the new Firefox out for a spin.
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Re: QFT of Charged n-Particle: Towards 2-Functorial CFT

I am currently in Aarhus, Denmark, visiting Jens Fjelstad with whom I am developing the above ideas.

We made more progress with getting the full Reshetikhin-Turaev 3d TFT functor and its boundary CFT from the holonomy of a “differential nonabelian 3-cocycle” on the worldvolume.

I am beginning to incorporate this into the big-picture discussion developing in

On Σ\Sigma-models and nonabelian differential cohomology

Posted by: Urs Schreiber on April 9, 2008 1:23 AM | Permalink | Reply to this
Read the post Sigma-Models and Nonabelian Differential Cohomology
Weblog: The n-Category Café
Excerpt: Notes on nonabelian differential cohomology and its application to classical and qantum parallel transport.
Tracked: April 11, 2008 8:33 AM

Re: QFT of Charged n-Particle: Towards 2-Functorial CFT

Recall this bit from the above entry:

2nd Statement.

This propagation 2-functor is the component map of a transformation of propagation 3-functors of the corresponding 3-dimensional topological field theory.

At Strings, Fields, Topology I asked Chris Schommer-Pries if he has an idea how to push this idea. He told me, to my pleasant surprise, that as a bypoduct of his thesis on a generators-and-relations construction of extended Bord 2Bord_2, section 4.5, he is pretty close to formalizing and proving that, indeed, the FFRS decoration prescription is precisely the data of a transformation between Reshitikhin-Turaev extended TQFT 3-functors.

And that he will talk about that tomorrow.

The main thing that I was missing here, to go beyond noticing that the transformatoin components pictured above carry the right data to give the FFRS prescription, was the intricate details of the naturality condition on such a transformatoin between 3dTQFT 3-functors. It’s a careful analysis of these naturality conditions using the generators-and-relations prescription that gives the full picture.

What is still missing for the full formalization of the above statement is the full control over the generators-and-relations construction of extended Bord 3Bord_3. But, as Chris indicated convincingly, it is from his 2d results already pretty obvious that and how this works.

Posted by: Urs Schreiber on June 10, 2009 1:28 PM | Permalink | Reply to this

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