Dijkgraaf-Witten Theory and its Structure 3-Group

November 6, 2006

Dijkgraaf-Witten Theory and its Structure 3-Group

Posted by Urs Schreiber

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Chern-Simons theory, for every choice of compact Lie group $G$ and class $\tau \in H^4(B G,\mathbb{Z})$ , is a theory of volume holonomies. Therefore one might want to understand it in terms of parallel 3-transport # with respect to a suitable structure 3-group.

A toy example for Chern-Simons theory is Dijkgraaf-Witten theory. This instead depends on a group 3-cocycle $\alpha$ of a finite group $G$ .

As recently mentioned here, there are attempts to categorify Dijkgraaf-Witten theory by suitably replacing $G$ by some $n$ -group.

But do we even understand the ordinary theory in natural terms?

In particular, since Dijkgraaf-Witten theory, too, is a theory of parallel 3-transport, it should really come from a 3-group itself already. A 3-group, that is, which is naturally obtained from an ordinary finite group $G$ and a group 3-cocycle. And preferably in such a way that it illuminates the structure of Chern-Simons theory itself.

Which 3-group is that?

After briefly recalling the idea of Dijkgraaf-Witten theory, I shall argue that the 3-group in question is the inner automorphism 3-group

(1)

\mathrm{INN}(G_\alpha) \,,

where $G_\alpha$ is the skeletal weak 2-group whose group of objects is $G$ , whose group of morphisms is $U(1)$ and whose associator is determined by the given group 3-cocycle $\alpha \in H^3(G,U(1))$ .

This gives a concise way to say what Dijkgraaf-Witten theory is. It also fits in nicely with the claim # that the 3-group $\mathrm{INN}(\mathrm{String}(G))$ governs Chern-Simons theory - since $\mathrm{String}(G)$ # is essentially the Lie analog of $G_\alpha$ .

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The setup we are dealing with in Dijkgraaf-Witten theory was introduced on pp. 42-46 of

Robbert Dijkgraaf & Edward Witten
Topological Gauge Theories and Group Cohomology
Commun. Math. Phys. 129 (1990), 393.
(pdf).

In its simplest form, it defines a map from 3-manifolds to the abelian group $U(1)$ , induced by any $U(1)$ -valued group 3-cocycle $\alpha \in H^3(G,U(1))$ .

Recall that this simply means that $\alpha$ is a map

(1)

\alpha : G \times G \times G \to U(1)

which satisfies the cocycle condition

(2)

\delta \alpha(g,h,k,l) = 1 \,,

meaning that

(3)

\alpha(g,h,k)\alpha(g,h k,l)\alpha(h,k,l) = \alpha(g h,k,l)\alpha(g,h,k l) \,,

for all $g,h,k,l \in G$ .

Consider now a 3-manifold $X$ . Choose any oriented triangulation $T$ , decomposing $X$ into tetrahedra.

A (necessarily flat, since $G$ is finite) $G$ -connection on $X$ is now represented by an assignment of elements $g(v_1,v_2) \in G$ to edges $v_1 \to v_2$ of the triangulation, such that all triangles commute.

The flatness condition ensures that the coloring of all tetrahedra by group elements is fixed already if three of the tetrahedron’s edges are labelled.

Given any such flat connection, we can hence assign to each tetrahedron an element in $U(1)$ , by evaluating our group cocycle $\alpha$ on three such group elements.

Dijkgraaf-Witten theory is the study of the functional obtained by sending each such colored triangulated X to the product of these elements of $U(1)$ over all tetrahedra of the triangulation.

Now I reformulate this procedure as the computation of the 3-transport in a principal 3-bundle with 3-connection on $X$ .

It is a theorem (see corollary 44 of HDA V for details) that every 2-group is equivalent to a 2-group $G_\alpha$

- whose group of objects is given by an ordinary group $G$

- which is skeletal as category (all isomorphic objects are in fact equal)

- whose automorphism group of every object is an abelian group $A$

- whose nonvanishing associator (a map sending three objects to a morphism) is a group 3-cocycle $\alpha \in H^3(G,A)$ .

Hence the basic data on which Dijkgraaf-Witten theory is built is precisely that classifying 2-groups (up to equivalence).

We are therefore very tempted to try to see if the 3-group that we expect is canonically associated to the Dijkgraaf-Witten model is obtained from $G_\alpha$ .

I think to make contact with the more general case of Chern-Simons theory, we want to look at $\mathrm{INN}(G_\alpha)$ , the 3-group of inner automorphisms of $G_\alpha$ .

But for the present purpose it is helpful to first restrict attention to the much “smaller” and more accessible 3-group

(4)

G'_3 \subset \mathrm{INN}(G_\alpha)

which has objects and 1-morphisms those of $G_\alpha$ , and which has precisely one 2-morphism between any ordered pair of parallel 1-morphisms.

A principal 3-transport with values in this 3-group is a lax functor

(5)

\mathrm{tra} : P_1(X) \to \Sigma(G'_3) \,.

Here I write $P_1(X) = (X\times X \stackrel{\to}{\to} X)$ for the pair groupoid of $X$ , and $\Sigma(G'_3)$ for the 3-category which has a single object “ $\bullet$ ” such that $\mathrm{Hom}(\bullet,\bullet) = G'_3$ .

The functor $\mathrm{tra}$ associates elements $g(v_1,v_2)$ of $G$ to edges $v_1 \to v_2$ such that all triangles

(6)

\array{ {}^{g(v_1,v_2)}\nearrow \searrow^{g(v_2,v_3)} \\ \stackrel{g(v_1,v_3)}{\to} }

commute.

This commutativity is a consequence of the fact that $G_\alpha$ is skeletal. Being lax, the functor $\mathrm{tra}$ also associates an element in $U(1)$ to each triangle, representing a 2-morphism (the compositor) from $g(v_1,v_2)g(v_2,v_3)$ to $g(v_1,v_3)$ - but since the source and target of that 2-morphism must coincide, it requires $g(v_1,v_2)g(v_2,v_3) = g(v_1,v_3)$ .

Furthermore, $\mathrm{tra}$ associates a unique element in $U(1)$ to every tetrahdron,

(7)

\mathrm{tra} \;\; : \;\; \left( \array{ v_2 &\to & v_3 \\ \downarrow &\nearrow& \downarrow \\ v_1 &\to& v_4 } \;\; \Rightarrow \;\; \array{ v_2 &\to & v_3 \\ \downarrow &\searrow& \downarrow \\ v_1 &\to& v_4 } \right) \;\; \mapsto \;\; \mathrm{tra}(v_1,v_2,v_3,v_4) \in \mathrm{Mor}_3(\Sigma(G'_3))

representing the unique 3-morphism filling it.

The crucial point is that the image of this tetrahedron involves the nontrivial associator of $G_\alpha$ , which relates

(8)

(g(v_1,v_2)g(v_1,v_2))g(v_3,v_4) \stackrel{\alpha(g(v_1,v_2),g(v_1,v_2),g(v_3,v_4))}{\to} g(v_1,v_2)(g(v_1,v_2))g(v_3,v_4))

the boundary of the left hand side to the boundary of the right hand side.

Both 1-morphisms here are actually equal. Still, by the general rules, they are related by a nontrivial associator.

If we assume for the moment that $\mathrm{tra}$ is such that the coloring of all faces is by the neutral element in $U(1)$ , then we find that the 3-morphism assigned by our 3-transport to any tetrahedron is precisely the appropriate Dijkgraaf-Witten amplitude.

But even is the faces are colored nontrivially we get the Dijkgraaf-Witten volume holonomy: the contribution of the faces cancels out if the boundary of the 3-manifold we do the parallel transport over vanishes. Only the contribution of the associator remains. That’s the Dijkgraaf-Witten amplitude of a closed manifold.

So we can concisely define the Dijkgraaf-Witten amplitude of a flat connection on $X$ as the volume holonomy of a transport 3-functor with values in $\Sigma(G'_3)$ .

And the same argument applies, I think, also if we use the full 3-group $G_3 = \mathrm{INN}(G_\alpha)$ (which should come from something like a weak 2-crossed module of groups, which looks like $(U(1) \to U(1)\times G \to G)$ (two copies of $U(1)\to G$ , one shifted in degree)).

Finally, notice that the 2-group $\mathrm{String}_G$ is, for (simply connected, compact, simple) Lie groups $G$ essentially what $G_\alpha$ is for finite groups. This is most manifest when one looks at the skeletal version of the Lie 2-algebra of $\mathrm{String}_G$ : that’s $\mathrm{Lie}(G)$ of objects, $\mathrm{Lie}(U(1))$ of morphisms and the only nontrivial structure is the associator given by a Lie algebra cocycle $H^3(\mathrm{Lie}(G),\mathbb{R})$ .

Posted at November 6, 2006 5:55 PM UTC

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18 Comments & 4 Trackbacks

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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Please excuse my stepping back right to the beginning, but I haven’t thought much about 3-transport in a 3-connection except in broad outlines… Thanks for describing it.

So I guess I really have a question for John about the background to this. I’ve been wanting for a while to understand better how the associator for this 2-group $G_{\alpha}$ is related to the 3-cocycle in $H^3(G,U(1))$ . This gave me an incentive to go back and think about it.

In the 2004/5 sessions of John’s QG seminar, we discussed the Dijkgraaf-Witten model as related to an extended TQFT coming from categorification of the Fukuma-Hosono-Kawai construction. In that case, it’s clear that to the interior of a tetrahedron has to be assigned the associator for the two composition operations on the front and back faces. So when I went to look up Dijkgraaf and Witten’s paper, I was confused to find it presented in terms of this cocycle.

There are a few differences in presentation - one is straightforward, namely that in the seminar we were thinking of the extended TQFT as giving an operator for a cobordism, as opposed to giving amplitudes for manifolds. This is okay, since for closed manifolds, the operator (from $\mathbf{C}$ to $\mathbf{C}$ ) can be interpreted as a complex number, and indeed it’ll be an amplitude.

The other is the one that I’m less sure of. It has something to do with the fact that in the seminar, we were talking about edges labelled by elements of the 2-algebra $Vect[G]$ , and it’s clear what this is. The product in this 2-algebra is a convolution product, and the standard associator is inherited from the associator for tensor products of vector spaces. One can twist this associator using a group cocycle, multiplying the maps of vector spaces by a constant - the cocycle condition guarantees the result still satisfies the pentagon identity and is an associator.

So if we’re given:

an associator for $G_{\alpha}$ , which is a map $\alpha : G^3 \rightarrow U(1)$ giving a morphism for each triple of objects (I’m following you in ignoring the point of view where there is one object, a layer below the elements of $G$ and this is really an associator for composition of morphisms)
a standard associator for $Vect[G]$ (which may as well be the canonical one derived from the associator in $Vect$ )

then we get a new associator for $Vect[G]$ . The Dijkgraaf-Witten amplitude for a given relates the two different 2-morphisms we get by using the two different associators.

I’m not entirely clear on the connection between the 2-algebra and the 2-group point of view, beyond what I said above. There’s some kind of action of $G_{\alpha}$ on $Vect[G]$ - is there an illuminating way to put this?

Posted by: Jeff Morton on November 7, 2006 9:27 AM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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Hi Jeffrey,

thanks for your comment!

Probably there are various different ways to think about Dijkgraaf-Witten theory.

The point of view I was advocating above was supposed to support, in turn, a certain point of view on Chern-Simons theory that I find interesting.

I am afraid I am only aware of the QG sessions which discussed the Fukuma-Hosono-Kawai model with the algebra being a group algebra, but not what came afterwards.

In my personal universe of ideas, I think of FHK not as a 2-transport, but as the local gluing data of a flat vector 2-transport (as first indicated here).

This is one more example for different valid points of view that, however, lead one to different modifications and generalizations.

the 2-algebra $\mathrm{Vect}[G]$

I am sure I could find this discussed in the seminar notes available on John’s website, but maybe you could tell me:

I am guessing that this is supposed to denote the category of $G$ -graded vector spaces, which we can think of as having objects being (direct sums of) pairs

(1)

V_g = (V,g)

with $V$ an ordinary vector space and $g$ an element in $G$ and with the tensor product being the combination of the tensor product of vector spaces and the product in the group

(2)

V_{g} \otimes W_{h} := (V\otimes W)_{g h} \,.

If so, I might be able to make a connection with the point of view I was talking about:

I was trying to identify the structure 3-group describing the principal (in the sense of principal bundles) parallel transport encoded by the Dijkgraaf-Witten amplitude.

But we will want to pass from this principal transport to a vector transport by choosing a representation of our structure group.

Now, as I have said elsewhere (introduction of these notes for instance), given a 2-group $G_2$ with object space $G$ and group of morphisms starting at the identity being $H$ , together with an ordinary rep $\rho$ of $H$ , we get a 2-rep of $G_2$ in

(3)

\mathrm{Bim}(\mathrm{Vect}) \subset {}_{\mathrm{Vect}}\mathrm{Mod} \,,

i.e. a 2-functor

(4)

\tilde \rho : \Sigma(G_2) \to \mathrm{Bim}(\mathrm{Vect}) \,.

This sends the single object of $\Sigma(G_2)$ to the algebra $A$ generated by the image of $\rho$ .

It sends any 1-morphism $g\in G$ of $\Sigma(G_2)$ to the $A$ - $A$ bimodule

(5)

A_g \,,

which is, as an object, $A$ itself, with the ordinary left $A$ action and the right action twisted by $g$ .

Finally, it sends 2-morphisms in $\Sigma(G_2)$ to bimodule homomorphisms.

In the above notes I provide the details of this for the case that $G_2$ is strict. But I think one can easily generalize this to the case that $G_2$ is for instance the skeletal weak 2-group $G_\alpha$ .

I have tried to show recently # that by taking inner automorphisms of $G_2$ and inner endomorphisms of $\mathrm{Vect}$ , the above construction lifts from 2-reps of 2-groups to 3-reps of their inner automorphism groups.

You see, if we now compose the principal 3-transport functor that I was talking about with this 3-rep, then the result is a 3-functor which

- assigns vector spaces $V_g$ indexed by a group element $g \in G$ to edges (if the dimension of $V$ is greater than one we are talking about lax 2-reps)

- assigns to composite edges the tensor product $V_g \otimes W_h = (V\otimes W)_{g h}$

and so on.

So maybe the prescription discussed in the QG seminar can be understood as coming from the 2-vector (3-vector, really) transport associated to the principal 3-transport that I was discussing by means of the above rep.

Posted by: urs on November 7, 2006 10:36 AM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

Yes, Vect[G] is the category of G-graded vector spaces, which we write in that slightly funny way to emphasize that it’s the group 2-algebra, analogous to the group algebra C[G], which is the major example we used for the semisimple algebra the FHK construction for a TQFT takes as input.

I think your answer, that the relation between the two pictures can be seen as corresponding to the relation between transport in a principal bundle and a vector bundle, is exactly what I meant by “an illuminating way” to say it. So now I’m going to try to look at the rest of it more carefully. Thanks!

Posted by: Jeff Morton on November 8, 2006 3:18 AM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

Jeff wrote:

Yes, $Vect[G]$ is the category of $G$-graded vector spaces, which we write in that slightly funny way to emphasize that it’s the group 2-algebra, analogous to the group algebra $\mathbf{C}[G]$, …

When posting your comment, you forgot to choose a “Text Filter” which allows you to display TeX - and when you previewed your comment you apparently didn’t notice you were getting the ugly dollar signs above: uncompiled TeX. Unfortunately this is not something I can change now - an annoying bug in this system. So, I’ll just edit your entry to make those ugly dollar signs go away.

In the future, try the text filter “Itex to MathML with parbreaks” - then your TeX will compile. For further info, or if you want to ask questions, go here.

I should post something more interesting - the questions you and Urs are discussing are really fascinating! But, I’m too tired right now. I need to go home, watch the election returns, and go to sleep. So: more later.

Posted by: John Baez on November 8, 2006 6:11 AM | Permalink | Reply to this

Comment text-filters

Since user-selectable comment text-filters are not a stock feature of MovableType (or, for that matter, any other blogging system), there’s no interface to change it in the MT Administrative page that you have access to.

I can change it directly in the MySQL database.

Someday, maybe, I will get around to hacking this into the MT Admin interface. But there’s a long list of other, more pressing features to implement …

Posted by: Jacques Distler on November 8, 2006 7:51 AM | Permalink | PGP Sig | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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[…] emphasize that it’s the group 2-algebra […]

By the way: that we can use any ordinary group algebra in Fukuma-Hosono-Kawai is due to the fact that these algebras are in fact special symmetric Frobenius algebras. (“Special” here refers to the bubble move.)

So: is $\mathrm{Vect}[G]$ “2-Frobenius” in some sense?

Posted by: urs on November 8, 2006 10:10 AM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

Lax 2-reps - tell me more.
What about lax 2-irreps?
Do we get the same list of irreps?
but with lax structure added as 2-reps?

The tensor product satisfies your equality on the nose?
as opposed to laxly?

What becomes of the famous 6j’s?
Since the Biedenharn-Elliott identities say the pentagon
is satisfied strictly, when we laxify, do we get the next
associahedron K_5 satisfied strictly?

Posted by: jim stasheff on November 20, 2006 3:08 PM | Permalink | Reply to this

lax and pseudo 2-reps

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Lax 2-reps - tell me more.

I don’t have a full theory of lax 2-reps of any sort. But I can spell out the example that I mentioned above.

Let $V$ be a (complex, say) vector space equipped with the structure of an associative unital algebra $m : V \otimes V \to V$ , $i : \C \to V$ .

Let $G$ be some group.

For any $g \in G$ , let $V_g$ be the same vector space as before, but regarded as in degree $g$ in $\mathrm{Vect}[G]$ .

Then we get a lax 2-rep

(1)

\rho : \Sigma(G) \to \Sigma(\mathrm{Vect}[G])

by sending

(2)

\rho : (\bullet \stackrel{g}{\to} \bullet) \mapsto V_g

and taking the compositor to be given by the product in $V$ :

(3)

\rho(g)\otimes \rho(g') \to \rho(g,g') \,.

What Kapranov and Ganter considered were pseudo 2-reps of 1-groups (where the compositor is an isomorphism). See their definition 4.1

In the case that $\mathrm{dim}(V) = 1$ the above could yield an example for a pseudo 2-rep.

Do we get the same list of irreps?

For pseudo 2-reps, one finds that these are essentially ordinary reps together with a group 2-cocycle. That 2-cocycle is nothing but the compositor. It’s cocycle condtion is nothing but the associativity of the compositor. See section 5 of Kapranov and Ganter.

What becomes of the famous 6j’s? Since the Biedenharn-Elliott identities say the pentagon is satisfied strictly, when we laxify, do we get the next associahedron K_5 satisfied strictly?

I don’t know! There is more work on 2-reps that I am aware of and/or that I have absorbed. Maybe somebody else can say something about this.

Posted by: urs on November 20, 2006 3:29 PM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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Sorry to take an infinite time to reply!

I’ve been wanting for a while to understand better how the associator for this 2-group $G_\alpha$ is related to the 3-cocycle in $H^3(G,U(1))$ .

Okay. I haven’t read everything you guys said about this question, but I guess you know the quick answer is “they’re basically the same thing”.

The fun part is working out the details.

We start with any 3-cocycle $\alpha : G^3 \to \mathrm{U}(1)$

Since the 3-cocycle equation is exactly the pentagon identity, we instantly get a skeletal monoidal category with $G$ as objects and $\mathrm{U}(1)$ as the endomorphisms of any object, with $\alpha$ as the associator. Since everything is invertible, this monoidal category is a 2-group. This 2-group is what Urs is calling $G_\alpha$ .

But, we can also use $\alpha$ as the associator in other contexts. Namely, we can start with a strict version of the monoidal category $\Vect$ , and use it to define a strict monoidal category of $G$ -graded vector spaces, $\mathrm{Vect}[G]$ . Here the associator is the identity. Then, replace this boring associator by $\alpha$ , where we treat the phase $\alpha(g,g',g'') \in \mathrm{U}(1)$ as a linear map

(1)

\alpha(g,g',g'') : V(g) \otimes V(g') \otimes V(g'') \to V(g) \otimes V(g') \otimes V(g'')

in the obvious way: a phase times the identity. Now we have a “twisted” version of the monoidal category $\mathrm{Vect}[G]$ - a new monoidal category.

This new monoidal category may deserve to be called “ $\mathrm{Vect}[G_\alpha]$ ”, but I don’t know precisely what this notation means - that is, how generally we can use it.

So, I guess the interesting formal question is: what process are we using here to get various different monoidal categories with “the same” associator?

I don’t know the answer, but I suspect we’re using the obvious group homomorphism $\mathrm{U}(1) \to \mathrm{Aut}(1_{\mathrm{Vect}})$ sending phases into natural automorphisms of the identity functor on Vect.

Here’s an interesting puzzle: is $\mathrm{U}(1) \cong \mathrm{Aut}(1_{\mathrm{Vect}})?$

Posted by: John Baez on November 14, 2006 12:44 AM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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So, I guess the interesting formal question is: what process are we using here to get various different monoidal categories with “the same” associator?

Above # I suggested that this process is “representation”.

“ $\mathrm{Vect}[G_\alpha]$ ” arises, it seems, when we represent $G_\alpha$ on

(1)

\mathrm{Bim} \stackrel{\subset}{\to} 2\mathrm{Vect}

using a variant of the “canonical 2-rep” #.

Posted by: urs on November 14, 2006 10:13 AM | Permalink | Reply to this

Read the post Chern-Simons Lie-3-Algebra inside derivations of String Lie-2-Algebra
Weblog: The n-Category Café
Excerpt: The Chern-Simons Lie 3-algebra sits inside that of inner derivations of the string Lie 2-algebra.
Tracked: November 7, 2006 8:57 PM

Read the post Flat Sections and Twisted Groupoid Reps
Weblog: The n-Category Café
Excerpt: A comment on Willerton's explanation of twisted groupoid reps in terms of flat sections of n-bundles.
Tracked: November 8, 2006 11:45 PM

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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I have a burning question about this stuff, since I’m giving a talk about higher gauge theory on Friday. Maybe you already answered this question somewhere, in which case you might just point me to the answer.

What physical theory is the basic example of a higher gauge theory with the 2-group $\mathrm{String}_G$ as its “gauge 2-group”?

You’re trying to convince us that Chern-Simons theory has a gauge 3-group built starting from $\mathrm{String}_G$ . Since Chern-Simons theory is a 3d TQFT, maybe I should look for a closely related 2d TQFT (or conformal field theory) to find a theory with $\mathrm{String}_G$ as its gauge 2-group.

The obvious candidate is the $G/G$ gauged WZW model (or just the WZW model, if we want a conformal field theory).

So, is this right?

Hmm, I bet you answered this already somewhere.

Posted by: John Baez on November 14, 2006 12:18 AM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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What physical theory is the basic example of a higher gauge theory with the 2-group $\mathrm{String}_G$ as its “gauge 2-group”?

I have convinced myself that we need to be a little careful with what we mean by the gauge $2$ -group $G_2$ of some theory.

You probably have in mind that the connection of the theory is locally a 2-functor taking values in $\Sigma(G_2)$ .

This, however, is only a special case of what one might more generally mean. As you know, the “fake flatness” constraint this 2-functor automatically obeys is an indication for that.

More generally, we should say # (following Danny’s observation of how $n$ -connections relate to higher Schreier theory # ) that a 2-connection with structure 2-group $G_2$ is locally a pseudo-functor to the 3-group

(1)

\Sigma(\mathrm{INN}(G_2)) = \mathrm{Inn}(G_2) \,.

I was collecting evidence that the gauge theory with structure 2-group being $G_2 = \mathrm{String}_G$ in this last sense is precisely $G$ -Chern-Simons theory.

In fact, I think the inclusion #

(2)

\mathrm{cs}(\mathrm{Lie}(G)) \subset \mathrm{Lie}( \mathrm{INN}(\mathrm{String}_G))

proves that the structure 3-group of Chern-Simons theory is at least a sub-3-group of $\mathrm{INN}(\mathrm{String}_G)$ . And considerations in terms of transition 1-gerbes with connection (as here ) show that this sub 3-group must be larger than $\mathrm{String}_G$ itself.

Of course

(3)

\mathrm{String}_G = (1 \to \hat \Omega_G \to P G) \subset \mathrm{INN}(\mathrm{String}_G) \,.

In fact I expect that we even have an equivalence

(4)

\mathrm{cs}(\mathrm{Lie}(G)) \simeq \mathrm{Lie}( \mathrm{INN}(\mathrm{String}_G)) \,,

but when I tried to construct this equivalence I had to notice that I had difficulties describing 2-morphisms of Lie 3-algebras in terms of homotopies of FDA chain maps (not that there is a conceptual problem - I simply was not up to it).

So: I think $\mathrm{String}_G$ gauge theory is the theory of Chern-Simons 2-gerbes, namely of $\mathrm{INN}(\mathrm{String}_G)$ -transport.

Posted by: urs on November 14, 2006 10:05 AM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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The obvious candidate is the $G/ G$ gauged WZW model (or just the WZW model, if we want a conformal field theory).

Sorry, I realize I probably did not really answer your question.

I see where you are coming from when your propose this candidate model. I am not really sure, though.

It is certainly right that on the boundary of Chern-Simons theory we have a WZW theory, in a way.

But right now I feel a little unsure with saying that this WZW theory is a $\mathrm{String}_G$ gauge theory.

The funny thing is (as you know, but others reading this here might not) that the WZW theory rather lives “inside” $\mathrm{String}_G$ in a peculiar way, related to the fact that the 2-group $\mathrm{String}_G$ is, as a groupoid with monoidal structure, nothing but the extended groupoid picture # of a bundle gerbe on $G$ with multiplicative structure.

The group of based paths $P G$ plays the role of the surjective submersion $Y \to G$ and the central extension

(1)

U(1) \to \hat \Omega G \to \Omega G

provides precicely a circle bundle on

(2)

Y^{[2]} = (P G)^{[2]} = \Omega G \,.

Finally, the “vertical” product in the 2-group yields the multiplicativity of this line bundle on $Y^{[3]}$ .

WZW theory is a theory of maps into $G$ that couple to this gerbe over $G$ .

So in that sense WZW theory “lives inside” $\mathrm{String}_G$ instead of being a $\mathrm{String}_G$ gauge theory.

I would say.

Posted by: urs on November 14, 2006 10:33 AM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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Thanks very much for your replies. I will do my best to say something about this in my talk, and point the listeners to you for more details.

Pretty soon I’ll put up some transparencies for my talk. You’ll know everything I say, but there’s some stuff about $BF$ theory that nobody seems to discuss except us. Namely: let $G$ be a compact simple Lie group, let $A$ be a $G$ -connection on a 4-manifold $M$ , and let $B$ be a $\mathfrak{g}$ -valued 2-form. The action for $BF$ theory with cosmological term in 4 dimensions is $\int_M \mathrm{tr}(B \wedge F + \frac{1}{2} B \wedge B)$ where $F$ is the curvature of $A$ . The resulting field equations are $B = F,$ $d_A B = 0.$ (Given the first, the second actually follows from the Bianchi identity.)

So, we higher gauge theory experts instantly note that given these field equations, the pair $(A,B)$ is a 2-connection with gauge 2-group $\mathbf{G}$ . Here $\mathbf{G}$ corresponds to the obvious crossed module $G \stackrel{1}{\to} G .$

This is old stuff, at least for us. But now combine it with your newer ideas! Two ideas seem to follow:

To get holonomies in $BF$ theory “off shell”, when the field equations do not hold. it seems we’ll need to use your trick of replacing the relevant 2-group $\mathbf{G}$ with the 3-group $INN(\mathbf{G})$ (or something like that).
There’s a tight relation between 4d BF theory with cosmological term and 3d Chern-Simons theory: the latter appears as a kind of “boundary field theory” of the former. This suggests a possible relation between the 2-group I’m calling $\mathbf{G}$ (built from $G$ as discussed above) and the 2-group you’re calling $\mathrm{String}_G$ (built from $G$ in a seemingly very different way).

But, it’s difficult to see the relation, since $\mathbf{G}$ is so much simpler than $\mathrm{String}_G$ !

Posted by: John Baez on November 15, 2006 3:56 AM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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So, we higher gauge theory experts instantly note that given these field equations, the pair $(A,B)$ is a 2-connection with gauge 2-group $\mathbf{G}$

On the other hand, one might also notice the following:

The action of WZW theory locally involves a term of the form

(1)

\int_\Sigma B \,,

where $B$ is a 2-form. Hence we realize that globally this term is really the surface holonomy of a 1-gerbe.

The action of CS theory locally involves a term of the form

(2)

\int_X \mathrm{CS}(A) \,,

where $\mathrm{CS}(A)$ is a Chern-Simons 3-form. Hence we realize that the action is really the volume holonomy of a 2-gerbe.

Now, the action of BF-theory involves a term of the form

(3)

\int_M \mathrm{tr}(B \wedge F + \frac{1}{2}B \wedge B) \,.

Hence we realize that globally this action is really the 4-volume holonomy of a 3-gerbe … or wait, do we?

I’d be tempted to think about it this way. And when you say

There’s a tight relation between 4d BF theory with cosmological term and 3d Chern-Simons theory: the latter appears as a kind of “boundary field theory” of the former.

this seems to be consistent with thinking of BF theory as a 3-gerbe theory.

Actually, I wasn’t aware of this relation of BF to CS. But now that you mention it, it’s sort of obvious: because on-shell the BF integrand is just the Pontryagin 4-form

(4)

\mathrm{tr}(F \wedge F) \,,

and this is indeed the curvature 4-form of the Chern-Simons 3-form.

Hm, I will want to think about this. Incidentally, in the last couple of days I talked with several people about whether or not there is a 3-gerbe theory that is to CS like CS is to WZW. Probably the relation BF/CS will be subtly different in nature from that of CS/WZW, but still.

To get holonomies in BF theory “off shell”, when the field equations do not hold. it seems we’ll need to use your trick of replacing the relevant 2-group $\mathbf{G}$ with the 3-group $INN(\mathbf{G})$ (or something like that).

Yes, maybe. I need to think about this. From what I just said about 3-gerbes it would follow that we should instead be looking for a suitable 4-group.

The obvious candidate for that 4-group would be

(5)

\mathbf{INN}(G_3) \,,

whjere $G_3$ is the structure 3-group of Chern-Simons theory (something like $G_3 = \mathrm{INN}(\mathrm{String}_G)$ ).

The nature of these “inner automorphisms” $n$ -groups is precisely such that at top level they provide for the curvature of connections with values in the underlying $(n-1)$ -group. That’s what makes interpreting connections in terms of Schreier theory work.

And just for the record, it might be worthwhile to recall why we restrict to inner automorphisms: the Atiyah groupoid sequence associated to a $G$ -principal bundle $P \to X$ is

(6)

P \times_G G \to P \times_G P \to (X\times X) \,,

where the action on $G$ in forming $P \times_G G$ is by conjugation. Then Schreier theory combined with Danny’s insight tells us that a connection on $P$ is a pseudofuntor $(\mathrm{tra},\mathrm{curv})$

(7)

\array{ && \mathrm{AUT}(P \times_G G) \\ & \swarrow && \nwarrow^{(\mathrm{tra},\mathrm{curv})} \\ P \times_G G &\to& P \times_G P &\to& (X\times X) } \,,

where $\mathrm{AUT}(P \times_G P)$ is the 2-groupoid whose objects are points in $X$ , whose morphisms are isomorphisms between fibers of $P \times_G G$ over these points and whose 2-morphisms are natural transformations of these (where we think of the groups $P_x \times_G G$ as 1-object categories).

So we do have outer automorphisms available in $\mathrm{AUT}(P \times_G P)$ - but the pseudofunctor $(\mathrm{tra},\mathrm{curv})$ does not have them in its image!

The reason is that the transport

(8)

\mathrm{tra} : (x \to y) \mapsto (P_x \stackrel{\mathrm{tra}_{x,y}}{\to} P_y)

acts on the $P$ -factor of $P_x \times_G G$ , not directly on the $G$ -factor. So if you locally identitfy all the fibers $P_x$ by choosing a local section, we find that $\mathrm{tra}$ acts by conjugation on $G$ as we use the coequalizer to transform

(9)

(P_x \stackrel{\mathrm{tra}(x,y)}{\to} P_y)\times_G G

(10)

P_x \times_G (G \stackrel{\mathrm{Ad}_{g}}{\to} G) \,,

for some $g\in G$ determined by $\mathrm{tra}$ and our choice of local section.

Of course this argument should be made more systematic such that we can have more trust in it as we categorify the setup. But for the moment that’s my explanation of why we want to say that a $G_n$ -connection is locally an $(n+1)$ -functor to $\mathrm{INN}(G_n)$ .

And I guess we could keep on going like this.

A $G_2$ -transport should locally be a 3-functor to

(11)

\mathrm{INN}(G_2) \,,

thus being a fake flat $\mathrm{INN}(G)_2$ -transport (the curvature transport of the original $G_2$ -connection).

But the most general (not fake flat) $\mathrm{INN}(G_2)$ -transport should then be a 4-functor to the 4-group

(12)

\mathrm{INN}(\mathrm{INN}(G_2)) \,.

I guess we can continue this way.

Having said all this, I’d be tempted to propose that 4-dimensional BF-theory for gauge group $G$ is a theory of pseudo-3-functors to

(13)

\mathrm{INN}(\mathrm{INN}( \mathrm{String}_G)) \,.

In as far as this is correct, it would in fact be a generalization of the way BF theory is usually stated, analogous to how a gerbe connection is a generalization of a globally defined 2-form.

What one should do to test this proposal: proceed along the lines how I constructed the Lie 3-algebra $\mathrm{cs}(g)$ from the Lie 2-algebra $g_k$ #. There are only a handful of tensors on the corresponding Lie algebras that yield terms that one can add to the equations defining the corresponding FDA, and there are lots of constraints coming from the grading and the nilpotency of the differential. As a result only a small number of coefficients (likely just one) are actually not determined by the general structure.

Posted by: urs on November 15, 2006 9:43 AM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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I wrote:

I’d be tempted to propose that 4-dimensional BF-theory for gauge group G is a theory of pseudo-3-functors to $\mathrm{INN}(\mathrm{INN}(\mathrm{String}_G))$ .

Not sure if it is helpful at all: but in as far $\mathrm{INN}(\mathrm{String}_G)$ is related to the (?) M-theory 3-group # it would suggest that the corresponding 4-group would correspond to - F-theory.

But what do I know…

Posted by: urs on November 15, 2006 6:27 PM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

I don’t see any cosmological constant,
just plain BF theory - no?

In re: off-shell, that’s a major aspect of
the Batalin-Vilkovisky constructions for the
Lagrangian case with symmetries and even more visibly forthe Batalin-Fradkin- Vilkovisky construction in the constrained Hamiltonian case.

I would guess any comparison would require
dg-categorification.

Posted by: jim stasheff on November 20, 2006 3:34 PM | Permalink | Reply to this

Re: Dijkgraaf-Witten Theory and its Structure 3-Group

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I don’t see any cosmological constant, just plain BF theory - no?

John is the expert for that, but while I am online and he is not yet, I hope it’s okay if a provide a link or two.

To get Einstein-Hilbert gravity from BF theory one needs to deform it a little by adding extra terms to the Lagrangian.

For a discussion of how this may be related to the cosmological constant, you might try this and the literature given there.

There would be more to say. But not by me. :-)

Posted by: urs on November 20, 2006 4:39 PM | Permalink | Reply to this

Read the post Oberwolfach CFT, Tuesday Morning
Weblog: The n-Category Café
Excerpt: On Q-systems, on the Drinfeld Double and its modular tensor representation category, and on John Roberts ideas on nonabelian cohomology and QFT.
Tracked: April 3, 2007 2:10 PM

Read the post Dijkgraaf-Witten and its Categorification by Martins and Porter
Weblog: The n-Category Café
Excerpt: On Dijkgraaf-Witten theory as a sigma mode, and its categorification by Martns and porter.
Tracked: January 5, 2008 3:23 AM

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November 6, 2006