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November 5, 2007

2-Vectors in Trondheim

Posted by Urs Schreiber

Tomorrow early morning Konrad Waldorf and I will catch a plane to Trondheim where we have been invited to spend a couple of days with Prof. Nils Baas (known to nn-Café regulars from

Detecting Higher Order Necklaces
Topology in Trondheim and Kro, Baas & Bökstedt on 2-Vector Bundles
What Does the Classifying Space of a 2-Category Classify?
Seminar on 2-Vector Bundles and Elliptic Cohomology, I, II).

Konrad and I will also give a 1+1 hour talk on 2-transport, Konrad probably with an emphasis on (nonabelian) surface holonomy and WZW terms, and I probably with an emphasis on 2-vector bundles and the corresponding 2-vector transport.

You can find Konrad’s slides from previous talks along these lines here and here.

In that context I wanted to go over my notes on 2-Vector transport and Line bundle gerbes and incorporate a polished version into my slide set String- and Chern-Simons nn-transport today.

But then, after I spent the better part of the day with teaching, right when I wanted to get back to my office to do some real work, they were — evacuating the building! Because some stupid emergency power supply was found to be defunct and life in the presence of a defunct backup power supply regarded to be so dangerous that they decided to shut down everything.

This means no further slides and some of the links provided here temporarily broken (since linking to a shut down server…).

But here is a little blogged discussion of how a rank-1 2-vector bundle, when regarded as a transport 2-functor, is trivialized by a twisted ordinary vector bundle and has descent data given by a line bundle gerbe.

If you replace everywhere in the discussion the algebras (Morita equivalent to the ground field) appearing with Clifford algebras, you pass from line 2-bundles to more interesting 2-vector bundles and obtain the kind of phenomenon that we talked about in Higher Clifford Algebras.

I’ll decide, just for definiteness, to work once and for all over complex finite dimensional vector spaces, which I take to live in the category Vect. \mathrm{Vect} \,. A 2-vector space for me is a Vect\mathrm{Vect}-module category. I am restricting attention to those 2-vector space which admit a basis which means that they look like Mod A \mathrm{Mod}_A for some (finite dimensional, complex in my setup) algebra AA. (That’s because Mod A=Hom(ΣA,Vect)\mathrm{Mod}_A = \mathrm{Hom}(\Sigma A, \mathrm{Vect}) just like an ordinary vector space VV with basis SS is VHom(S,)V \simeq \mathrm{Hom}(S, \mathbb{C})).

One point I might emphasize is that this means we are looking at something more general that Kapranov-Voevodsky 2-vector spaces, which are those 2-vector spaces of the form Mod n \mathrm{Mod}_{\mathbb{C}^{\oplus n}} only. The following simple application is supposed to illustrate what happens when we don’t restrict to the leftmost part of the chain of inclusions KV2VectBim:=2Vect b2Vect. \mathrm{KV}2\mathrm{Vect} \hookrightarrow \mathrm{Bim} := 2\mathrm{Vect}_{b} \hookrightarrow 2\mathrm{Vect} \,.

So lets look at line 2-bundles (with connection and parallel transport). This are 2-vector bundles P 2(X) tra 2Vect b \array{ P_2(X) \\ \downarrow^{\mathrm{tra}} \\ 2\mathrm{Vect}_{b} } with local structre (see The First Edge of the Cube for the general formalism used here) given by the canonical 2-representation (html) of shifted U(1)U(1):

ρ:ΣΣU(1)2Vect b \rho : \Sigma \Sigma U(1) \to 2\mathrm{Vect}_b

ρ: Id c Id c . \rho : \array{ & \nearrow \searrow^{\mathrm{Id}} \\ \bullet &\Downarrow^c& \bullet \\ & \searrow \nearrow_{\mathrm{Id}} } \;\;\; \mapsto \;\;\; \array{ & \nearrow \searrow^{\mathbb{C}} \\ \mathbb{C} &\Downarrow^{\cdot c}& \mathbb{C} \\ & \searrow \nearrow_{\mathbb{C}} } \,.

This simply meas that the typical fiber looks like \mathbb{C} thought of as a placeholder for Mod =Vect\mathrm{Mod}_{\mathbb{C}} = \mathrm{Vect}, i.e. like the canonical 1-dimensional 2-vector space, and that the local transitions of this fiber are given by the canonical action of ΣU(1)\Sigma U(1) on this.

But this then also means that the generic fiber of our 2-vector bundle is a gadget equivalent to our typical fiber. Since the equivalence is equivalence internal to 2Vect b:=Bim2\mathrm{Vect}_b := \mathrm{Bim}, this means it is Morita equivalence of algebras.

Notice that the ground field is Morita equivalent to all algebras of the form End(V)\mathrm{End}(V), with the weak equivalence induced by the bimodules

VEnd(V) \mathbb{C} \stackrel{V}{\to} \mathrm{End}(V)

and

End(V)V * \mathrm{End}(V) \stackrel{V^*}{\to} \mathbb{C}

satisfying

VEnd(V)V * \mathbb{C} \stackrel{V}{\to} \mathrm{End}(V) \stackrel{V^*}{\to} \mathbb{C} \simeq \mathbb{C} \stackrel{\mathbb{C}}{\to} \mathbb{C}

and

End(V)V *VEnd(V)End(V)End(V)End(V) \mathrm{End}(V) \stackrel{V^*}{\to} \mathbb{C} \stackrel{V}{\to} \mathrm{End}(V) \simeq \mathrm{End}(V) \stackrel{\mathrm{End}(V)}{\to} \mathrm{End}(V)

(all of this taking place in Bim\mathrm{Bim}).

So we find that the existence of a smooth local ρ\rho-trivialization of our 2-vector bundle

P 2(X) tra 2Vect b \array{ P_2(X) \\ \downarrow^{\mathrm{tra}} \\ 2\mathrm{Vect}_{b} }

namely a completion to a square

P 2(Y) π P 2(X) triv t tra ΣΣU(1) ρ 2Vect b \array{ P_2(Y) &\stackrel{\pi}{\to}& P_2(X) \\ \downarrow^{\mathrm{triv}} &\Downarrow^t& \downarrow^{\mathrm{tra}} \\ \Sigma \Sigma U(1) &\stackrel{\rho}{\to}& 2\mathrm{Vect}_{b} }

is (possibly recall the discussion at Higher Clifford Algebras at this point)

a vector bundle VY V \to Y over the “cover” space YY (some surjective submersion π:YX\pi : Y \to X) arising as the components of the local trivialization transformation tt

t 1:(yY)V x(End(V)tra(x)) t^{-1} : (y \in Y) \mapsto \;\; \mathbb{C} \stackrel{V_x}{\to} (\mathrm{End}(V) \simeq \mathrm{tra}(x))

which identitfies each algebra coming from the fibers of our 2-vector bundle, after being pulled back to YY, with the endomorphism algebra of a fixed vector space VV.

From this local trivialization, we obtain now, by the general nn-transport nonsense, the transition

g:=π 2 *t 1π 1 *t g := \pi_2^* t^{-1} \circ \pi_1^* t

Over points yY [2]y \in Y^{[2]}, its components are the vector spaces

g y=L y:=V π 2(y)tra(π(π 1(y)))V π 1(y) *. g_y = \mathbb{C} \stackrel{L_y}{\to} \mathbb{C} := \mathbb{C} \stackrel{V_{\pi_2(y)}}{\to} \mathrm{tra}(\pi(\pi_1(y))) \stackrel{V^*_{\pi_1(y)}}{\to} \,.

These are 1-dimensional and equipped with a canonical product operation

π 12 *Lπ 23 *Lπ 13 *L \pi_{12}^* L \otimes \pi_{23}^* L \simeq \pi_{13}^* L

coming from forming the obvious filled triangle from the transition functions, as described in section: Parallel nn-transport in String- and Chern-Simons nn-transport.

In a flattened-out notation circumventing having to draw these triangles here in MathML, the operation consists of cancelling the center piece

End(V)V *VEnd(V)End(V)End(V)End(V) \mathrm{End}(V) \stackrel{V^*}{\to} \mathbb{C} \stackrel{V}{\to} \mathrm{End}(V) \simeq \mathrm{End}(V) \stackrel{\mathrm{End}(V)}{\to} \mathrm{End}(V)

in

π 12 *Lπ 23 *L =V π 1(y)tra(π(y))V π 2(y) *V π 2(y)tra(π(y))V π 3(y) * V π 1(y)tra(π(y))V π 3(y) * \begin{aligned} \pi_{12}^* L \otimes \pi_{23}^* L &= \mathbb{C} \stackrel{V_{\pi_1(y)}}{\to} \mathrm{tra}(\pi(y)) \stackrel{V^*_{\pi_2(y)}}{\to} \mathbb{C} \stackrel{V_{\pi_2(y)}}{\to} \mathrm{tra}(\pi(y)) \stackrel{V^*_{\pi_3(y)}}{\to} \mathbb{C} \\ & \simeq \mathbb{C} \stackrel{V_{\pi_1(y)}}{\to} \mathrm{tra}(\pi(y)) \stackrel{V^*_{\pi_3(y)}}{\to} \mathbb{C} \end{aligned}

So we see: the descent data on YY for the local trivialization of the line 2-bundle we started with is nothing but a line bundle gerbe.

Moreover, we see that the local trivialization t 1:(yY)V x(End(V)tra(x)) t^{-1} : (y \in Y) \mapsto \;\; \mathbb{C} \stackrel{V_x}{\to} (\mathrm{End}(V) \simeq \mathrm{tra}(x)) we started with is the corresponding gerbe module (this is the part we miss when working not with the full 2Vect b2\mathrm{Vect}_b): a vector bundle on YY which descends to a twisted vector bundle on XX whose twist s precisely our bundle gerbe.

(From what I have said so far it is not clear that it is necessary for tt to define a smooth vector bundle on YY, but is sufficient for LL to be smooth. So if VV wasn’t in the first place, we can replace it after the fact with one that is.)

The whole discussion straightforwardly incorporates the 2-transport, too. See 2-Vector Transport and Line Bundle Gerbes. That’s the toy example of rank-1 2-vector bundles in terms of all of 2Vect b2\mathrm{Vect}_b.

And now I’ll go to bed. Need to get up real early tomorrow.

Posted at November 5, 2007 7:28 PM UTC

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1 Comment & 1 Trackback

Re: 2-Vectors in Trondheim

The a posteriori slides for my talk in Trondheim can now be found in

- section: Parallel nn-transport

- subsection: Miscellanea

- subsection: Associated nn-transport

of String- and Chern-Simons nn-Transport

(Hm, does anyone know if it is possible to link directly into a hyperlinked pdf file??)

I am glad to be able to say that people are now seriously looking into solving The first nn-Café Millenium Prize.

Posted by: Urs Schreiber on November 12, 2007 8:17 PM | Permalink | Reply to this
Read the post Comparative Smootheology
Weblog: The n-Category Café
Excerpt: A survey and comparison of the various notions of generalized smooth spaces, by Andrew Stacey.
Tracked: January 3, 2008 11:01 PM

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