### 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 $n$-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 $n$-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
$\mathrm{Vect}
\,.$
A 2-vector space for me is a $\mathrm{Vect}$-module category. I am restricting attention to those 2-vector space which admit a *basis* which means that they look like
$\mathrm{Mod}_A$
for some (finite dimensional, complex in my setup) algebra $A$. (That’s because $\mathrm{Mod}_A = \mathrm{Hom}(\Sigma A, \mathrm{Vect})$ just like an ordinary vector space $V$ with basis $S$ is $V \simeq \mathrm{Hom}(S, \mathbb{C})$).

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 $\mathrm{Mod}_{\mathbb{C}^{\oplus n}}$ 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 $\mathrm{KV}2\mathrm{Vect} \hookrightarrow \mathrm{Bim} := 2\mathrm{Vect}_{b} \hookrightarrow 2\mathrm{Vect} \,.$

So lets look at *line* 2-bundles (with connection and parallel transport). This are 2-vector bundles
$\array{
P_2(X)
\\
\downarrow^{\mathrm{tra}}
\\
2\mathrm{Vect}_{b}
}$
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 $U(1)$:

$\rho : \Sigma \Sigma U(1) \to 2\mathrm{Vect}_b$

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

This simply meas that the *typical fiber* looks like $\mathbb{C}$ thought of as a placeholder for $\mathrm{Mod}_{\mathbb{C}} = \mathrm{Vect}$, 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 $\Sigma U(1)$ 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 $2\mathrm{Vect}_b := \mathrm{Bim}$, this means it is **Morita equivalence** of algebras.

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

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

and

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

satisfying

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

and

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

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

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

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

namely a completion to a square

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

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

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

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

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

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

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

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

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

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

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

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

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

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

in

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

So we see: the descent data on $Y$ 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
$t^{-1} : (y \in Y) \mapsto \;\; \mathbb{C} \stackrel{V_x}{\to} (\mathrm{End}(V) \simeq \mathrm{tra}(x))$
we started with is the corresponding *gerbe module* (this is the part we miss when working not with the full $2\mathrm{Vect}_b$): a vector bundle on $Y$ which descends to a twisted vector bundle on $X$ whose twist s precisely our bundle gerbe.

(From what I have said so far it is not clear that it is *necessary* for $t$ to define a *smooth* vector bundle on $Y$, but is sufficient for $L$ to be smooth. So if $V$ 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 $2\mathrm{Vect}_b$.

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

## Re: 2-Vectors in Trondheim

The

a posteriorislides for my talk in Trondheim can now be found in- section:

Parallel $n$-transport- subsection:

Miscellanea- subsection:

Associated $n$-transportof String- and Chern-Simons $n$-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 $n$-Café Millenium Prize.