## March 22, 2007

### Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

#### Posted by Urs Schreiber

The open charged 2-particle looks like $a \to b$ and its quantization, $q(\mathrm{tra})$, assigns to it a morphism $f$ of its 2-vector space $V \stackrel{f}{\to} V$ of 2-states $\psi : I \to q(\mathrm{tra})$ each of which is a generalized element $\psi : (a \to b) \;\; \mapsto \;\; \array{ I &\stackrel{\mathrm{Id}}{\to}& I \\ \psi(a) \downarrow \;\, &\;\;\Downarrow^{\psi(a\to b)}& \;\, \downarrow \psi(b) \\ V &\stackrel{f}{\to}& V } \,.$

When the 2-particle is charged under a line 2-bundle (a line bundle gerbe) the 2-vectors $\psi(a)$ and $\psi(b)$ are Chan-Paton bundles on D-branes, also known as modules for that gerbe.

The space of states is acted on 2-linearly by pull-push through spans $\array{ && \mathrm{hist} \\ & \swarrow &&& \searrow \\ \mathrm{conf} &&&&& \mathrm{conf} }\,,$ which may encode operation like time evolution or gauge transformations like T-duality.

In a chosen 2-basis for $V$, which is an algebra, 2-states appear as modules and 2-linear maps appear as bimodules.

The former fact harmonizes with the term “gerbe module” used for D-branes. In that sense, these bimodules could be addressed as bi-branes.

This is the language now chosen in

Fuchs, Schweigert, Waldorf
Bi-branes: Target Space Geometry for World Sheet topological Defects
Bi-branes: Target Space Geometry for World Sheet topological Defects.

Like an ordinary brane – at least in its geometric incarnation as a subspace with Chan-Paton bundle on it – is a submanifold of target space over wich the Kalb-Ramond field strength (the curvature of the gerbe) trivializes, a bi-brane is defined to be a submanifold of two different target spaces, over which the difference of two KR-fields trivializes.

While only very briefly toughed upon in the above paper, this is the familiar central structure of interest in topological T-duality, in which case the bi-brane bundle is the Poincaré-line bundle.

In fact, the condition on the KR fields now given for bi-branes is known in topological T-duality, as for instance discussed on p. 5 of

Peter Bouwknegt, Keith Hannabuss, Varghese Mathai
T-duality for principal torus bundles and dimensionally reduced Gysin sequences
hep-th/0412268.

New constructions along these lines, with Courant algeboroids and their morphisms, encoding gerbes and their morphisms, are in preparation by Cavalcant & Gualtieri: T-duality with NS-flux and generalized complex structures.

Posted at March 22, 2007 9:16 PM UTC

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### Re: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

This is interesting and I feel somehow related to some work I’m doing on orthogonal hermite polynomials in 2-variables and pointwise fourier integral transforms and diagonalization via projective integral transforms via the Gram-Schmidt process. All my work is done via hypergeometric functions, generating functions, and quite a bit of analytic number theory. The nice thing is that every single step of the way I am only working with integer sequences and renormalized power series expansions. So little ole me has been able to do this stuff concretey without all this abstract nonsense. So, why is all this abstract stuff so popular when number theory is so much more concrete and applicable?

Posted by: Stephen Crowley on March 22, 2007 10:51 PM | Permalink | Reply to this

### two things and 2-things

[…] polynomials in 2-variables […]

I am a little worried that what you mean are polynomials in two variables, i.e. elements in some $K[x,y]$.

Is that so? In that case I’d unfortunately have to disappoint you: when we say “2-vectors” we don’t just mean two vectors (as in: “a pair of vectors”), but a notion of vector that is to an ordinary vector like a category is to a set.

On the other hand, if you really do mean polynomials in categorified variables, then I’d love to know more details on what exactly it is you are considering!

I should maybe remark that, after all, there is often a simple special case of a 2-something which does appear more or less as nothing but two somethings.

For instance, the 2-vector spaces $\mathbf{V}$ described and studied by John Baez and Alissa Crans in HDA IV are , in particular, two vector spaces $V_0$ and $V_1$ $\mathbf{V} = \{V_0, V_1\} \,,$ namely a vector space, $V_0$, of objects and a vector space, $V_0 \oplus V_1$ of morphisms.

What makes the two vector spaces a 2-vector space is extra structure on this, given by maps $s,t : V_0 \oplus V_1 \to V_0$ and $i : V_0 \to V_0 \oplus V_1$ and $\circ : (V_0\oplus V_1)^{[2]} \to V_0 \oplus V_1$satisfying the usual laws which say that $\circ$ is the composition of morphism which have source $s$ and target $t$.

Similar statements hold for 2-groups. A (strict) 2-group $\mathbf{G}$ is, in particular, two groups $\mathbf{G} = (G_0, G_1) \,.$

Again, the reason for this is that a strict 2-group, like a Baez-Crans 2-vector space, is an internal category: BC 2-vector spaces are categories internal to ordinary vector spaces, while strict 2-groups are categories internal to ordinary groups.

Since a category consists, in particular, of two objects, namely an object of objects and an object of morphisms, this gives rise to the phenomenon that 2-things often look like two things .

Sometimes (rarely, but sometimes), it indeed happens that people first study a theory of two things (like of two variables) with extra structure and properties around, and only later realize (or somebody else does, usually ;-) that what they are really studying is a 2-thing.

A nice example of this are 2-class functions and their categorical interpretation.

Like a class function on a group is a function $g \mapsto f(g)$ of a single variable that is invariant under conjugation, an $n$-class function $(g_1,g_2,\cdots, g_n) \mapsto f(g_1,g_2,\cdots, g_n)$ is a function of $n$-tuples of (pairwise commuting) group elements, that is invariant under simultaneous conjugation of these group elements.

So an $n$-class function is, in particular, a funcion of $n$-variables. People found that to be of use in certain contexts.

Then along came Ganter and Kaparanov and showed that like class functions come from traces of representations of ordinary groups, 2-class functions can be seen as coming from 2-traces of (2-)groups!

I think there are more examples of this kind. In fact, here we enjoy going through standard literature and trying to spot conglomerates of structures that secretly arrange themselves into single, unified $n$-categorical structures.

A more intricate example of this, for instance, is the local data for connections on gerbes. This comes in large parts of the literature as a vast array of various $p$-forms with values in various groups and Lie algebras. But staring at this structure for a while reveals that all this mess are nothing but the components of one single morphism between higher-groupoids.

There are more examples like that, but I think I’ll stop here.

If you knew all this and really do mean that you are working with categorified variables then I apologize for wasting your time with this lengthy reply here, hoping that maybe others find it helpful.

Posted by: urs on March 23, 2007 11:27 AM | Permalink | Reply to this

### Re: two things and 2-things

Is the $n$-class function story expected to continue? Are 3-class functions expected to come from 3-traces of (3-)groups?

And what happens with 2-traces of more general 2-groups ($H \to G$)?

Posted by: David Corfield on March 23, 2007 1:35 PM | Permalink | Reply to this

### n-class functions and n-traces

Is the $n$-class function story expected to continue? Are 3-class functions expected to come from 3-traces of (3-)groups?

And what happens with 2-traces of more general 2-groups $(H \to G)$?

This are interestring questions!

I haven’t seriously thought about them yet.

But probably Bruce Bartlett has. He is our local expert on Kaparanov-Ganter 2-traces.

Posted by: urs on March 23, 2007 3:01 PM | Permalink | Reply to this

### Re: n-class functions and n-traces

Since I’ve been sneakily co-opted into this conversation, let me add my two cents worth.

Is the n-class function story expected to continue? Are 3-class functions expected to come from 3-traces of (3-)groups?

It seems that the pattern should indeed continue - that’s the way I understand it, at least. One thing though - I understand it as n-class function coming from n-representations of groups, not just from a n-group. I guess that’s what you guys meant as well; I mean what is a “trace” of a group element?

Lets recall the pattern. You can take the character of an ordinary representation of a groupoid $G$ to get a class function, i.e. a section of the inertia groupoid $\Lambda G$:

(1)$\chi : Rep(G) \rightarrow \Gamma_{\Lambda G}.$

In particular, if $G$ is a group, taking the character of a representation of $\Lambda G$ gives us a section of $\Lambda^2 G$:

(2)$\chi : Rep(\Lamdba G) \rightarrow \Gamma_{\Lambda^2 G}$

A section of $\Lambda^2 G$ is just a 2-class function on the group.

Similarly, you can take the 2-character of a 2-representation of a group $G$ to get a representation of $\Lamdba G$:

(3)$\chi : 2Rep(G) \rightarrow Rep(\Lambda G)$

Thus if you take the ordinary character of a 2-character you get a 2-class function:

(4)$2Rep(G) \rightarrow Rep(\Lamdba G) \rightarrow \Gamma_{\Lambda^2 G}$

Similarly, one expects that the 2-character of a 2-representation of $\Lambda G$ will give us a rep of $\Lambda^2 G$:

(5)$\chi : 2Rep (\Lambda G) \rightarrow Rep(\Lambda^2 G)$

Taking the character again will give us a section of $\Lambda^3 G$, i.e. a 3-class function .

By the way, $2Rep (\Lambda G)$ is a really cool 2-category to consider. Just like $Rep(\Lambda G)$ is a braided monoidal category, $2Rep(\Lambda G)$ is a braided monoidal 2-category with duals! Anyone want to help me make this rigourous?

Posted by: Bruce Bartlett on March 23, 2007 4:15 PM | Permalink | Reply to this

### Re: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

Ok, so I’ll make this more concrete. The hermite polynomials can be expressed via hypergeometric 2F0 functions and multiplicatively renormalized.

Nobuhiro Asai, Izumi Kubo, Hui-Hsiung Kuo, Multiplicative Renormalization And Generating Functions: II, Taiwanese Journal of Mathematics 8 (4) (2004) 593–628.

$\hat{\Psi}_n (x) = \frac{\Psi_n (x)}{\sqrt{\lambda_n}} = \frac{_2 F_0 \left( \frac{[- \frac{n}{2}, \frac{1}{2} - \frac{n}{2}]}{\emptyset}, - \frac{2 t^2}{x^2} \right) x^n}{\sqrt{t^{2 n} n!}}$

Which is orthnormal with respect to the measure on x, e.g., the inner products are given by the Kronecker delta

$\left\langle \hat{\Psi}_n (x,t), \hat{\Psi}_m (x,t) \right\rangle_{\mathcal{H}_{\mu_{\sigma^2}}} = \int_{- \infty}^{\infty} \hat{\Psi}_n (x,t) \hat{\Psi}_m (x,t) d \mu_{\sigma^2 (x,t)} = \delta_{n, m}$

Now we apply the pointwise Fourier transform to each element in this real orthonormal set (a discrete infinite vector of continuous vectors or distribution functions).

$T_n^{\Psi} (x, t) = \int_{- \infty}^{\infty} \hat{\Psi}_n (x, t) e^{- i x y} \d x$

The generating function for this transform can be found.. derived it heuristically and its unkown to me whether there is a systematic way to do it.

In any case, after the fourier transform the set is no longer orthogonal or normal, so it can be renormalzed again, and then the Gram-Schmidt orthogonalization process can be applied by taking the complex inner products using the complex conjugates and sucessively projecting the 0th vector onto the 1st, then the 1st projection onto the 2nd, 2nd onto 3rd, etc.

This process is burdensome but I wrote some maple code to do it fairly quickly and then work on deriviving generating functions again heuristically. I noticed that the imaginary component of the even fourier transforms =0 forall n, and the real part of the odd dimensional transforms =0 forall n, Based on this I decmoposed the transform into even and odd componets, and apply the Gram-schmidt process to each piece and then divide the dimension by 2 and sum to get an infinite seqeuence of complex orthonormal distribution functions which rapidly approach 0, and t always stays real. It’s the space variable that is complexified.

Posted by: Stephen Crowley on March 23, 2007 5:20 PM | Permalink | Reply to this

### Re: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

Okay, thanks for the details.

Now I am sure that, concerning your original remark #

This is interesting and I feel somehow related to some work I’m doing

I have to say, unfortunately, that, no, your work does not seem to be related to 2-vector spaces.

Sorry…

Posted by: urs on March 23, 2007 5:35 PM | Permalink | Reply to this

### Re: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

Thanks for the clarification.. how can I be sure?

Is a 2-vector not related to a ‘symmetric bimeasure’ ?

Schroeder has done a lot of work in this area. see http://www.springerlink.com/content/tu1864h0354k5375/

specifically, http://www.math.uni-hamburg.de/home/schreiber/FRSfrom2Transport.pdf

on page 30 where it talks about “2.4.2 Adjunctions using Duality and Projection”

I’m interpreting the pointwise countable Fourier transforms in terms of the Bochner integral or the ‘Morse-Transue’ integral, and ‘weakly harmonizable’ stochastic processes, Haar/Radon Measures, etc.

I believe the things im studying are the ‘Bochern-Riesz means’ related to the Riesz-Fischer theorem.

I have the great book ‘Intruduction to Geometric Probability’ by Klain and Rota where I’ve found their presentation of ‘valuations on polyconvex lattices’ and the Euler characteristic very nice. Specfically, Section 8.2 on ‘Even and Odd Valuations’, which I believe is exactly what I’ve found with the real/even odd/imaginary decomposition of the Fourier transform I’ve found. The hypergeometric function I listed above was studied by Wiener (although not in this hypergeometric form) and he called it the ‘polynomial chaos’, and the closely related ‘Differential Space’.

So does Defintion 16 on the symmetric Frobenius algebra not meet my ‘sequential orthnormal complex projections’ ?

Posted by: Stephen Crowley on March 23, 2007 8:35 PM | Permalink | Reply to this

### 2-vector spaces

Is a 2-vector not related to a ‘symmetric bimeasure’?

Not that I knew!

There are many flavors of 2-vector spaces. It would be very nice if you could explicitly point me to the one that you have in mind.

What I know is that the “continuously infinite” version of Kapranov-Voevosky 2-vector spaces are of the form $\mathrm{Hilb}^X$, where $X$ is a measure space. (See this.)

Is that what you are thinking of?

Posted by: urs on March 26, 2007 9:14 AM | Permalink | Reply to this

### Re: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

I think this is mostly off-topic, aside from the word “gerbe”: some time ago you had an interesting post on the String Coffee Table about the n-cubed scaling in the theory on a stack of 5-branes. In your opinion, is the n-cubed scaling now understood, from the perspective of work on gerbes or 2-gauge theories or anything else along these lines? Has any of this shed light on what the right degrees of freedom are to see the counting explicitly? Any links would be much appreciated.

Posted by: Matt Reece on March 25, 2007 8:07 PM | Permalink | Reply to this

### Re: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

I am not aware of any concrete progress in these questions. I also haven’t thought about it much since then. One day I should again.

Posted by: urs on March 26, 2007 7:38 AM | Permalink | Reply to this
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