## April 29, 2005

### PSM and Algebroids, Part II

#### Posted by Urs Schreiber

Everybody is getting excited about Hitchin et al’s ‘generalized geometry’ (see also Luboš’ blog entry). I should hence hurry up with my PSM and Algebroids program, since that’s closely related.

[Administrative Comment: Those of you not blessed with a Mac will have problems with the display of some of the math characters below. Try this workaround and see if it helps.]

I am currently visiting LMU Munich, where I gave a seminar talk on - guess what, right - on ‘Nonabelian Strings’. After the talk I was asked if I see a relation of this catgeorified stuff to algebroids.

Last time I had mentioned that algebroids should be another special case of what are called $k$-tuply stabilized Lie $n$-algebras, which again are nothing but ${L}_{\infty }$ algebras which come from a graded vector space which is nontrivial at most in degrees $k-1,k,k+1,\dots ,n$ (if I counted correctly).

Lie 2-algebras are just another special case. Furthermore, bundle gerbes have a lot to do with groupoids and hence it would not be surprising if it turned out that a connection on a bundle gerbe could be regarded as an algebroid-valued entity, somehow, maybe, sort of.

So how precisely do algebroids appear as ${L}_{\infty }$ algebras, anyway? Bransilav Jurčo pointed me to the paper

D. Roytenberg & A. Weinstein
Courant Algebroids and Strongly Homotopy Lie Algebras
math.QA/9802118

Therein we have a theorem 4.3 which says that every Courant algebroid gives rise to a $k=0,n=1$ ${L}_{\infty }$ algebra (in the terminology of HDA6) and, furthermore, a theorem 3.3. which says that every Lie bialgebroid gives rise to a Courant algebroid.

I’ll have to look up some simple but annoying detail concerning the definition of Lie bialgebroids, so this is postponed to part III. But here is the definition of what a Courant algebroid is:

A Courant algebroid is much like a Lie algebroid, but with a little more structure.

Definition.

A Courant algebroid is a vector bundle $E\to M$ which has, like a Lie algebroid, a skew-symmetric bracket $\left[\cdot ,\cdot \right]$ on its space $\Gamma \left(E\right)$ of sections and an anchor map $\rho :E\to \mathrm{TM}$ to the tangent bundle. In addition, however, there is now a bilinear form $〈\cdot ,\cdot 〉:\Gamma \left(E\right)×\Gamma \left(E\right)\to {C}^{\infty }\left(M\right)$ $\left({e}_{1},{e}_{2}\right)↦{\beta }_{\mathrm{ij}}{e}^{i}{e}^{j}$ These have to satisfy the conditions of a Lie algebroid in a somwhat modified form. Consider the functions $𝒟:{C}^{\infty }\left(M\right)\to \Gamma \left(E\right)$ $f↦\frac{1}{2}{\beta }^{-1}\cdot {\rho }^{*}\mathrm{df}$ and $T:\Gamma \left(E\right)×\Gamma \left(E\right)×\Gamma \left(E\right)\to {C}^{inft}\left(M\right)$ $\left({e}_{1},{e}_{2},{e}_{3}\right)↦〈\left[{e}_{1},{e}_{2}\right],{e}_{3}〉+\mathrm{cyclic}$ Using these the properties of a Courant algebroid are now the following:

1. The Jacobiator is given by $𝒟\circ T$: $\left[\left[{e}_{1},{e}_{2}\right],{e}_{3}\right]+\left[\left[{e}_{2},{e}_{3}\right],{e}_{1}\right]+\left[\left[{e}_{3},{e}_{1}\right],{e}_{2}\right]=𝒟T\left({e}_{1},{e}_{2},{e}_{3}\right)$

2. The anchor is (still) a homomorphism $\rho \left(\left[{e}_{1},{e}_{2}\right]\right)=\left[\rho \left({e}_{1}\right),\rho \left({e}_{2}\right)\right]$

3. The Leibnitz rule acquires an extra term: $\left[{e}_{1},f{e}_{2}\right]=f\left[{e}_{1},{e}_{2}\right]+\rho \left({e}_{1}\right)\left(f\right){e}_{2}-〈{e}_{1},{e}_{2}〉𝒟f$

4. and then there is this funny condition $〈𝒟g,𝒟g〉=0$

5. and this ever funnier one $\rho \left(e\right)\left(〈{h}_{1},{h}_{2}〉\right)=〈\left[{e}_{1},{h}_{1}\right]+𝒟〈e,{h}_{1}〉,{h}_{2}〉+〈{h}_{1},\left[e,{h}_{2}\right]+𝒟〈e,{h}_{2}〉〉$ which is kind of a product rule.

That’s the definition. The idea how to turn this into an ${L}_{\infty }$-algebra is to start with the complex of vector spaces

$0\to \mathrm{ker}\left(𝒟\right)\stackrel{{d}_{2}=\iota }{\to }{C}^{\infty }\left(M\right)\stackrel{{d}_{1}=𝒟}{\to }\Gamma \left(E\right)\to \mathrm{coker}\left(𝒟\right)$ and let the ${L}_{\infty }$/algebra operator ${l}_{2}$ be the bracket on $\Gamma \left(E\right)$ and be the action of $\rho \left(e\right)$ on $f\in {C}^{\infty }\left(M\right)$ on $\Gamma \left(E\right)×{C}^{\infty }\left(M\right)$ and let $T$ still be the Jacobiator.

I am running out of time once again…

Posted at April 29, 2005 6:30 PM UTC

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## 5 Comments & 3 Trackbacks

### Re: PSM and Algebroids, Part II

Hi Urs,

I was just glancing at this generalized geometry, plus the discussion on Lubos’ blog and get a strong sense of deja vu. Haven’t we been talking about these ideas for the past couple of years? In fact, aren’t some of these concepts in our paper (at least in the discrete world)?

What is so special about it that has apparently gotten some people excited?

Just curious.

Thanks,
Eric

Posted by: Eric on May 2, 2005 9:16 PM | Permalink | Reply to this

### Re: PSM and Algebroids, Part II

Hi Eric,

it is true that the prominent role played by the bundle $\mathrm{TM}\oplus {T}^{*}M$ in generalized geometry is related to the fact that there are left- and rightmoving supercharges on the string’s worldsheet such that there is a ${\mathrm{Cl}}_{n,n}$ Clifford algebra which is equivalent to the algebra of $n$ form creation and $n$ form annihilation operators.

I was always fond of the fact that the linear combinations $G±i\overline{G}$ of the supercharges are hence nothing but deformed deRham operators on loop space acting on differential forms on loop space.

This is based on precisely the same simple idea that is for instance expressed in section 2.3 of Marco Gualtieri’s thesis as the fact that ‘differential forms are the spinors of $\mathrm{Cl}\left(\mathrm{TM}\oplus {T}^{*}M\right)$’.

And, true, we have played around with precisely this fact a lot over the years. For instance I note that what Marco Gualtieri calls the ‘$B$-transform’ in example 2.10 on p. 2.9 is precisely the operation which I discussed at the above link that when conjugating the supercharges with it gives rise to those supercharges describing that $B$-field background geometry.

Also, at the above link and in hep-th/0401175 I was discussing what Gualtieri calls the ‘$\beta$-transform’ (example 2.11). At that time I was puzzled by the physical interpretation of that transform and was discussing how it seems to have something to do with reversing the sign of the dilaton and hence maybe with S-duality. As far as I am aware now people interested in generalized geometry are speculating that it may have something to do with making the target space non-commutative (regarding the 2-vector $\beta$ in the context of deformation quantization or something).

(Does anyone know in which papers this ‘$\beta$-transform’ appeared first?)

But of course there is more in ‘generalized geometry’ than we had discussed. :-)

Posted by: Urs Schreiber on May 4, 2005 4:39 PM | Permalink | Reply to this

### Re: PSM and Algebroids, Part II

But of course there is more in ‘generalized geometry’ than we had discussed. :-)

I thought there might be :D

Posted by: Eric on May 4, 2005 6:54 PM | Permalink | Reply to this

### Re: PSM and Algebroids, Part II

But of course there is more in ‘generalized geometry’ than we had discussed. :-)

I thought there might be :D

On the opther hand, the same could apply the other way round, couldn’t it?

It seems to me that precisely the field which I called $C$ here and in hep-th/0401175 and whose physical interpretation was not fully clear, is what Kapustin is trying to related to nonvommutative branes in

A. Kapustin: A-branes and Noncommutative Geometry hep-th/0502212

based on

A. Kapustin: Topological Strings on Noncommutative Manifolds hep-th/0502212

In my formalism it seemed there were indications that the $C$ field comes from the $B$ field under a transformation which also inverts the couplingt constant.

I wonder how this relates to what Kapustin does.

Posted by: Urs Schreiber on May 4, 2005 8:32 PM | Permalink | Reply to this

### Re: PSM and Algebroids, Part II

About the beta transform, I’m not sure where it appeared first, but you can also see some instances of it in

math.DG/0309013

and

math/0405303

–Oren

Posted by: Oren Ben-Bassat on May 22, 2006 3:14 AM | Permalink | Reply to this
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