The String Coffee Table

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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 $L_{\mathrm{\infty }}$ 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 $L_{\mathrm{\infty }}$ 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 $k=0,n=1$ $L_{\mathrm{\infty }}$ 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 $[\cdot ,\cdot ]$ on its space $\Gamma (E)$ of sections and an anchor map $\rho :E\to \mathrm{TM}$ to the tangent bundle. In addition, however, there is now a bilinear form $$\langle \cdot ,\cdot \rangle :\Gamma (E)\times \Gamma (E)\to C^{\mathrm{\infty }}(M)$$ $$(e_1,e_2)\mapsto {\beta }_{\mathrm{ij}}{e}^i{e}^j$$ These have to satisfy the conditions of a Lie algebroid in a somwhat modified form. Consider the functions $$𝒟:C^{\mathrm{\infty }}(M)\to \Gamma (E)$$ $$f\mapsto \frac{1}{2}{\beta }^{-1}\cdot {\rho }^{*}\mathrm{df}$$ and $$T:\Gamma (E)\times \Gamma (E)\times \Gamma (E)\to C^{inft}(M)$$ $$(e_1,e_2,e_3)\mapsto \langle [e_1,e_2],e_3\rangle +\mathrm{cyclic}$$ Using these the properties of a Courant algebroid are now the following:

1. The Jacobiator is given by $𝒟\circ T$: $$[[e_1,e_2],e_3]+[[e_2,e_3],e_1]+[[e_3,e_1],e_2]=𝒟T(e_1,e_2,e_3)$$

2. The anchor is (still) a homomorphism $$\rho ([e_1,e_2])=[\rho (e_1),\rho (e_2)]$$

3. The Leibnitz rule acquires an extra term: $$[e_1,f{e}_2]=f[e_1,e_2]+\rho (e_1)(f)e_2-\langle e_1,e_2\rangle 𝒟f$$

4. and then there is this funny condition $$\langle 𝒟g,𝒟g\rangle =0$$

5. and this ever funnier one $$\rho (e)(\langle h_1,h_2\rangle )=\langle [e_1,h_1]+𝒟\langle e,h_1\rangle ,h_2\rangle +\langle h_1,[e,h_2]+𝒟\langle e,h_2\rangle \rangle$$ which is kind of a product rule.

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

$$0\to \ker (𝒟)\stackrel{d_2=\iota }{\to }{C}^{\mathrm{\infty }}(M)\stackrel{d_1=𝒟}{\to }\Gamma (E)\to \mathrm{coker}(𝒟)$$ and let the $L_{\mathrm{\infty }}$/algebra operator $l_2$ be the bracket on $\Gamma (E)$ and be the action of $\rho (e)$ on $f\in C^{\mathrm{\infty }}(M)$ on $\Gamma (E)\times C^{\mathrm{\infty }}(M)$ and let $T$ still be the Jacobiator.

I am running out of time once again…