Question

The number $12$ (or, probably we shall say Bernoulli numbers in general) appears in many places in Mathematics, sometimes leading to unexpected connections between different topics.

For instance, some time ago there was a very interesting explanation for

1) its occurrence in the Todd class

and

2) its occurrence in the Euler-Maclaurin formula

in terms of Riemann-Roch for toric varieties, as explained in:

http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch

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My question is, will there be some relation between 1) and

3) its occurrence in the Baker-Campbell-Hausdorff formula.

I guess this might be related to some explicit local expressions in some method of proving the index theorem on Lie groups, or even the Duflo map (which I don't really understand).

Thank you very much.

Answer 1

The answer to your question is the following: given two non-commutative variables $x$ and $y$ one has $$ log(e^xe^y)=x+e^{ad_x}\frac{ad_x}{e^{ad_x}-1}(y)+O(y^2) $$

It is not the appearance of $12$ that is intriguing, but the appearance of the Todd series in algebraic geometry. It suggests that there is a group hidden somewhere... and this is indeed the case. This group is the derived loop space of your favorite algebraic variety $X$, and its tangent Lie algebra is the shifted tangent sheaf $T_X[-1]$, with Lie bracket given by the Atiyah class (the fact that the Atiyah class gives rize to a Lie structure was discovered by Kapranov).

The universal enveloping algebra of this Lie algebra is the Hochschild complex of $X$. One then gets a nice dictionnary between the Lie side and the algebraic geometry side. E.g.:

• any object in the derived category of $X$ turns out to be a representation of this Lie algebra.

• Poincare-Birkhoff-Witt is Hochschild-Kostant-Rosenberg.

• the Duflo isomorphism is the Kontsevich-Caldararu isomorphism between the Harmonic and Hochschild structures.

• there is also an relation between closed embeddings in algebraic geometry and inclusions of Lie algebras.

• ...