# Todd class and Baker-Campbell-Hausdorff, or the curious number $12$

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:

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.

• Isn't this a numerology question?! At least you don't ask why 12 is close to an integer... :-) – Wadim Zudilin Jul 15 '10 at 7:27
• Hi Wadim, I edited the question make it somehow more "feasible" :-) – Bo Peng Jul 15 '10 at 7:42
• A baker's dozen is 13. I don't know about Campbell and Hausdorff. – Tom Goodwillie Jul 15 '10 at 11:11
• Possibly related mathoverflow.net/questions/9220/… – SandeepJ Jul 15 '10 at 14:11
• Have you looked at the Wikipedia article, en.wikipedia.org/wiki/Baker-Campbell-Hausdorff_formula? Please, clarify in what way the explanation there involving the expansion of $\operatorname{ad}(X)/(e^{\operatorname{ad}(X)}-1)$ found there is lacking? – Victor Protsak Jul 15 '10 at 17:56

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.

• ...

• In item 1 "derived category of X " do you mean current sheaves on X? PS very useful answer! – Alexander Chervov Dec 5 '12 at 19:14
• Coherent sheaves – Alexander Chervov Dec 5 '12 at 19:43
• I actualy mean quasi-coherent sheaves (the action of $T_X[-1]$ on a given one $E$ is given by the Atiyah class of $E$). – DamienC Dec 5 '12 at 20:07