⇤Math diary
May 12, 2024: Euler-Maclauren formula and Riemann Roch
I found this paper RIEMANN SUMS OVER POLYTOPES by Guilleman and sternberg. It deals with a generalization of the Euler-Mclaurauen formula. For an elementary summary of the Euler-Mclauren formula, see Some Riemann Sums Are Better Than Others by Victor W. Guillemin & Daniel W. Stroock. The euler-mclauren formula computes the error of a Riemann sum:
\[\int_a^b f\mathrm{d} x - \sum_{i=0}^{N} \frac{1}{N} f\left(\frac{i}{N}\right) =- \sum_{1\leq k< l} (-1)^{k-1} \frac{b_{2k}}{N^{2k}} (f^{(2k-1)}(1)-f^{(2k-1)}(0)) - \frac{b_{2l}}{N^{2l}} f^{(2l)}(\xi)\]where $b_{2k}$ is the bernoulii numbers, and $\xi$ is some point in the range of $[0,1]$. Essentially, the error of the riemann sum for equally spaced points is computed to arbitrary order by the taylor series of $f$ at the endpoints. The extra remainder term is to be expected from the usual taylors theorem. This is not true for non-equally spaced points.
One implicaiton f this is, for a periodic function, the Riemann sum converges exponentially fast. A periodic function has all derivatives at 0 and 1 the same, so the main error term will vanish. The convergence will be $O(N^{-\infty})$, faster than any polynomial in $1/N$. We can prove this directly using fourier series. We can explicitly compute the Riemann sum of $e^{inx}$, compare that to the fourier series of $f$, and use the schwartz inequality.
The bernoulii numbers are (roughly) defined by the coefficients of the talyor series of the todd function
\[\tau = \frac{x}{1-e^{-x}}\]So another way of writing the euler-Mackleauren formula is to say the error tem is equal to
\[\tau \left(\frac{1}{N} \frac{\mathrm{d}}{\mathrm{d}x}\right) f\]The todd function appears in many places. Here is a list off the top of my head:
- The derivative of the exponential map in Lie theory satisfies
- We can rewrite the Weyl character formula to feature the todd function
where $W$ is the weyl group and, $\rho = \frac{1}{2}\sum_{\alpha \in R^+} \alpha$ is the half sum of positive roots. The demoninator looks like the formula for the Todd class. Rewriting,
\[\chi(e^{it}) = \prod\limits_{\alpha \in R^+} \frac{\tau(i\langle\alpha,t \rangle)}{i\langle\alpha,t \rangle} \sum\limits_{w \in W}e^{i \langle w.(\lambda+\rho) - \rho,t \rangle }\]Though I’m not sure how helpful this is… If you prove weyl charecter formula using weyls integral formula, the Weyl denominator shows up as the jacobian of the map from the group to its maximal torus. This, in turn, comes from the derivative of the exponential map above.
- The Hirzebrush-Riemann-Roch formula states that the holomorphic euler charecteristic of a holomrophic vector bundle $H$ is
where $\text{ch}(E)$ is the chern charecter of $E$, and $\text{td}$ is the todd class of $TM$. To define the todd class, we apply the splitting princple. Construct a manifold $F$ with a projection $\pi:F \to M$ such that $\pi^* TM$ splits into a sum of line bundles $L_i$. Then, the todd class is defined as $\prod \tau(c_1(L_i))$, for $\tau$ the todd function. The todd class appears in a similar way in the full atiyah-singer index formula.
- The todd function appears in the propegator for the 1D harmonic oscillator:
This is why the todd function appears in the atyiah-singer index formula. The heat kernel proof of the index formula reduces the index to a computation of the supertrace, which is the integral of the propegator over the diagonal. When evaluated at long times, this computes the analytic index. When evaluated at short times, this computes a number of integrals of curvature, which form charecteristic classes. To compute the short time propagator for an elliptic operator, we reduce it to a locally quadradic potential on a laplacian, which is exactly the harmonic oscillator. So, the harmonic oscillator’s propagator along the diagonal gives us the needed charecteristic classes.
- The todd function shows up as the euler class of the normal bundle of $M$ inside of $LM$. This arises in the loop group proof for the index theory of dirac operators. In this setting, the index theorem becomes the localization of the path integral on $LM$ to the constant paths $M\subset LM$, which can be computed using duestermaat heckman for the $S^1$ action. see entries 12 of my diary.
For more sources and information, see this stackoverflow post on the relationship between the euler-mclauren and Riemann-Roch. This is a wide area of research, coming from connections to toric varieties.
This makes me wonder if there is a direct proof of the Euler-Maclauren formula using Riemann roch on the sphere.