⇤Math diary
May 13, 2024: 32
To continue from entry 31, I’m interested in why the todd class keeps showing up. Heres one good point of view I saw on math overflow. Here is a very rough approximation: A map from the cobordism ring to another ring $R$ is called a genus. Some examples:
- The total chern class is an addictive genus
- The K-theory class is a multiplicative genus Any two genus are related by a third genus. In this case, the genus taking $K$ theory to the total chern class is the todd function. We can explicitly see Since it takes a multiplicative ring to an additive ring, it is a sort of Logarithm. This connects the appearance as the logarithm of the exponential map to the Hirzebruch-Riemann-roch appearance. . There is another proof that goes straight from the derivative of the exponential map to the index formula. This is the proof of the index formula by Berline and Vergne, in A computation of the equivariant index of the Dirac operator. This follows the same start as the heat kernel proof. But, to compute the diagonal of the heat kernel, it doesnt use the clifford algebra symbol calculus like Geltzer’s standard proof. instead, it uses the riemnnian exponential map on the principle $O(n)$ bundle associated to the riemannian structure. Through this, the lie theorey exponential map comes in. This gives rise to the todd function.
I think I can prove the euler mclauren formula using Riemann-Roch on a sphere. Then, the “localization” phemoenona (where the error is entirely determined by the germ at $0,1$) are a manifestation of symplectic localization to fixed points. The integral points being special comes from quantization perspective on Riemann Roch for $\mathbb{P}^1$. Perhaps i can relate the euler mclauren formula of $[0,\infty)$ to the quantum mechanics of the harmonic oscillator on $\mathbb{C}$, using the segal-bergmann transform. Idk yet. It’d be fun to give a talk on this.