The Duistermaat Heckman localization formula states how integrals over symplectic spaces with Hamiltonian $U(1)$ group actions.

$$ \int_M \frac{\omega^n}{n!} e^{-\mu} = \sum_{x_i \text{ fixed}} \frac{e^{-\mu(x_i)}}{e(x_i)} $$

Here $M$ is a symplectic manifold and here are a few invariants:

  • $\omega$ is the symplectic form
  • $\mu$ is the moment map of the $U(1)$ rotation action
  • $x_i$ are the fixed points of the group action
  • $e(x_i)$ is the product of the weights of the rotation at fixed point $x_i$

In a few cases, we can produce formulas an engineer might recognize:

  • $M = \mathbb{R}^2 \simeq \mathbb{C}$ and $\omega = dx \wedge dy$
  • $U(1)$ the the rotation $z = x_1 + i x_2 \mapsto e^{i\theta}z$, the moment map is $\mu(z) = |z|^2 = x_1^2 + x_2^2$
  • Certainly $\mathbb{R}^{2n}$ is analogous.

The fixed point of the rotation is $x_i = 0$. This leads to the formula for the Gaussian integral:

$$ \int (dx \wedge dy) \,e^{-(x^2 + y^2)} = \frac{e^{-\mu(0)}}{e(0)} = \frac{1}{2\pi}$$

I'm not actually sure that $e(0) = 2\pi$ but I'm guessing. There's not too many cases were all the ingredients are known. It is possible to write a manifold (or orbifold or variety) with moment map $\mu$.

My understanding is that we can rarely write $\omega$ down explicitly. For the sphere $S^2 = \{ x^2 + y^2 + z^2 = 1\}$ there's a symplectic form $\omega = \frac{dx \wedge dy}{z}$ which is invariant under rotating aroun the $z$ axis.

Are there other cases where the DH formula specializes in a particularly nice way? I'm guessing that typically we bypass explicitly computing $\omega$.

  • $\begingroup$ Flipping through this article of Álvaro Pelayo, but still inconclusive. Also ICM 2006 article of Michéle Vergne. $\endgroup$ – john mangual Aug 7 '18 at 18:44

Nice examples are worked out in Audin (2004, §VI.3.d), Arvanitoyeorgos (1999)(pdf), McDuff-Salamon (1998, §5.6). It’s not true that $\omega$ is rarely explicit: e.g. on all coadjoint orbits (including $\smash{S^2}$) it is, and DH gives a formula of Harish-Chandra for their Fourier transforms; see Berline-Vergne (1983), Vergne (1983), or Guillemin-Sternberg (1984, ends of §§33 and 34).

  • $\begingroup$ i believe you. i have to read these and verify what you are saying $\endgroup$ – john mangual Aug 8 '18 at 20:57

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