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In the famous 1984 paper "The Moment Map and Equivariant Cohomology" by Atiyah and Bott, an equivariant de Rham theory was presented in relation to the Duistermaat–Heckman formula $$ \int_M e^{-itf} \frac{\omega^n}{n!} = \sum_p \frac{e^{-itf(p)}}{(it)^n \text{e}(p)},$$ where $(M,\omega)$ is a symplectic manifold and $f: M \to \mathbb{R}$ is the moment map on $M$ coming from a circle action.

This sparked my curiosity in the physical applications of equivariant cohomology (i.e., equivariant de Rham theory), especially the above formula.

What are real world applications of the Duistermaat–Heckman formula?

By real world applications, I mean physical applications in classical mechanics, statistical mechanics, quantum mechanics, or quantum field theory, but probably not string theory. I would appreciate if someone could provide real world applications of the Duistermaat–Heckman formula.

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  • $\begingroup$ Why don't you upvote the accepted answer? $\endgroup$ – Arnold Neumaier Dec 9 '19 at 9:41
  • $\begingroup$ @ArnoldNeumaier Your answer is definitely helpful and that's why I accepted it. I thought "upvote" is more for the others. $\endgroup$ – Yuhang Chen Dec 9 '19 at 10:55
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Here are some applications that might qualify for your criterion of 'real world':

Yasui, Y., & Ogura, W. (1996). Vortex filament in a three-manifold and the Duistermaat-Heckman formula. Physics Letters A, 210(4-5), 258-266.

Karki, T., & Niemi, A. J. (1994). On the Duistermaat-Heckman Integration Formula and Integrable Models. arXiv preprint hep-th/9402041.

Bismut, J. M. (2011). Duistermaat–Heckman formulas and index theory. In Geometric Aspects of Analysis and Mechanics (pp. 1-55). Birkhäuser Boston.

Zhang, L., Jiang, Y., & Wu, J. (2019). Duistermaat–Heckman measure and the mixture of quantum states. Journal of Physics A: Mathematical and Theoretical, 52(49), 495203.

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