\\section{Duistermaat-Heckman Formula}
\nIn this section, we consider the case of that $\\left(M^{2l},\\omega\\right)$ is a symplectic manifold. Let $(M,\\omega)$ be a symplectic manifold with $\\omega$ is a symplectic structure. It means
\n\\begin{enumerate}
\n \\item $\\omega$ is a non-singular 2-form. i.e. If for any $Y\\in \\Gamma(TM)$ there always have $\\omega(X,Y)=0$, then $X=0$.
\n \\item $\\rd \\omega=0$.
\n\\end{enumerate}
\nAssume $\\omega$ is $S^1$-invariant and the $S^1$-action on $(M,\\omega)$ is Hamiltonian. It means there exists a smooth function $\\mu\\in C^{\\infty}(M)$ such that $\\rd \\mu=i_K\\omega$. the $\\mu$ is called with momentum map. We still assume that the zero set of $K$ is discrete.
\n\\begin{thm}
\n\\[\\int_{M}\\exp\\left(\\sqrt{-1}\\mu\\right)\\frac{\\omega}{(2\\pi)^ll!}
\n=\\left(\\sqrt{-1}\\right)^l\\sum_{p\\in\\mathrm{zero}(K)}\\frac{\\exp\\left(\\sqrt{-1}\\mu(p)\\right)}{\\lambda(p)}.\\]
\n\\end{thm}
\n\\begin{proof}
\nSince
\n\\begin{align*}
\n\\rd_{K}(\\omega-\\mu)=&(\\rd+i_K)(\\omega-\\mu)\\\\
\n=&\\rd\\omega+i_k\\omega-\\rd\\mu-i_K\\mu\\\\
\n=&\\rd\\omega+i_K\\omega-i_K\\omega=\\rd\\omega=0,
\n\\end{align*}
\none sees that $\\exp\\left(\\sqrt{-1}\\mu-\\sqrt{-1}\\omega\\right)$ is also $\\rd_K$-closed. Using Berline-Vergne localization formula, one has
\n\\[
\n\\int_{M}\\exp\\left(\\sqrt{-1}\\mu-\\sqrt{-1}\\omega\\right)=
\n(2\\pi)^l\\sum_{p\\in\\mathrm{zero}(K)}\\frac{\\exp\\left(\\sqrt{-1}\\mu(p)\\right)}{\\lambda(p)}
\n\\]i.e.,
\n\\begin{align*}
\n\\int_M\\exp\\left(\\sqrt{-1}\\mu\\right)\\left(-\\sqrt{-1}\\right)^l\\frac{\\omega^l}{l!}
\n=&(2\\pi)^l\\sum_{p\\in\\mathrm{zero}(K)}\\frac{\\exp\\left(\\sqrt{-1}\\mu(p)\\right)}{\\lambda(p)}\\\\
\n\\int_M\\exp\\left(\\sqrt{-1}\\mu\\right)\\frac{\\omega^l}{(2\\pi)^ll!}
\n=&\\left(-\\sqrt{-1}\\right)^{-l}\\sum_{p\\in\\mathrm{zero}(K)}\\frac{\\exp\\left(\\sqrt{-1}\\mu(p)\\right)}{\\lambda(p)}\\\\
\n=&\\left(\\sqrt{-1}\\right)^{l}\\sum_{p\\in\\mathrm{zero}(K)}\\frac{\\exp\\left(\\sqrt{-1}\\mu(p)\\right)}{\\lambda(p)}
\n\\end{align*}
\n\\end{proof}
\nAbout the generalization of above theorem in the case where the zero set of $K$ may not be discrete you can see[DH]: “J.J.Duistermaat and G.Heckman, Onthevaroation inthe cohomology of the symplectic from of the reduce phase space”.
\n\\section{Bott’s Original Idea}
\nFrom Bismut’s lemma, If $\\rd_K\\Omega =0$, one has
\n\\[
\n\\int_M\\omega=\\int_M\\omega\\exp\\left(-T\\rd_K\\theta\\right), \\quad for~~any~~T>0.
\n\\]when $\\mathrm{zero(K)}=\\emptyset $,
\n\\[
\n\\int_M\\omega=\\lim_{T\\rightarrow+\\infty}\\int_M\\omega\\exp\\left(-T\\rd_K\\theta\\right)=0.
\n\\]Bott’s original proof is different from above. Now we give a description of Bott’s idea.
\nLet $\\omega$ be a $\\rd_K$-closed form on $M$ with $K$ has no zeros on $M$. Since $\\rd_K(\\theta)=|K|^2+\\rd\\theta=|K|^2\\left(1+\\frac{\\rd\\theta}{|K|^2}\\right)$, by $\\frac{1}{1+x}=\\sum_{i=0}^{\\infty}(-1)^ix^i$, we can define
\n\\[
\n\\left(\\rd_K\\theta\\right)^{-1}=
\n\\frac{1}{|K|^2}\\left[\\sum_{i=0}^{\\infty}(-1)^i\\left(\\frac{\\rd\\theta}{|K|^2}\\right)^i\\right].
\n\\]From $K$ is $S^1$-invariant, the $\\theta$ is also $S^1$-invariant. It shows $\\mathcal{L}_K\\theta=0$, i.e., $\\rd_K^2\\theta =0$. one can verify that
\n\\begin{align*}
\n\\rd_K\\left[\\rd_K^{-1}\\wedge\\theta\\wedge\\omega\\right]=&
\n\\rd_K\\left(\\rd_K\\theta\\right)^{-1}\\wedge\\theta\\wedge\\omega+
\n\\left(\\rd_K\\right)^{-1}\\theta \\wedge\\rd_K\\theta\\wedge\\omega-\\left(\\rd_k\\theta\\right)^{-1} \\wedge\\theta\\wedge\\rd\\omega\\\\
\n=&\\left(\\rd_K\\theta\\right)^{-2}\\wedge \\left(\\rd_K\\theta\\right)^2+\\omega-0\\\\
\n=&\\omega.
\n\\end{align*}
\nFrom this formula and the Stokes formula, one can directly calculate
\n\\[
\n\\int_M\\omega=0.
\n\\] <\/p>\n","protected":false},"excerpt":{"rendered":"
\\section{Duistermaat-Heckman Formula} In …<\/p>\n