Consider a compact Riemannian manifold of even dimension $n$ admitting a $U(1)$ action. If the fixed points of the action are isolated, then Witten [1; eq. 35] gives the character-valued index of the Dirac operator:


$I(\theta)= \left(\frac{i}{2}\right)^{n/2}\sum\_\limits{i}{\rm Tr} \thinspace e^{i\theta Q^{(i)}}\prod\_\limits{a}\frac{1}{\sin(r\_a^{(i)}\theta/2)}$.


The sum is over fixed points of the $U(1)$ action. The $Q^{(i)}$ are eigenvalues of the $U(1)$ generator at the fixed points and the $r\_a^{(i)}$ are rotation angles characterizing the $U(1)$ generator when it is expressed as a vector field in local coordinates near the fixed points.


Witten remarks: "We shall not write down here the more general formula that holds if the zeros of $K$ [the $U(1)$ generator] are not isolated. This formula involves weighting the factors of $1/\sin(r\theta/2)$ by the number of zero modes of a certain Dirac operator on the fixed point set."



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> My question is: what is the more general formula and/or where can I find it?
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[1] Witten, Edward. "Fermion quantum numbers in Kaluza-Klein theory." Supergravities in Diverse Dimensions: Commentary and Reprints (In 2 Volumes). 1989. 1412-1462.



Index formula with nonisolated fixed points
Atiyah-Segal-Singer Theorem, Theorem 6.16 in 
*Berline, Nicole; Getzler, Ezra; Vergne, Michèle*, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften. 298. Berlin etc.: Springer-Verlag. vii, 369 p. (1992). [ZBL0744.58001](https://zbmath.org/?q=an:0744.58001).


