I want to compute the following integral: $$ I = \int_0^{2\pi} d\theta_1\dots \int_0^{2\pi} d\theta_n \exp\left(-\sum_{ij} M_{ij} e^{i(\theta_j-\theta_i)}\right) $$ where $M$ is a Hermitian matrix. Writing $\psi = (\psi_1,\dots,\psi_n)$, with $\psi_i = e^{i\theta_i}$, we see that this is a type of Gaussian integral, where the integration is over variables $\psi_i$ on the unit circle instead of over the real line: $$ I = \oint_{S^1} d\psi_1\dots \oint_{S^1} d\psi_n \, e^{- \psi^\dagger M \psi}. $$ We can not resort to contour deformation arguments, because the integrand is not meromorphic. I have tried to map the circle to the real line using a Möbius transformation, but that renders the integral hard to compute. Is there a way to evaluate this integral, or relate it to the standard complex multi-dimensional Gaussian?
1 Answer
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Not sure this will work, but consider integrating each $\psi=x+iy$ over the real and imaginary coordinates $x,y$, while keeping the circle constraints via Dirac Deltas $\delta(\psi^* \psi-1)$, then use the Fourier representation of the Delta to convert it into another Gaussian. That is: $$\int \prod_kdp_k\int \prod_k dx_k dy_ke^{i\sum_kp_k(x_k^2+y_k^2-1)}e^{-\psi^* M \psi}.$$ This integral looks tractable, or at least the Gaussian part is.
There may be a Jacobian involved due to the Delta being over $(|\psi|^2-1)$, and may need some careful analysis.
