We consider the imaginary Gaussian multiplicative chaos, i.e. the complex Wick exponential
\mu_\beta := :e^{i\beta \Gamma(x)}:
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for a log-correlated Gaussian field Gamma in d /geq 1 dimensions. We show that for any nonzero and bounded test function f, the complex-valued random variable
has a smooth density w.r.t. the Lebesgue measure on
. Our main tool is Malliavin calculus, which seems to be well-adapted to the study of (complex) multiplicative chaos. To apply Malliavin calculus to imaginary chaos, we develop some estimates on imaginary chaos that could be of independent interest.
