  In this article, we study complex Gaussian multiplicative chaos. More
precisely, we study the renormalization theory and the limit of the exponential
of a complex log-correlated Gaussian field in all dimensions (including
Gaussian Free Fields in dimension 2). Our main working assumption is that the
real part and the imaginary part are independent. We also discuss applications
in 2D string theory; in particular we give a rigorous mathematical definition
of the so-called Tachyon fields, the conformally invariant operators in
critical Liouville Quantum Gravity with a c=1 central charge, and derive the
original KPZ formula for these fields.
