Does Kähler structure on $X$ imply Kähler structure on the loop space ($LX$) of $X$? Since the loop space of $X$ is the space of maps from the circle $S^1$ to $X$, I suspect one may use the pullback via the evaluation map $e:LX\rightarrow X$ of the closed Kähler form $\omega$ on $X$ to obtain a closed twoform $e^*\omega$ on $LX$. Am I correct?
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See Part I of Sergeev's book... (he talks more about Lie groups)  see also Henrich Falk's diplomarbeit (2009, Berlin) page 26

$\begingroup$ Could you kindly point out the section I should be looking for, I can't seem to find it. $\endgroup$ – Meer Ashwinkumar Nov 28 '15 at 17:01

$\begingroup$ @MeerAshwinkumar An even more explict description is in the second reference... $\endgroup$ – Igor Rivin Nov 28 '15 at 18:39

2$\begingroup$ Igor, your references do not really resolve the question which OP is asking. The issue of integrability of the almost complex structure is a delicate one: For the loop space of a 3dimensional Riemannian manifold (the case I thought about) the answer is that the almost complex structure is formally integrable but, in general, not integrable, see Lempert's paper. What happens when the target is Kahler I am not sure. $\endgroup$ – Misha Nov 28 '15 at 23:45