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 two-form $e^*\omega$ on $LX$. Am I correct?
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$\begingroup$ For what it is worth, five years later: you seem to be trying to arrive at the transgression map that sends differential $p$-forms on $X$ to $(p-1)$-forms on $LX$, so the Kaehler form on $X$ would go to a 1-form on $LX$, rather than a 2-form. You might be interested in the following string theory reference on Kahler structure on $LX$ and papers referencing that: inspirehep.net/literature/235315 $\endgroup$– AlexArvanitakisApr 16, 2020 at 22:34
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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
answered Nov 28, 2015 at 16:52
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$\begingroup$ Could you kindly point out the section I should be looking for, I can't seem to find it. $\endgroup$ Nov 28, 2015 at 17:01
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$\begingroup$ @MeerAshwinkumar An even more explict description is in the second reference... $\endgroup$ Nov 28, 2015 at 18:39
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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 3-dimensional 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. $\endgroup$– MishaNov 28, 2015 at 23:45