Notations:

Let $X$ be a manifold, and denote by $LX := C^\infty(S^1,X)$ its loop space. For a loop $\gamma \in LX$ we can think at the tangent space of $LX$ at the point $\gamma$ as the space of loops that are 'arbitrarily closed' to $\gamma$. This motivates the following definition

$$T_{\gamma}LX := \Gamma(S^1, \gamma^*TX),$$

where $\Gamma(S^1, \gamma^*TX)$ denote the space of section of the pullback bundle $\require{AMScd}$ \begin{CD} \gamma^*TX @>>> TX\\ @V V V @VV V\\ S^1 @>>\gamma> X. \end{CD} This is the description of the tangent bundle $TLX \to LX$ of the loop space of $X$. By definition a $k$-differential form on $LX$ is a section of the $k^{th}$ exterior power of $TLX$$TLX^*$, the cotangent bundle. $$\Omega^k(LX):= \Gamma(LX, \Lambda^kTLX).$$$$\Omega^k(LX):= \Gamma(LX, \Lambda^kTLX^*).$$

Question:

How is the exterior differential $d$ defined for differential forms on loop space? \begin{CD} \Omega^k(LX) @>d>> \Omega^{k+1}(LX) \end{CD}