14. Let the angle between vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ be $\frac{\pi}{3}$, the angle between vectors $\boldsymbol{c}-\boldsymbol{a}$ and $\boldsymbol{c}-\boldsymbol{b}$ be $\frac{2 \pi}{3}$, $|\boldsymbol{a}-\boldsymbol{b}|=5$, and $|\boldsymbol{c}-\boldsymbol{a}|=2 \sqrt{3}$. Then the maximum value of $\boldsymbol{a} \cdot \boldsymbol{c}$ is
14. 24 .

Let $\overrightarrow{O A}=a, \overrightarrow{O B}=b, \overrightarrow{O C}=c$. Then
$$
|\overrightarrow{A C}|=|c-a|=2 \sqrt{3},|\overrightarrow{A B}|=|a-b|=5 \text {. }
$$

Also, $\angle A O B=\frac{\pi}{3}, \angle A C B=\frac{2 \pi}{3}$, at this time, $O, A, C, B$ are concyclic.
By the Law of Sines, we get
$$
\sin \angle A B C=\frac{3}{5} \Rightarrow \cos \angle A B C=\frac{4}{5} \text {. }
$$

In $\triangle A C O$, since $\angle A O C=\angle A B C$, by the Law of Cosines, we have
$$
\begin{array}{l}
A C^{2}=|\boldsymbol{a}|^{2}+|\boldsymbol{c}|^{2}-2|\boldsymbol{a}||\boldsymbol{c}| \cos \angle A O C \\
\Rightarrow 12 \geqslant 2|\boldsymbol{a}||\boldsymbol{c}|-\frac{8}{5}|\boldsymbol{a}||\boldsymbol{c}| \\
\Rightarrow|\boldsymbol{a}||\boldsymbol{c}| \leqslant 30 \\
\Rightarrow \boldsymbol{a} \cdot \boldsymbol{c}=|\boldsymbol{a}||\boldsymbol{c}| \cos \angle A O C \leqslant 24 . \\
\text { When } \angle A C O=\frac{\pi}{4}+\frac{1}{2} \arctan \frac{4}{3} \text {, the above inequalities hold with equality. }
\end{array}
$$

When $\angle A C O=\frac{\pi}{4}+\frac{1}{2} \arctan \frac{4}{3}$, the above inequalities hold with equality.
Therefore, the maximum value of $a \cdot c$ is 24.