  This paper was inspired by four articles: surface cluster algebras studied by
Fomin-Shapiro-Thurston \cite{fst}, the mutation theory of quivers with
potentials initiated by Derksen-Weyman-Zelevinsky \cite{dwz}, string modules
associated to arcs on unpunctured surfaces by
Assem-Br$\ddot{u}$stle-Charbonneau-Plamondon \cite{acbp} and Quivers with
potentials associated to triangulated surfaces, part II: Arc representations by
Labardini-Fragoso. \cite{lf2}. For a surface with marked points ($\Sigma,M$)
Labardini-Fragoso associated a quiver with potential $(Q(\tau),S(\tau))$ then
for an ideal triangulation of ($\Sigma,M$) and an ideal arc Labardini-Fragoso
defined an arc representation of $(Q(\tau),S(\tau))$. This paper focuses on
extent the definition of arc representation to a more general context by
considering a tagged triangulation and a tagged arc. We associate in an
explicit way a representation of the quiver with potential constructed
Labardini-Fragoso and prove that the Jacobian relations are met.
