Concept:
Ancient Indian scholars and authors contributed significantly to the fields of dance, music, and aesthetics through their classical treatises. Matching authors with their famous works is important in the study of Indian performing arts.
Step 1: Match Dattila with his work.
\[
\text{Dattila}
\]
is associated with:
\[
\textit{Nritta Lakshanam}
\]
Thus:
\[
A \rightarrow III
\]
Step 2: Match Nandikeshwara with his work.
\[
\text{Nandikeshwara}
\]
is the author of:
\[
\textit{Abhinaya Darpanam}
\]
Thus:
\[
B \rightarrow IV
\]
Step 3: Match Sharangadeva with his work.
\[
\text{Sharangadeva}
\]
wrote the famous musical treatise:
\[
\textit{Sangeet Ratnakar}
\]
Thus:
\[
C \rightarrow II
\]
Step 4: Match Jayasenapati with his work.
\[
\text{Jayasenapati}
\]
is associated with:
\[
\textit{Nritta Ratnavali}
\]
Thus:
\[
D \rightarrow I
\]
Step 5: Write the final matching sequence.
The correct matching is:
\[
A\text{-}III,\ B\text{-}IV,\ C\text{-}II,\ D\text{-}I
\]
Step 6: Compare with the given options.
Among the given choices:
\[
(A)\ A\text{-}I,\ B\text{-}II,\ C\text{-}IV,\ D\text{-}III
\]
\[
(B)\ A\text{-}III,\ B\text{-}IV,\ C\text{-}II,\ D\text{-}I
\]
\[
(C)\ A\text{-}II,\ B\text{-}III,\ C\text{-}I,\ D\text{-}IV
\]
\[
(D)\ A\text{-}IV,\ B\text{-}I,\ C\text{-}III,\ D\text{-}II
\]
the correct answer is:
\[
\boxed{A\text{-}III,\ B\text{-}IV,\ C\text{-}II,\ D\text{-}I}
\]
Hence:
\[
\boxed{\mathrm{(B)}}
\]