The ancient dance teachers or gurus of Bharatanatyam are popularly known as:

The correct answer is Nattuvanars. A nattuvanar is a dance master or guru of Bharatanatyam. The nattuvanar trains the dancer, composes or arranges the choreography, and conducts the performance on stage by reciting the rhythmic syllables (sollukattu) and playing the cymbals called talam. In the traditional system, the nattuvanar was a central figure, both as teacher and as conductor.

In the past, the nattuvanars came mainly from hereditary families of musicians and dance teachers, who passed down their knowledge from generation to generation. They were closely connected with the devadasis, the women temple dancers who performed Sadir, the dance from which modern Bharatanatyam developed. The most famous nattuvanars in the history of Bharatanatyam are the Tanjore Quartet, four brothers named Chinnaiah, Ponnaiah, Sivanandam and Vadivelu, who lived in the early 19th century at the court of Thanjavur. They organised the Bharatanatyam recital (margam) into the order still followed today, with items such as Alarippu, Jatiswaram, Shabdam, Varnam, Padam and Tillana. In the 20th century, nattuvanars such as Meenakshisundaram Pillai of Pandanallur taught leading dancers such as Rukmini Devi Arundale. Today, a nattuvanar is still part of every traditional Bharatanatyam orchestra.

About the other options:

• Chakyars are the traditional performers of Chakyar Koothu and Koodiyattam, the ancient Sanskrit theatre of Kerala.
• Maharis were the women temple dancers of the Jagannath temple at Puri in Odisha, whose dance contributed to Odissi.
• Bhagavatars are performers and narrators in traditions such as Kuchipudi, Yakshagana (where the narrator is called the Bhagavata) and the Bhagavata Mela of Tamil Nadu.