The identities for elliptic gamma functions discovered by Felder and Varchenko [8] are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in $3$ -dimensional space. The language of stacks and gerbes gives a natural framework for a systematic description of these identities and their domain of validity. A triptic curve is the quotient of the complex plane by a subgroup of rank three. (It is a stack.) Our identities can be summarized by saying that elliptic gamma functions form a meromorphic section of a hermitian holomorphic abelian gerbe over the universal oriented triptic curve