Classification of two-dimensional conformal supergravity theories with finite-dimensional algebras

We present a list of all the finite supersymmetric extensions of the two-dimensional (2D) conformal algebra SO(2,2), which could lead to 2D superconformal gravity theories. The on-shell matter multiplets of the algebras allowing canonical spin-0 and -1/2 matter are constructed. Using these multiplets the Weyl and Yang-Mills anomalies are calculated. There are many new models. One new model is free of both Yang-Mills and Weyl anomalies, SU(1,1) x SU(2chemical bond1,1), but only the algebras SU(1,1)2, OSP(1chemical bond2)2, OSP(2chemical bond2)2, SU(1,1) x OSP(1chemical bond2), OSP(1chemical bond2) x OSP(2chemical bond2), and SU(1,1) x OSP(2chemical bond2) lead to models free of all anomalies. These models correspond to the known string models