Covariant effective action and one-loop renormalization of 2D dilaton gravity with fermionic matter

Two dimensional dilaton gravity interacting with a four-fermion model and scalars is investigated, all the coefficients of the Lagrangian being arbitrary functions of the dilaton field. The one-loop covariant effective action for 2D dilaton gravity with Majorana spinors (including the four-fermion interaction) is obtained, and the technical problems which appear in an attempt at generalizing such calculations to the case of the most general four-fermion model described by Dirac fermions are discussed. A solution to these problems is found, based on its reduction to the Majorana spinor case. The general covariant effective action for 2D dilaton gravity with the four-fermion model described by Dirac spinors is given. The one-loop renormalization of dilaton gravity with Majorana spinors is carried out and the specific conditions for multiplicative renormalizability are found. A comparison with the same theory but with a classical gravitational field is done. PACS: 04.50, 03.70, 11.17. On leave of absence from and permanent address: Department E.C.M., Faculty of Physics, Barcelona University, Diagonal 647, 08028 Barcelona, Spain; e-mail: eli @ ebubecm1.bitnet 2 E-mail: odintsov @ theo.phys.sci.hiroshima-u.ac.jp