Effective action for membrane dynamics in DLCQ M theory on a two-torus

The effective action for the membrane dynamics on the background geometry of the N-sector discrete light-cone quantization (DLCQ) M theory compactified on a two-torus is computed via supergravity. We compare it to the effective action obtained from the matrix theory, i.e., the (2 + 1)-dimensional supersymmetric Yang-Mills (SYM) theory, including the one-loop perturbative and full nonperturbative instanton effects. Consistent with the DLCQ prescription of M theory in the manner of Susskind, we find the precise agreement for the finite N sector (off-conformal regime), as well as for the large N limit (conformal regime), providing us with a concrete example of the correspondence between the matrix theory and the DLCQ hi theory. Nonperturbative instanton effects in the SYM theory conspire to yield the eleven-dimensionally covariant effective action. [S0556-2821(98)50224-X].