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 $\mathrm{}(2+1)\mathrm{}$-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 M theory. Nonperturbative instanton effects in the SYM theory conspire to yield the eleven-dimensionally covariant effective action.