Effective Action for Membrane Dynamics in DLCQ $M$ theory on a Two-torus

TL;DR

This paper computes the membrane effective action in DLCQ M-theory on a two-torus using supergravity and compares it with matrix theory results, confirming their agreement and illustrating the matrix/M-theory correspondence.

Contribution

It provides a detailed comparison between supergravity and matrix theory effective actions for membranes in DLCQ M-theory, including non-perturbative effects, confirming their agreement.

Findings

Exact agreement between supergravity and matrix theory for finite N

Agreement extends to large N in the conformal regime

Non-perturbative instanton effects restore eleven-dimensional covariance

Abstract

The effective action for the membrane dynamics on the background geometry of the $N$-sector 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 non-perturbative instanton effects. Consistent with the DLCQ prescription of $M$ theory {\em a la} 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. Non-perturbative instanton effects in the SYM theory conspire to yield the eleven-dimensionally covariant effective action.