Spherical membranes in Matrix theory

TL;DR

The paper investigates spherical membranes in uncompactified Matrix theory and demonstrates that, at large $N$, the one-loop Matrix potential between objects reproduces the long-range supergravity potential after removing gravitational radiation, supporting the BFSS conjecture. It provides a general formula for the one-loop potential and proves two theorems: (i) any finite-size classical membrane has the correct leading interaction with a graviton at large $N$, and (ii) two uncorrelated objects with that property interact correctly with each other. At finite $N$, the leading large-$N$ agreement holds for sphere–graviton configurations, but subleading $1/N$ corrections introduce discrepancies with naive DLCQ supergravity. The work also outlines a framework to study Schwarzschild black holes in Matrix theory via collapsing spherical membranes and discusses implications for DLCQ M-theory and black hole physics.

Abstract

We consider membranes of spherical topology in uncompactified Matrix theory. In general for large membranes Matrix theory reproduces the classical membrane dynamics up to 1/N corrections; for certain simple membrane configurations, the equations of motion agree exactly at finite N. We derive a general formula for the one-loop Matrix potential between two finite-sized objects at large separations. Applied to a graviton interacting with a round spherical membrane, we show that the Matrix potential agrees with the naive supergravity potential for large N, but differs at subleading orders in N. The result is quite general: we prove a pair of theorems showing that for large N, after removing the effects of gravitational radiation, the one-loop potential between classical Matrix configurations agrees with the long-distance potential expected from supergravity. As a spherical membrane shrinks, it eventually becomes a black hole. This provides a natural framework to study Schwarzschild black holes in Matrix theory.