More on Membranes in Matrix Theory

We study noncompact and static membrane solutions in Matrix theory. Demanding axial symmetry on a membrane embedded in three spatial dimensions, we obtain a wormhole solution whose shape is the same with the catenoidal solution of Born-Infeld theory. We also discuss another interesting class of solutions, membranes embedded holomorphically in four spatial dimensions, which are 1/4 BPS. Matrix theory [1], formulated from the U(N) gauge supersymmetric quantum mechanics describing motions of N D0-branes, is now well established as a nonperturbative formulation of M-theory. The basic properties of M-theory which could be readily checked were that it becomes 11 dimensional supergravity at low energy, and it contains extended objects like membrane and its magnetic dual five-brane. And the dynamics of D-particles of Type IIA superstring theory converges to the supermembrane action in the lightcone frame, formulated in [2]. In addition to the rather trivial flat branes, compact branes, in particular with spherical topologies were obtained and studied also [3, 4, 5]. These branes with finite size are not static, but oscillating in time. Recently an interesting class of static membrane solutions was presented in [6], viz. holomorphically embedded membranes. In this note we investigate the possibility of another static solution of Matrix theory. We find one embedded in three dimensions which describes wormhole membrane, made by a string between branes and antibranes. Similar system was already discussed using Born-Infeld (BI) theory [7, 8], which is low energy effective theory for Dp-brane. It is amusing to see that our solution has the same shape with the electrostatic solution of three dimensional Born-Infeld theory, describing two D2-branes connected by a throat. In this note it is sufficient to consider only the bosonic sector of the Matrix theory lagrangian, L = Tr ( 1 2g (DtX) + 1 4g ([X ,X ]) ) , (1) where XI , I = 1, . . . , 9 are N ×N hermitian matrices and g is the string coupling constant. DtX ≡ ∂tX − i[A,X] where A = X0 is the gauge field of our gauged quantum mechanics. It is well known that Matrix theory, in the large N limit, converges to the 11 dimensional supermembrane theory in lightcone frame. In other words, Eq.(1) becomes L = p11 2 ∫ dpdq(DtX) + 1 4p11 ∫ dpdq({XI ,XJ})2 (2) Address as of October 1st, 1998: Department of Physics, Queen Mary and Westfield College, Mile End Road, London E1 4NS UK; [email protected]