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October 1, 2004

BPS Droplets

A while ago, David Berenstein suggested a matrix model description of the 1/2-BPS sector of N=4 SYM. The idea is to pick a U(1)SU(4) RU(1)\subset SU(4)_R, generated by JJ and the Hamiltonian

(1)limϵ0H ϵ=ΔJ+ϵΔϵ \underset{\epsilon\to 0}{\lim} H_\epsilon = \frac{\Delta-J + \epsilon \Delta}{\epsilon}

When reduced on ×S 3\mathbb{R}\times S^3, this theory is equivalent to a gauged large-NN matrix model with a harmonic oscillator potential. The dynamics of the eigenvalues of the matrix model reduces to a theory of NN free fermions in a harmonic oscillator potential.

The ground state of AdS5×S5 is the filled Fermi sea, a circular disk in phase space. Ripples on the Fermi surface correspond to 1/2-BPS supergravity excitations (JNJ\ll N). Isolated droplets above or holes below the Fermi surface correspond to giant gravitons (JNJ\sim N).

Lin, Lunin and Maldacena have a very beautiful recent paper, in which they extend this picture to arbitrary droplets (JN 2J\sim N^2). They provide a detailed map between states of the Fermi theory and IIB supergravity geometries with SO(4)×SO(4)×SO(4)\times SO(4)\times \mathbb{R} isometry and 16 supercharges. The supergravity solutions are specified by a function z(x 1,x 2,y)z(x_1,x_2,y), obeying

(2) i 2z+y y( yzy)=0 \partial_i^2 z + y\partial_y\left( \textstyle{\frac{\partial_y z}{y}}\right)=0

The corresponding supergravity solution is nonsingular, provided z(x 1,x 2,0)=±1/2z(x_1,x_2,0)=\pm 1/2. So we need to specify, as a boundary condition at y=0y=0, those regions in the x 1x_1-x 2x_2 plane where z=+1/2z=+1/2 and those where z=1/2z=-1/2. Equivalently, we specify the boundary between these two regions — a shape for the Fermi surface.

The topology of the Fermi surface determines the topology of the supergravity solution; the detailed geometry of the Fermi surface determines the geometry of the supergravity solution.

LL&M extend this prescription to compactifications of M-theory on AdS4×S7 and AdS7×S4. Really nice stuff.

Posted by distler at October 1, 2004 10:54 AM

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