Musings

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 $U(1)\subset SU(4)_R$, generated by $J$ and the Hamiltonian

When reduced on $\mathbb{R}\times S^3$, this theory is equivalent to a gauged large-$N$ matrix model with a harmonic oscillator potential. The dynamics of the eigenvalues of the matrix model reduces to a theory of $N$ 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 ($J\ll N$). Isolated droplets above or holes below the Fermi surface correspond to giant gravitons ($J\sim N$).

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

The corresponding supergravity solution is nonsingular, provided $z(x_1,x_2,0)=\pm 1/2$. So we need to specify, as a boundary condition at $y=0$, those regions in the $x_1$-$x_2$ plane where $z=+1/2$ and those where $z=-1/2$. 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.