Take type IIB string theory on AdS5 x S5 or its pp-wave limit. Both of them have the maximal number of 32 supercharges. Is there some interesting generalization of these two geometries?
The answer is: yes, there is. Both of these geometries have at least the SO(4) x SO(4) x R isometry. The pp-wave is a limit of the the anti de Sitter space. Moreover, the pp-wave limit has a Z_2 symmetry exchanging the two SO(4) factors - this symmetry is broken by the anti de Sitter space. Is there some geometric heuristic picture how to visualize these two geometries?
Yes, there is. You can imagine
- the AdS5 x S5 space as a black disk drawn on a white paper
- the pp-wave limit of it is a paper whose lower half-plane is black
For any black-and-white picture that you can draw on the plane, there exists a solution of type IIB string theory - more or less, it's a geometry - with 16 supercharges. How do you construct it? Well, parameterize the ten-dimensional space by the following coordinates:
- time "t"
- three coordinates labeling the three-sphere S^3 number one
- three coordinates labeling the three-sphere S^3 number two
- a coordinate "y" which is kind of "radial"
- two coordinates "x_1, x_2" spanning a plane that you imagine to be analogous to the "x-p" phase space
The black regions in the "x_1, x_2" plane represent the Fermi liquid known from the matrix description of two-dimensional string theory. You may imagine that the two-dimensional string theory is embedded into the ten-dimensional type IIB string theory as a subsector. Analogous constructions, although possibly slightly less exciting ones, exist for other geometries - like the Anti de Sitter space solutions of M-theory.
So how many SUSY solutions of type IIB string theory did they obtain by this construction? A huge number. First of all, for every different topology of the black-and-white picture (a different number of "droplets" etc.), one obtains a different topology of the spacetime. If all droplets are large and their boundaries kind of straight, the curvature of the spacetime will also be small. The spacetime curvature becomes large if the droplets approach one another - a droplet eaten by a bigger droplet on the black-and-white picture describes topology changing transitions.
Even if you fix the topology, the shape of the droplet can be anything you want - and you obtain different geometries. In this sense, their Ansatz has infinitely many parameters. If you describe a boundary of a droplet as a function "x_2(x_1)" of one variable, for every function of one variable you will obtain one solution. A huge number. Of course, all these solutions have different asymptotics.
This continuously infinite number of parameters of the class of the solutions is analogous to Mathur et al. who construct their revolutionary solutions that are meant to describe the black holes, although they have neither horizon nor singularity. In that case, the solutions are also parameterized by a function of one variable - describing a shape of a string - that is dualized by various dualities to obtain a solution that looks like a black hole outside, but whose interior is very different.
Is any black-and-white picture allowed? One can see that the areas of all droplets must be actually integers (in some proper units of areas on the "x_1, x_2" plane). This requirement arises from quantization of the fluxes. In the "AdS_5 x S_5" solution, for example, the black-and-white picture is a black disk. Its area is proportional to "N", the five-form flux through the five-sphere. The classical geometry is only appropriate if the droplets are large and their curvature is small.
Therefore it sounds reasonable to imagine that the "x_1, x_2" plane is noncommutative, like a phase space, and the quantum of the area is a single cell of this phase space. The function "z" that equals +1/2 in the black regions and -1/2 in the white regions could really be a function on a non-commutative space that satisfies "z*z=1/4" where "*" is the non-commutative star-product. Anyone has a way to see that such a description is possible? There could be some "dual" object - like a D3-brane that can wrap either of these spheres S^3. The coordinates "x_1, x_2" would be fields living on the worldvolume of this dual object, and one should be able to show that they don't commute and the commutator is the right c-number. Such an object could be in various states, and a lowest energy state would correspond to the field "z(x_1,x_2)" that describes the black-and-white picture. Note that in the normal picture of string theory, "z" parameterizes the geometry (and the RR field strengths) and therefore we treat it as a closed string field. Near y=0, however, there could be a dual way to describe physics in which geometry comes from quantization of this "new kind of object" that sees a non-commutative "x_1, x_2" plane.
Any comments related to this paper are welcome.