Recently it has been shown in [1] that the half-BPS IIB supergravity solutions, which are asymptotically AdS5×S5 and preserve an O(4)×O(4) symmetry of the asymptotic isometry group, are in one-to-one correspondence with semiclassical configurations of free fermions in a harmonic oscillator potential. This result is yet another striking evidence of the AdS/CFT correspondence [2], since the free fermions are equivalent to [3] the half-BPS sector of the super Yang-Mills theory.

The correspondence between the supergravity configurations and semiclassical fermion configurations is based on a proposed identification between a supergravity mode u(x1, x2) with the phase space density u(q, p) of the free fermions, where x1, x2 are two of the coordinates of the LLM geometry and q, p are coordinates of the phase space of the free fermions. The present work began with the questions (a) how two coordinates of space time can become phase space (noncommutative) coordinates and (b) whether one can derive the noncommutative dynamics directly from supergravity.

2. The Moduli Space of 1/2-BPS Supergravity

As shown in [1], the half-BPS geometries (with O(4)×O(4) symmetry) are characterized by a single function z0(x1, x2) ≡ z(x1, x2, y= 0) (see eqs. (2.5)-(2.15) of [1]). The moduli space of these solutions is the space of z0’s, subject to the following regularity and topological constraints.

Z0 can only be either 1/2 or −1/2, that is 1 z0(x1, x2) =− 1 2 χRi + 1 2 χ˜Rj (2.2) where the x1, x2 plane is tessellated by the regions Ri, ˜Rj, with z0=−1/2 in Ri and z0= 1/2 in the ˜Rj.

3. Quantization of Half-BPS Vacua

We will treat the function u as the collective coordinate of the space of half-BPS configurations (with O(4)×O(4) symmetry). The space of u’s can be discussed in terms of orbits of a specific u0 under the action of the group of area-preserving diffeomorphisms in two dimensions.

Alternatively, u can be parameterized as in (2.5). By choosing generic enough regions Ri, we can describe all functions u subject to the constraints. This is the discussion we will use in this section to quantize the space of u’s.

4. Collective Coordinate Action

In this section, we will identify the u-functional integral with the classical limit of a functional integral describing free fermions in a harmonic oscillator.

5. Equivalence to Fermion Path Integral

Our previous analysis has shown that, after quantization, the supergravity sector of free fermions in a harmonic oscillator potential corresponds to a fermion path integral, which reveals the non-commutative nature of the configuration space in the half-BPS sector.

6. Conclusion

We've discussed the collective coordinate quantization of the half-BPS geometries of Lin, Lunin, and Maldacena. We've demonstrated that the resulting functional integral on the plane of a single function corresponds to the classical limit of a functional integral for free fermions in a harmonic oscillator. This reveals new insights on counting supersymmetric configurations in supergravity.

7. Acknowledgements

We would like to thank the Perimeter Institute of Theoretical Physics, Ontario, Canada for hosting this research.