Baby universes in string theory

We argue that the holographic description of four-dimensional Bogomol’nyi-Prasad-Sommerfield black holes naturally includes multicenter solutions. This suggests that the holographic dual to the gauge theory is not a single $\mathrm{Ad}{\mathrm{S}}_2\times S^2$ but a coherent ensemble of them. We verify this in a particular class of examples, where the two-dimensional Yang-Mills theory gives a holographic description of the black holes obtained by branes wrapping Calabi-Yau cycles. Using the free fermionic formulation, we show that $O(e^{-N})$ nonperturbative effects entangle the two Fermi surfaces. In an Euclidean description, the wave function of the multicenter black holes gets mapped to the Hartle-Hawking wave function of baby universes. This provides a concrete realization, within string theory, of effects that can be interpreted as the creation of baby universes. We find that, at least in the case we study, the baby universes do not lead to a loss of quantum coherence, in accord with general arguments.

Figure 1

The spectrum of two-dimensional Yang-Mills is described by a system of $N$ free fermions. Depicted are the ground state (a) and small excitations above the ground state (b), given by fluctuations of the two Fermi surfaces at $p=\pm N/2$.

Figure 2

In the large $N$ limit the two Fermi surfaces at $p=N_{+}$ and $p=-N_{-}$ decouple and we obtain two independent relativistic free fermion field theories.

Figure 3

The chiral product ${\psi }_{N_{+}}{\overline{\psi }}_{N_{-}}$ overcounts states. As illustrated in (a), it contains configurations where $n_{-}$ negative-momentum holes are brought to the top Fermi level and $n_{+}$ positive momentum holes are brought to the bottom level. These states, that look like (b) in the Yang-Mills theory, have already been counted and therefore have to be subtracted from the chiral product.

Figure 4

The fundamental recursion relation: (a) The configurations that we have overcounted create a bubble of holes in the middle of the Fermi sea. (b) These states can be considered as a tensor product of two copies of the system, where the first factor consists of the usual particles, but the second factor is made out of holes.

Figure 5

The metric (4.18) is sketched in (a), it describes an asymptotically flat space-time with multiple black holes with charges $(P_i^I,Q_{iI})$ ($i=1,\dots ,n$). In the near-horizon limit (b) this geometry gives a disjoint sum of $n$ $\mathrm{Ad}{\mathrm{S}}_2\times S^2$ geometries.

Figure 6

As a function of Euclidean time, which can be identified with the local entropy $S(x)$ of the metric (4.18), the geometry describes the branching off of baby universes.

Figure 7

As $S\to \infty$, the number of branches increases to $n$, the number of black hole centers.