## Saturday, April 30, 2005 ... //

### Stringy Baby Universes

Robbert Dijkgraaf, Rajesh Gopakumar, Hiroši Ooguri, and Cumrun Vafa (DGOV) have extended their previous work about the relations between topological string theory, two-dimensional Yang-Mills theory, and Hartle-Hawking states

to non-perturbative effects in Yang-Mills theory. The most relevant previous blog article about the topic is

Note that Savas Dimopoulos has used this term with an incorrect meaning (anthropic haystack) but we obviously mean the more correct one. ;-) Everyone who wants to read about the Baby Universes that are advertised in the title is encouraged to be extremely patient. Although the new work is very interesting, let me be rather brief. Imagine that you want to count the index of a (3+1-dimensional) black hole which is really a D4-brane wrapped on some 4-cycle of a six-dimensional Calabi-Yau space - a manifold which is nothing else than a four-dimensional fiber bundle over the two-torus. If the word "index" sounds too abstract, replace it by "the number of microstates with some minus signs".

If you accept the word "index" anyway, you are counting the supersymmetric (BPS) sector of this theory, and it is a usual story that the BPS sector of a higher-dimensional theory may be described by a non-supersymmetric lower-dimensional theory. In this particular case, the relevant lower-dimensional theory is nothing else than two-dimensional Yang-Mills theory compactified on the same two-torus.

Looking at two-dimensional Yang-Mills

Now, you might think that two-dimensional gauge theory must be terribly boring. The number of transverse physical excitations of a photon (or a gluon) is "D-2=0", for example. It does not have any other fields, like matrix string theory which is a two-dimensional gauge theory with matter, that would allow the theory to describe infinitely many states and their interactions (for example the whole type IIA string theory, in the case of matrix string theory). Nevertheless, you may still compute its partition sum as a function of the number of colors "N", the coupling constant, and the area of the torus. Don't forget that this partition sum is computing an index of the higher-dimensional black hole.

It's been shown roughly a decade ago that as far as this partition function goes, two-dimensional Yang-Mills theory is equivalent to a system of free fermions that fill a band of states with energies between "-N/2" and "+N/2" (let me ignore the integrality vs. half-integrality properties of "N"). This band has two Fermi surfaces: one of them is up (near "+N/2"), and one of them is down (near "-N/2"). The partition function is really a sum over possible excitations of these two Fermi surfaces.

Note that if "N" is large, these two Fermi surfaces are very far apart and almost decoupled. Consequently, the partition sum of the free fermions factorizes into a product

• Z_{Yang-Mills} = Z_{up} Z_{down}.

Moreover, "Z_{up}" and "Z_{down}" are very similar and essentially complex conjugates to each other. That's not the end of the "entropic principle" story: Z_{Yang-Mills} may be interpreted, for large "N", as the black hole entropy, while "Z_{up}" and "Z_{down}" are the partition sums "Z_{top}" of topological string theory on the Calabi-Yau manifold that describes our black hole and its (the partition sum's) complex conjugate. This was the essential point of the work by Ooguri, Strominger, and Vafa: the exponentiated black hole entropy may be computed as the squared absolute value of the partition sum of topological string theory.

In terms of the two-dimensional Yang-Mills variables, the black hole partition function becomes the Yang-Mills partition function. The partition sum of two-dimensional Yang-Mills theory may be written not only using free fermions, but more generally also as a sum over irreducible representations "Rep"

• Z_{Yang-Mills} = Sum_{Rep} Tr_{Rep} exp[-C_2(Rep) A (g^2) N]

where "C_2" is the second Casimir of the representation, "A" is the area of the two-torus, "g" is the coupling constant (whose dimension is "mass"), and "N" is the number of colors. For non-toroidal topologies, an extra factor "dim(Rep)^{chi}" with the exponent "chi" being the Euler character of the surface would have to be added to the sum. Nevertheless, for large "N", most irreducible representations have a huge Casimir that kills their contribution to the sum. The "small" Casimir irreps of "SU(N)" can be obtained from the tensor product of a "small" representation constructed by tensoring (drawing a Young diagram) from the fundamental representation, and another "small" representation obtained from the antifundamental representation in the same way. The Casimir is then a sum of two pieces, the summation over "Rep" factorizes into a summation over "Rep_{fun}" and an independent summation over "Rep_{antifun}". Finally, the partition sum itself factorizes in such a way that the last two displayed formulae agree.

Non-decoupling of two surfaces

Nevertheless, the two Fermi surfaces are not quite decoupled for finite "N" and there are correlations. For example, if you fix the total number of fermions "N", a missing group of electrons near "+N/2" must be accompanied by added fermions near "-N/2". These correlations modify the partition sum by "exp(-N)" effects - which are non-perturbative effects with respect to a "1/N" expansion and can be neglected for large "N". DGOV have the full expression for the partition sum, and therefore they can evaluate it including these tiny effects. The partition sum of Yang-Mills then contains not only terms of the type "Z_{top}^2" but also higher powers of "Z_{top}", so to say, and the exponential suppression of "exp(-kN)" arises because of the same reason that makes the total exponentiated entropy of several black holes negligible compared to a single black hole with the "total" mass: single black hole is entropically preferred and splitting it into pieces is unlikely and exponentially suppressed by entropy counting.

If we avoid the term "Baby Universe", the black hole partition sum may be visualized as a sum of partition sums of "K" black holes, where higher values of "K" are discouraged exponentially. However, every single black hole among these "K" objects has its own near-horizon geometry which is an independent "AdS2 x S2 x Calabi-Yau" universe. Consequently, the partition sum of Yang-Mills theory may be viewed as a gauge-theoretical dual of a system of many "AdS2 x S2" universes - the baby Universes. Andy Strominger and his collaborators also liked to play with these "AdS2 x S2 x Calabi-Yau" disconnected backgrounds. DGOV explain that this multiplicity of Universes does not destroy the coherence in a single Universe.

Another interesting subtlety is that the term in the partition sum coming from "K" disconnected Universes is weighted by a rather unusual factor - the Catalan number "C_{K}" (1, 1, 2, 5, 14, 42, 132, ... if we start from "K=0") that measures the number of planar trees whose endpoints are the given Universes. (They can be written as "(2K)! / (K)! (K+1)!".) For every tree like that, one can construct a corresponding "tree-like" solution of supergravity that are really generated by multi-centered black hole solutions. The appearance of this Catalan number may be interpreted as some new obscure kind of "statistics" that remembers the "origin": for Bose-Einstein and Fermi-Dirac statistics of the Universes, we would obtain simpler factors.

I am still confused about some interpretational issues. These Baby Universe effects only become important for small values of "N" which is exactly where the geometry (and even the "counting of the number of universes) is fuzzy. I don't know how could one ever extract the information about multiple large independent universes from the partition sum - and its generalizations.