Compactify type IIB on a Calabi-Yau three-fold. You obtain an N=2 theory in d=4 dimensions. It has "(h^12 + 1)" N=2 vector multiplets - the gauge fields are the Ramond-Ramond four-form contracted over "2(h^12 + 1)" existing three-cycles of the three-fold, and we erase the factor of two because the corresponding five-form field strength is self-dual.
In the vector multiplets you find scalars. These scalars are nothing else than the parameters of the complex structure - the shape of the Calabi-Yau, so to say. (The sizes of two-dimensional submanifolds are described by Kähler moduli.) You may parameterize the complex structure by the "periods" of the holomorphic three-form. What do I mean? Every Calabi-Yau space has a closed, holomorphic (3,0) form (antisymmetric tensor with three indices) called Omega, and you may integrate it over three-cycles.
To do so, it's useful to pick a good basis of the three-cycles. It is always possible to choose the cycles "A^I" and choose the following parameters to determine the complex structure:
- X^I = Int(over A^I) Omega
That's nice - these periods are "(h^12 + 1)" complex parameters. There are also corresponding dual three-cycles "B_J" defined so that the intersection numbers are
- #(A^I, B_J) = delta^I_J
You may want to say that the periods of "Omega" over "B_J" are new parameters of the shape, but they're not independent. They can be called "F_J":
- Int(over B_J) Omega = F_J
and this "F_J" is actually always the derivative of the prepotential:
- F_J = partial / partial X^J (F)
Note that the prepotential is the "N=2" generalization of the concept of the superpotential from "N=1" supersymmetric theories: it is also integrated over half of the superspace only ("d^4 theta" in this case). The prepotential is a holomorophic functions of the vector multiplets, in this case "X^I".
In the low energy effective action, you will find not only the simple term "int d^4 theta F", but also more complicated terms with the same chiral structure
- int d^4 theta F_h [(W_{ab})^2]^h
where "W_{ab}" is the superfield describing the graviphoton (the supersymmetric partner of the graviton with spin 1). It's been known for some time that the prepotential "F" as well as its cousins "F_h" may be extracted from topological string theory as the partition sum computed from genus "h" surfaces: this is the most well-known example how topological string theory is embedded into the full string theory. "F" itself corresponds to the sphere (classical topological string) and some insertions must be included into the path integral for genus "h=0,1" to get a nonzero result. Let me clarify that the text above was an introduction of the setup and its most important features, not yet the work of Ooguri-Strominger-Vafa.
Black holes
Now, choose some charges
- (p^I, q_I)
for your future black hole. You may divide them into "electric" charges "q_I" and "magnetic" charges "p^I". These integers may be interpreted as winding numbers of D3-branes over the "A^I, B_J" cycles, respectively. Try to find the black hole solutions that carry these charges, using the effective four-dimensional N=2 supergravity. It has been known for some time that whatever values of the complex structure at infinity (asymptotic values) you choose, it cannot stay constant and it will flow to fixed values at the horizon. One may see that the horizon values always satisfy
- Re (C X^I, C F_I) = (p^I, q_I)
where "C" is a constant that could be absorbed to the definition of "Omega". So the real parts of the periods are simply equal to the charges of the black holes. For any choice of "(p^I, q_I)", you may find a supersymmetric black hole solution of supergravity. And the black hole will have a nonzero entropy "S" - whose exponential is the number of microstates obtained from string theory.
The Ooguri-Strominger-Vafa (OSV) conjecture relates the prepotential to the number of states of the black hole. Let's first prepare some observables
- Prepare the total prepotential "F_{total}" that also contains the "W_{ab}" corrections. Substitute the value "W^2 = 256" for the value of the graviphoton. Substitute "p^I + i.phi^I" for the value of "X^I". Compute the imaginary part of the prepotential and call it "F(p, phi)".
OSV now tell you that
- F(p, phi) = Tr exp(-q_I phi^I)
where some uncertainty about the insertion of things like "(-1)^F" is implicitly contained. Note that the left hand side is constructed from the prepotential and its corrections while the right hand side depends on the exact counting of the black hole states (the entropy, including the corrections to the leading Bekenstein-Hawking entropy). The right hand side is a sum over all states with the right value of magnetic charges "p^I" but arbitrary values of the charges "q_I".
You may also write the summation over "q_I" as an integral as long as you insert "rho(q)" counting the density of states. In an exact treatment, "rho" would be a sum of delta functions, but you should smear it - I don't believe that anyone has computed this formula so exactly that it would give "see" the quantized values of "q". The integral over "q_I" may be interpreted as the Laplace transform. If you evaluate it using the saddle point evaluation, the integral will be dominated by one point that turns out to be the Legendre transform.
Note that "p^I" and "q_I" are kind of dual to each other, by the Dirac quantization rule. On the other hand, the chemical potential is also "dual" to "p^I" - and the Ooguri-Strominger-Vafa conjecture is based on mixing the role of "q_I" and "phi_I" that transform in the same way.
But OSV want you to calculate not only the Legendre transform (arising from the saddle point approximation), but also all the subleading corrections as powers of "1/Q" that follow from the full integral - the Laplace transform. This expansion converges quickly for large values of the charges "Q" which means "(p^I, q_I)" and the two sides should agree at every order. This conjectured identity is remarkable. It would be even more remarkable if we knew more accurately what the objects exactly mean - whether we should count all states, an index, or what it exactly is (in the simplest examples, there is no difference between these options). And most likely, the identity would become less remarkable - and simple to prove - if we knew really exactly what it says.
Atish Dabholkar had a good idea how to test this hypothesis using the heterotic string methods. Consider type IIA on a Calabi-Yau - this differs from the comments at the beginning of the article by mirror symmetry. Moreover, take the Calabi-Yau three-fold to be special, a "K3 x T2". Then you may use the string-string duality that implies that type IIA on K3 is the same thing as heterotic strings on T4. In the heterotic setup, you may consider heterotic strings that are wrapped on another circle and carry a momentum in such a way that the level-matching condition
- N_{L} - N_{R} + n.w + 1 = 0
allows you to choose no excitations on the supersymmetric side ("N_R=0") and an arbitrary number of left-moving excitations "N_L" on the bosonic side of the superstring. Such states will preserve the same amount of SUSY as the ground state (graviton), namely one half, and we can count their number by Cardy's techniques.
Note that in the string-string duality, the heterotic string is a NS5-brane of type IIA wrapped on the K3 (and vice versa, the fundamental IIA string is the heterotic fivebrane wrapped on T4). But because we have type IIA on extra T2, besides the K3, we may choose different circles to be the M-theoretical 11th circle. You either obtain NS5-branes on the K3, or - if you make the "9-11" flip and choose a different circle to be the eleventh one - you may also find another dual representation where the heterotic string is a D4-brane wrapped on the K3. Both NS5 and D4 come from M5 in M-theory. The wrapping number of the D4-brane on the K3 is the winding number of the heterotic string and the momentum of the heterotic string becomes the number of extra D0-branes that you must add. The number of D4-branes and D0-branes play a similar role to the electric and magnetic charges discussed previously.
Atish Dabholkar computed the degeneracy of the states, including the corrections to the leading term in Cardy's formula, and found agreement with the expression calculated from the prepotential. Note that he is comparing two expressions - and neither of them is the Bekenstein-Hawking entropy with stringy corrections included - the last mentioned approach was included to the game by Dabholkar, Kallosh, and Maloney whose black hole effectively modifies the Bekenstein-Hawking "A/4G" entropy to "A/2G" by the stringy corrections.
A more extensive verification was attempted by Dabholkar, Denef, Moore, Pioline who considered a more general setup, heterotic orbifolds etc. They had to "improve" the conjecture at several places to make it work - erasing inconvenient terms from the prepotential was one of their techniques. Because a description of this paper requires better setup to write math, I will leave the paper to my readers.
