Musings

What Greg and friends do, is address these questions for a class of Calabi-Yau’s for which the counting of (a certain subclass of) BPS states is well understood. They look at Type-IIA compactifications on orbifolds of $K3\times T^2$. These have dual descriptions as heterotic string theory on $T^6/\Gamma$. In addition to some $N=4$ examples, there is the celebrated $N=2$ example, the FHSV manifold, constructed by taking a $K3$ with an Enriques involution, and taking the quotient $M=(K3\times T^2)/\mathbb{Z}_2$, where $\mathbb{Z}_2$ acts as the Enriques involution on $K3$ and as $z\to -z$ on the $T^2$. This model has a well-understood heterotic dual, and there are a class of BPS states — the so-called Dabholkar-Harvey states — which are simply BPS perturbative string states on the heterotic side. Thus their degeneracies are computable by standard perturbative string techniques.

Now, there’s something a bit peculiar about the DH blackholes. Classically, in the string frame, they have zero horizon-area. Higher-derivative corrections to the supergravity action give them finite, but string-scale, horizon areas. The horizon area, as measured in the Einstein frame, is large, however, which is presumably what is physically relevant. And Greg and friends find no pathology in their computation.

The first thing that they find is that, to make any sense of the $\phi$-integral, the contour must be chosen to lie on the imaginary $\phi$-axis. Even with that choice, the integral diverges, unless one truncates $F$ to its “perturbative part”, $$F_{\text{pert}}(X^I,\lambda^2)= \frac{1}{6}D_{abc} \frac{X^a X^b X^c}{X^0} + \frac{1}{2}A_{a b} X^a X^b + \frac{1}{24} c_{2 a} \frac{X^a}{X^0} \frac{\lambda^2}{64}$$ throwing away all of the worldsheet instanton (Gromov-Witten) contributions, $$F_{\text{inst}}(X^I,\lambda^2)= - \frac{i}{2}\frac{\zeta(3)}{(2\pi)^3} (X^0)^2 + i \sum_{d,g \geq 0} N_{d,g} \frac{\lambda^{2g}}{(X^0)^{2g-2}}$$ In particular, there’s nothing left of $F_g$, for $g\geq 2$. One of the hopes of the OSV conjecture was that the LHS of the equation would somehow give a nonperturbative completion of the topological string partition function, which appears on the RHS. That hope seems to be dashed, if you have to throw away $F_g$ for $g\geq 2$ in order to define the integral.

Anyway, the task, now, is to compute the helicity supertraces $$\Omega_n(Q) = \frac{1}{2^n} \left(y \frac{\partial}{\partial y}\right)^n_{y=1} Tr_{\mathcal{H}_{\text{BPS}}(Q)} (-y)^{2 J_3}$$ (where $J_3$ is the generator of the massive little group in 4 dimensions) for the DH states of charge $Q=(p,q)$. $\Omega_{\text{abs}}=\dim (\mathcal{H}_{\text{BPS}}(Q))$ jumps around in some crazy fashion as you move about in the moduli space, and cannot be approximated by some smooth function. In an $N=2$ compactification, the first non-vanishing supertrace is $\Omega_2$. For $N=4$, it’s $\Omega_4$.

So one goes ahead and computes $\Omega_2(Q)$ in the heterotic dual description, and expands the result in a Rademacher series. “What’s that?” you ask. It’s an expansion in Bessel functions which conveniently captures the asymptotic behaviour of the density of states. You’re probably familiar with (at least the leading term of) the classic example of the number of (24-coloured) partitions of N. The degeneracies in the bosonic string partition function $$\Delta(\tau) = \frac{1}{q \prod_{n=1}^\infty (1-q^n)^{24} } = q^{-1}\sum_{N=0}^\infty P_{24}(N) q^N$$ have a (convergent) series expansion $$P_{24}(N) = \sum_{n=1}^\infty n^{-14} \hat{I}_{13}(\frac{4\pi}{n}\sqrt{N-1})$$ where $$\hat{I}_\nu(z) = 2\pi \left(\frac{z}{4\pi}\right)^{-\nu} I_\nu(z)$$ and the Bessel function, $I_\nu(z)$, has the asymptotic expansion $$I_\nu(z) \sim \frac{e^z}{\sqrt{2\pi z}} \left[1 - \frac{4\nu^2 -1}{8z} + \dots \right]$$ The first term in the series was discovered by Hardy and Ramanujan (1918); the full series was found by Rademacher (1938).

Anyway, you go ahead and evaluate $\Omega_2(Q)$ for the FHSV model and you find that, for charge vectors, $Q$, coming from the twisted sector of the heterotic orbifold, $$\Omega_2(Q) = -2^{-3} \hat{I}_7(4\pi \sqrt{Q^2/2}) + 2^{-11} i e^{i\pi Q^2} \hat{I}_7 (2\pi \sqrt{Q^2/2}) + \dots$$ If you compare the first term with the saddle-point evaluation of the integral (1) (where, to emphasize again, we just keep the “classical” piece of the Topological String Free Energy, dropping all the Gromov-Witten contributions), you get precise agreement. Not just for the leading exponential, but also for all the $1/|Q|$ corrections as well!

On the other hand, things are not so great in the untwisted sector First of all, for some $Q$, the BPS states fall into $N=4$ multiplets, and so $\Omega_2(Q)= 0$. In $N=4$ models (e.g. Type IIA on $K3\times T^2$), one finds agreement if one chooses instead $\Omega_4(Q) +\Omega_6(Q)$ to compare with (1). Perhaps that might work here as well, but it seems a little ad-hoc — changing the definition of $\Omega$, depending on the charge vector you wish to calculate it for.

For the untwisted sector BPS states which form $N=2$ multiplets, $$\Omega_2(Q) = 2^{-8} e^{2\pi i Q\cdot \delta}(1- e^{i\pi Q^2/2}) \hat{I}_7 (2\pi \sqrt{Q^2/2}) +\dots$$ where $\delta$ is the shift vector, which is part of the definition of the $\mathbb{Z}_2$ orbifold action on the heterotic side. This is exponentially smaller than the desired answer.

Indeed, that seems to be a generic feature of $N=2$ heterotic orbifold models. The leading contribution to $\Omega_2(Q)$ in the untwisted sector involves linear combination of $\hat{I}_{\frac{1}{2}(n_v+2)}(4\pi \sqrt{|\Delta_g|Q^2/2})$, where one sums over all twisted sectors of the orbifold for which the ground state energy, $\Delta_g = -1 +\frac{1}{2} \sum_j \theta_j(g)(1-\theta_j(g))$ is negative. Since $|\Delta_g|$ is always less than 1, the asymptotic degeneracy of $\Omega_2(Q)$ is always exponentially smaller than the desired value.

The situation seems quite muddy.

• To make any sense of the integral (1), one needs to truncate the topological string Free Energy to its classical piece.
• When one does that, one finds complete agreement with the leading exponential (and all its $1/|Q|$ corrections) behaviour of $\Omega_2(Q)$ for some charge vectors, $Q$.
• But one gets flagrant disagreement (not even the right exponent) for other choices of $Q$.

Greg’s been arguing, for a while, that the BPS degeneracies (and hence their asymptotics) ought to be rather subtle arithmetic functions of $Q$. That would make it rather unlikely that you could recover them from some smooth function, à la OSV. On the other hand, when it works, (1) works so nicely.