Musings

A single wrapped D6-brane, when lifted to M-theory is replaced by a Taub-NUT space. Taking the radius of the Taub-NUT space to infinity, one obtains a 5D blackhole preserving the same amount of supersymmetry. The degeneracies of those blackholes were computed in the classic paper by Strominger and Vafa and by Breckenridge et al. But the degeneracies can’t depend on the radius, a continuous parameter. So the 4D blackhole degeneracies, with $Q_6=1$, are given by the same formula. The result is invariant under only a subgroup of the 4D $U$-duality group (which preserves $Q_6=1$), but can be completed in a unique fashion to a formula invariant under the full $SL(2,\mathbb{Z})\times SO(6,22,\mathbb{Z})$ $U$-duality group.

The result, conjectured long ago by Dijkgraaf, Verlinde and Verlinde, can be written as $$\Omega(p,q) = \oint d\rho d\sigma d\nu \frac{e^{i\pi(\rho q_m^2 +\sigma q_e^2 +(2\nu-1)q_e\cdot q_m)}}{\Phi(\rho,\sigma,\nu)}$$ where $\Phi$ is the unique automorphic form¹ of weight 10 for $Sp(2,\mathbb{Z})$. The following combinations of charges are invariant under $SO(6,22,\mathbb{Z})$ $$\array{\arrayopts{\colalign{right left}} q_e^2&=2q_0 p^1 +C^{a b} q_a q_b + c^{i j}q_i q_j\\ q_m^2&=2p^0 q_1 + C_{a b}p^a p^b + c_{i j}p^i p^j\\ q_e\cdot q_m&= p^0 q_0 +p^1 q_1 -p^a q_a -p^i q_i }$$ where $C_{a b}$ is the intersection form on $H^2(K3)$ (which has signature (3,19) ) and $c_{i j}=\textstyle{\left(\array{0&\sigma_1\\ \sigma_1&0}\right)}$ (which has signature (2,2)).

In the IIA language,

• $q_0$ is the D0-brane charge
• $q_1$ is the number of D2’s wrapped on $T^2$
• $q_a$ are the D2’s wrapped on 2-cycles of the $K3$.
• $q_i$ are the momentum and winding modes of NS5’s wrapped on $K3\times S^1$
• $p^0$ is the number of wrapped D6-branes
• $p^1$ is the number of D4-branes wrapped on $K3$
• $p^a$ are D4-branes wrapped on $T^2\times$ a 2-cycle of $K3$
• $p^i$ are momentum and winding modes of fundamental strings wrapped on $T^2$.

Anyway, one wishes to Laplace-transform, $$Z(p,\phi)= \sum_q \Omega(p,q) e^{- \phi\cdot q}$$ and compare the result with the square of the topological string partition function, evaluated at the attractor values for the moduli, $|Z_{\text{top}}(p+i\phi/\pi)|^2$.

For $T^6$, $$Z_{\text{top}} = e^{F_{\text{top}}}, \qquad F_{\text{top}}=\frac{(2\pi i)^3 D_{A B C} t^A t^B t^C}{g_{\text{top}}^2}$$ and for $K3\times T^2$ case, $$F_{\text{top}}= \frac{(2\pi i)^3 C_{M N} t^M t^N t^1}{g_{\text{top}}^2} -24\log \eta(t^1)$$

where $$t^M = \frac{X^M}{X^0}=\frac{p^M+i\phi^M/\pi}{p^0+i\phi^0/\pi}$$ are the Kähler moduli and $$g_{\text{top}}= \frac{4\pi i}{X^0}= \frac{4\pi i}{p^0 +i\phi^0/\pi}$$ is the topological string coupling.

Shih and Yin carry out the computation for $p^0=0$. They find disagreement, even at the perturbative level, but the discrepancy can be summarized by a modified formula, $$Z(p,\phi) = \sum_{\phi\to \phi +2\pi i k}|Z_{\text{top}}(p+i\phi/\pi)|^2 g_{\text{top}}^{2(b_1-2)}V(X) + \mathcal{O}\left(e^{-V(X)/g_{\text{top}}^2}\right)$$ where $V(X)$ is the attractor value of the volume of $X=K3\times T^2$ (or $X=T^6$ in the $N=8$ case) and $b_1$ is the first Betti number³ of $X$.

It would be very interesting to see whether this persists for other $N=4$ models (orbifolds of the above), and it would be very nice to understand the origin of these corrections to OSV.