### Shih on OSV

David Shih visited with us this week, which gives me an excuse to talk about the checks he and collaborators have been able to carry out on the Ooguri-Strominger-Vafa conjecture. I’ve written about the work of Dabholkar *et al*, which focusses on “small” BPS blackholes which, in their examples, have heterotic duals, the Dabholhar-Harvey states (whose degeneracies are, therefore, exactly-known). In the end, that wasn’t a very satisfactory test, because — even in the large-charge limit — the volumes of some *cycles* on $X$ turned out to be small. There wasn’t an *a-priori* reason to *expect* agreement with OSV.

Shih, Strominger and Yin computed the degeneracy of 1/4 BPS blackholes in $N=4$ string theory (IIA compactified on $K3\times T^2$ with nonzero D0, D2, D4 and D6-brane charge, or one of its duals), and 1/8 BPS blackholes in $N=8$ string theory. These blackholes are “large,” already at the classical level.

Shih and Yin then went on to compare these degeneracy formulæ with OSV.

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 form^{1} 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 number^{3} 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.

^{1} The $4\times 4$ matrix, $\left(\array{A&B\\ C&D}\right)\in Sp(2,\mathbb{Z})$ acts on $\tau= \left(\array{\rho&\nu\\ \nu&\sigma}\right)$ as $\tau\to (A\tau +B)(C\tau+D)^{-1}$. The weight-10 modular form,

is the product of all the genus-2 theta functions for even spin structures.

^{2} Later, we’ll take the direct sum, and denote by $C_{M N}$, the quadratic form of signature (5,21), $C_{a b} \oplus c_{i j}$.

^{3} Note that extrapolating that the power of $g_{\text{top}}$ in the correction depends linearly on $b_1(X)$, based on only two data point, $X=K3\times T^2,\, T^6$, is a bit suspect.

## MathML

I think a while ago you asked to report problems with the display of mathematical characters.

On my Explorer+Mathplayer on WinXp setup the above text produces a red question mark for the first symbol right after the equality sign in your very first equation, as well as two red question marks right in front of the bracket term in your very last non-inline equation.