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February 27, 2005

Conversations with Greg

Greg Moore was in town for a few days, and we had — as always — some very interesting discussions. Among the topics was his recent paper with Dabholkar, Denef and Pioline.

I’ve talked before about the Ooguri-Strominger-Vafa proposal relating the entropy of a charged N=2N=2 black (which appears as a nontrivial solution of type-IIA strings compactified on some Calabi-Yau, MM) to the topological string partition function for the same Calabi-Yau. Specifically,

(1)

Ω(p,q)=dϕ|e iπ2F(p+iϕ,256)| 2e πqϕ \Omega(p,q) = \int d\phi \left|e^{i\frac{\pi}{2}F(p+i\phi,256)}\right|^2 e^{\pi q\cdot\phi} where (p I,q I)(p^I,q_I), I=0,,h 1,1(M)I=0,\dots,h^{1,1}(M) are the electric and magnetic charges, F(X I,λ 2)F(X^I,\lambda^2) is the holomorphic topological string free energy, and Ω\Omega is a microcanonical partition function — the number, or perhaps some index, of the number of states of charge (p,q)(p,q).

There are three questions about this formula

  1. What contour of integration should be chosen (if one exists) so that the integral is well-defined?
  2. Exactly what is Ω(p,q)\Omega(p,q) counting?
  3. Is the formula right?

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×T 2K3\times T^2. These have dual descriptions as heterotic string theory on T 6/ΓT^6/\Gamma. In addition to some N=4N=4 examples, there is the celebrated N=2N=2 example, the FHSV manifold, constructed by taking a K3K3 with an Enriques involution, and taking the quotient M=(K3×T 2)/ 2M=(K3\times T^2)/\mathbb{Z}_2, where 2\mathbb{Z}_2 acts as the Enriques involution on K3K3 and as zzz\to -z on the T 2T^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 ϕ\phi-integral, the contour must be chosen to lie on the imaginary ϕ\phi-axis. Even with that choice, the integral diverges, unless one truncates FF to its “perturbative part”, F pert(X I,λ 2)=16D abcX aX bX cX 0+12A abX aX b+124c 2aX aX 0λ 264 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 inst(X I,λ 2)=i2ζ(3)(2π) 3(X 0) 2+i d,g0N d,gλ 2g(X 0) 2g2 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 gF_g, for g2g\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 gF_g for g2g\geq 2 in order to define the integral.

Anyway, the task, now, is to compute the helicity supertraces Ω n(Q)=12 n(yy) y=1 nTr BPS(Q)(y) 2J 3 \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 3J_3 is the generator of the massive little group in 4 dimensions) for the DH states of charge Q=(p,q)Q=(p,q). Ω abs=dim( BPS(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=2N=2 compactification, the first non-vanishing supertrace is Ω 2\Omega_2. For N=4N=4, it’s Ω 4\Omega_4.

So one goes ahead and computes Ω 2(Q)\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 Δ(τ)=1q n=1 (1q n) 24=q 1 N=0 P 24(N)q N \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)= n=1 n 14I^ 13(4πnN1) P_{24}(N) = \sum_{n=1}^\infty n^{-14} \hat{I}_{13}(\frac{4\pi}{n}\sqrt{N-1}) where I^ ν(z)=2π(z4π) νI ν(z) \hat{I}_\nu(z) = 2\pi \left(\frac{z}{4\pi}\right)^{-\nu} I_\nu(z) and the Bessel function, I ν(z)I_\nu(z), has the asymptotic expansion I ν(z)e z2πz[14ν 218z+] 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 Ω 2(Q)\Omega_2(Q) for the FHSV model and you find that, for charge vectors, QQ, coming from the twisted sector of the heterotic orbifold, Ω 2(Q)=2 3I^ 7(4πQ 2/2)+2 11ie iπQ 2I^ 7(2πQ 2/2)+ \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|1/|Q| corrections as well!

On the other hand, things are not so great in the untwisted sector First of all, for some QQ, the BPS states fall into N=4N=4 multiplets, and so Ω 2(Q)=0\Omega_2(Q)= 0. In N=4N=4 models (e.g. Type IIA on K3×T 2K3\times T^2), one finds agreement if one chooses instead Ω 4(Q)+Ω 6(Q)\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 Ω\Omega, depending on the charge vector you wish to calculate it for.

For the untwisted sector BPS states which form N=2N=2 multiplets, Ω 2(Q)=2 8e 2πiQδ(1e iπQ 2/2)I^ 7(2πQ 2/2)+ \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 δ\delta is the shift vector, which is part of the definition of the 2\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=2N=2 heterotic orbifold models. The leading contribution to Ω 2(Q)\Omega_2(Q) in the untwisted sector involves linear combination of I^ 12(n v+2)(4π|Δ g|Q 2/2)\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, Δ g=1+12 jθ j(g)(1θ j(g))\Delta_g = -1 +\frac{1}{2} \sum_j \theta_j(g)(1-\theta_j(g)) is negative. Since |Δ g||\Delta_g| is always less than 1, the asymptotic degeneracy of Ω 2(Q)\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|1/|Q| corrections) behaviour of Ω 2(Q)\Omega_2(Q) for some charge vectors, QQ.
  • But one gets flagrant disagreement (not even the right exponent) for other choices of QQ.

Greg’s been arguing, for a while, that the BPS degeneracies (and hence their asymptotics) ought to be rather subtle arithmetic functions of QQ. 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.

Note:

I should have said that there’s an even more fundamental reason to wonder about the OSV formula. The FHSV manifold (and other, similar, examples) have a variety of torsion RR fluxes that can be turned on. Indeed, the heterotic model is supposed to be dual to the theory with a particular 2\mathbb{Z}_2 flux turned on. The RHS of (1) is supposed to be insensitive to turning on torsion RR flux. But the degeneracies of BPS states (the LHS) certainly does depend on what RR fluxes are turned on. These subtle torsion effects are what Greg and I spent most of our time talking about.
Posted by distler at February 27, 2005 1:59 AM

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