Counting black hole microstates as open string flux vacua
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1 Counting black hole microstates as open string flux vacua Frederik Denef KITP, November 23, 2005 F. Denef and G. Moore, to appear
2 Outline Setting and formulation of the problem Black hole microstates and open string flux vacua Counting BPS states (or open string flux vacua)
3 Setting and formulation of the problem
4 Setting IIA on Calabi-Yau X 4d N = 2 sugra + vect. mult. D6-D4-D2-D0 BPS bound st. BPS black holes with magn. (D-branes + gauge flux) and el. charges (p 0, p A, q A, q 0 )
5 General problem Count number of BPS states with given charge (p i, q i ), Ω(p, q), and compare with gravity prediction (beyond Bekenstein-Hawking). Conjecture [Ooguri-Strominger-Vafa]: Ω(p, q) = dφ e F(p,φ)+πq i φ i (+ exp. small) where F(p, φ) F top (g, t A ) + c.c., and F top = topological string free energy: F top = 1 g 2 D ABC t A t B t C + c 2A t A + Nβ h e2πiβ At A g 2h 2. h,β with following substitutions: g 4πi p 0 + iφ 0, ta pa + iφ A p 0 + iφ 0 Motivation: leading order saddle point approximation reproduces Bekenstein-Hawking-Wald entropy: dφ e F(p,φ)+πq i φ i e S BHW (p,q)
6 Specific problem To test conjecture, we need to find Ω(p, q) beyond leading order. Problem: difficult in general. Known cases: D4-D2-D0 dual to pert. heterotic states small black holes [Dabholkar,Dabholkar-Denef-Moore-Pioline] some D4-D2-D0 in noncompact CY (BH interpret.?) [Vafa,Aganagic-Ooguri-Saulina-Vafa] T 6, K3 T 2 [Dijkgraaf-Moore-Verlinde-Verlinde,Strominger-Shih-Yin] This talk: arbitrary D4-D2-D0 system on arbitrary, compact Calabi-Yau. Ω(p, q) in large q 0 (D0-charge) expansion, computable using... landscape techniques (flux vacua counting methods of [Ashok-Douglas, Denef-Douglas])
7 Specific problem for D4-D2-D0 For general D4-D2-D0: F(p, φ) = π φ 0 ( 1 6 (D ABC p A p B p C + c 2A p A ) D ABC p A φ B φ C ) 1 +O( eβ p/φ0 (φ 0 ) 2h ) Saddle point of OSV integral: φ 0 p 3, φ A D AB q B φ 0 q 0 So q 0 φ 0 0 instanton corrections exp. small. Hence we need to compute Z q Ω(p, q)e πφ q and show that this reduces to e F 0(p,φ) at small φ 0, where F 0 corresponds to F above without the instanton corrections.
8 Black hole microstates and open string flux vacua
9 From flux to charge Consider D4-brane wrapped on divisor P = p A J A, with N D0-branes bound to it and U(1) flux F turned on. Total D0-brane charge: where q 0 = N 1 2 F 2 χ 24 χ = P 3 + c 2 P = Euler characteristic of P Conserved D2-brane charges: q A = J A F. Here scalar product = intersection product on H 2 (P). Note: typically dim H 2 (P) dim H 2 (X ), so many different fluxes F can give rise to equal charges! need to count different flux realizations of given charge.
10 Supersymmetric configurations Supersymmetry requires [Marino-Minasian-Moore-Strominger]: F (0,2) = F (2,0) = 0 For generic fluxes F at generic points in the D4-brane deformation moduli space, this will not be satisfied. Exceptions: fluxes F which are pulled back from H 2 (X ) = H 1,1 (X ): for these, F (0,2) = 0 identically. But many F H 2 (P) not pulled back from H 2 (X ). Then condition F 0,2 = 0 imposes h 2,0 equations on the h 2,0 geometric moduli of P. generically restricts moduli to set of isolated points: open string flux vacua.
11 Divisor moduli Divisor P has deformation moduli space M, parametrized locally by coordinates z i, i = 1,..., n. 1-1 correspondence infinitesimal holomorphic deformations of P (given by holomorphic normal vector fields δ i n on P) and H 2,0 (P): n = h 2,0 (P). ω 2,0 i = Ω 3,0 δ i n
12 Special geometry structure Moduli space M has N = 1 special geometry [Lerche-Mayr-Warner]: Choose basis C α of H 2 (P), and corresponding 3-chains Γ α with Γ α P = C α (and possibly other, fixed, z-independent boundary components). Define chain periods Π α (z) Ω. Then Γ α(z) i Π α = Ω = δ i n Ω = δ i Γ α C α ω 2,0 i C α
13 Special geometry structure Natural Kähler metric on M determined by periods: g i j ω i ω j = iπ α Q αβ j Π β = i j (Π α Q αβ Π β ) P where Q αβ (Q αβ ) 1 and Q αβ C α C β, i.e. the intersection form on H 2 (P). i Π is period vector of (2, 0)-form, and by Griffiths transversality: i j Π (1, 1). By orthogonality of (2, 0) and (1, 1) forms, this implies e.g. i j Π α Q αβ k Π β = 0.
14 Susy conditions from superpotential Given flux F Poincaré dual 2-cycle Σ F on P. Expand Σ F = m α C α. Define superpotential W F (z) m α Π α (z). Then so i W = m C α ω i = ω i = α Σ F P F ω i i W (z) = 0 F 0,2 = 0 susy.
15 . Closed string landscape
16 Open string landscape Same form same techniques applicable.
17 Counting BPS states (or open string flux vacua)
18 Counting critical points At fixed F, number of isolated critical points of W F given by d 2n z δ 2n ( W F ) det i j W F 2. M Determinant ensures each isolated zero of the delta function contributes +1 to the integral. At any such critical point, the divisor is frozen, so the only remaining moduli are the positions of the N D0-branes bound to P. Upon quantization number (index) of susy ground states corresponding to this critical point = Euler characteristic of Hilbert scheme of N points on P. This is p χ (N), where N p χ (N)q N χ/24 = 1 η(q) χ
19 Black hole partition sum Using this, we get for the OSV partition sum Z = q Ω(p, q) e πφ0 q 0 πφ A q A = N,F M 1 = η χ (e πφ0 ) p χ (N) e πφ0 (N 1 2 F 2 χ 24 ) πφ F d 2n z δ 2n ( W F ) det i j W F 2 d 2n z M m δ 2n (m α i Π α ) det m α i j Π α 2 φ0 π e 2 Q αβm α m β πφ αm α
20 Gaussian form of Z Both the delta-function and the determinant can be rewritten as integrals of exponentials linear in m α : δ 2n (m α i Π α ) = d 2n λ e iπmα (λ i i Π α+ λī ī Πα) det m α i j Π α 2 = 1 π 2n d n θ d n ψ d n θ d n ψ Second integral is over fermionic variables. e πmα ( i j Π α θ i ψ j + ī j Π α θī ψ j ). Gaussian ensemble with boson-fermion-fermion interactions.
21 Large q 0 (small φ 0 ) approximation large fluxes m α continuum approximation: replace m d b 2 m. Integral over m α is straightforward Gaussian: Z = 1 η χ (e πφ0 ) d 2n z d 2n λ d n θ d n ψ d n θ d n ψ 1 ( ) 2 b2 /2 π 2n φ 0 e π 2φ 0 (Φα iλi i Π α i j Π αψ i θ j + c.c.) Q αβ (Φ β iλ i i Π β i j Π β ψ i θ j + c.c.) At first sight: 25 complicated cross terms in exponential??? But recall i Π (2, 0), i j Π (1, 1) and Φ (1, 1), and only products of (1,1) with (1,1) or (2,0) with (0,2) can be nonzero. Most cross terms are zero.
22 Geometrization The only nontrivial intersection products contributing are: with R the curvature of g. i Π α Q αβ j Π β = g i j i j Π α Q αβ k l Π β = R i kj l Hence exponential becomes simply e π φ 0 ( 1 2 Φ2 g i j λi λ j +R i kj l ψi ψ k θ j θ l ). Doing gaussian integrals over λ and ψ turns this in π n e π 2φ 0 Φ2 (det g i j ) 1 det(r i kj lθ j θl ) Integrated over θ and combined with d 2n z, this produces measure π n e π 2φ 0 Φ2 det R with R k i i 2 Rk ij l dzj d z l the curvature 2-form on M.
23 Final result After doing a modular transformation on η: 1 η χ (e πφ0 ) = ( ) φ 0 χ/2 1 2, η χ (e 4π φ 0 ) where χ = b 2 (P) 2b = P 3 + c 2 P, we get for small φ 0, Z ( φ 0 ) 1 b1 e π 2φ 0 Φ2 2 η χ (e 4π φ 0 ) ( ) φ 0 1 b1 ˆχ(M) exp 2 M 1 π n det R ( π φ 0 ( 1 6 (P3 + c 2 P) Φ2) ). where we defined the Euler characteristic ˆχ(M) of divisor moduli space as 1 ˆχ(M) det R. πn M
24 Comparison to OSV OSV prediction partition function derived from topological string, dropping all instanton corrections: ( π ( 1 Z OSV = exp φ 0 6 (P3 + c 2 P) + 1 ) 2 Φ2). our microscopic partition function at small φ 0 : ( ) φ 0 1 b1 Z = ˆχ(M) Z OSV 2 So essentially confirms conjecture in this regime, up to prefactor refinement. Recall that degeneracies are given as Laplace transform of Z, so this encodes an infinite series of 1/N corrections to the Bekenstein-Hawking entropy formula!
25 The ˆχ factor Results agree with [Shih-Yin] for X = T 6 and X = T 2 K3 in limit φ 0 0, provided For T 6 : ˆχ(p) = 1/(p) 3 with (p) 3 P 3 /6 so ˆχ = 1/p 1 p 2 p 3 if P = i p i[z i = 0] on (T 2 ) 3. For T 2 K3: ˆχ(p T 2, p K3 ) = ((p K3 ) 2 + 4)/2p T 2 Shih-Yin result is totally independent derivation, so this gives prediction for ˆχ. Nonintegral? = crazy? No! ˆχ(M) need not be integral, because M has singularities. Similar for closed strings: ˆχ(M) = 1/5 for mirror quintic. Problem: because of singularities, not known how to compute ˆχ directly. Are these values plausible?
26 The ˆχ factor For T 6, relevant moduli space = deformations of P modulo T 6 translations (since translations never get frozen by flux and trivial T 6 factor otherwise gives ˆχ = 0). Divisors in class P = (p 1, p 2, p 3 ) as above can be described as zero locus of linear combination of theta functions: Θ µ, p ( τ, z) P : p µ=1 a µ Θ µ, p ( τ, z) = 0 3 Θ µi,p i (τ i, z i ); Θ µ,p (τ, z) i=1 k µ+pz e πiτk2 +2πikz. Naive moduli space = {a µ }/C = CP p 1p 2 p 3 1, but there is residual Z 2 p 1 Z 2 p 2 Z 2 p 3 translation symmetry group to mod out: z i z i + n i +m i τ i p i acts as permutations and phase shifts on the a µ. M = CP p 1p 2 p 3 1 /Z 2 p 1 Z 2 p 2 Z 2 p 3.
27 The ˆχ factor Thus if ˆχ(CP p 1p 2 p 3 1 ) = χ(cp p 1p 2 p 3 1 ), we get ˆχ(M) = p 1p 2 p 3 (p 1 p 2 p 3 ) 2 = 1 p 1 p 2 p 3 = exactly as required for compatibility with [Shih-Yin]. Similar for T 2 K3, but now only residual translation symmetry for T 2 factor: ˆχ(M) = P c 2 P 12 pt 2 = (p K3) p 2 T 2 again exactly as required. Conjecture: For M = CP d, ˆχ(M) = d + 1.
28 Application to counting D7-D3 flux vacua By letting D-branes fill noncompact part of spacetime, (i.e. D0 D3, D4 D7), these computations can be adapted to counting open string flux vacua of IIB orientifold compactifications in weak string coupling limit. Large q 0 = large D3-tadpole L = χ(x 4 )/24. Related subtleties: divisor P constrained to be compatible with involution, and flux F must be odd. Joint treatment open + closed flux vacua at arbitrary coupling: F-theory on fourfold. OSV counts subsector (counting components weighted by their Euler characteristics). Main conclusion: working on black holes and the topological string = working on landscape statistics!
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