Black hole microstate counting

Transcription

1 Black hole microstate counting Frederik Denef Harvard & Leuven Varna, Sept 12, 2008

2 Outline Overview of approaches to black hole microstate counting Brief overview of some recent developments BPS bound states and wall crossing for dummies The OSV conjecture Wall crossing for non-dummies

3 Overview of some approaches

4 Bekenstein-Hawking entropy Schwarzschild: S = 4πGM 2 S BH = A 4G Extremal Reissner-Nordström: S = πgm 2 N=2 BPS: S = min πgm 2 stat. mech. explanation?

5 1. Quantizing field fluctuations in black hole background [ t Hooft 85] the good: S = const. A the bad: const. = (= Λ UV 1/G) the puzzling: depends on cutoff, number of species and other details (related: interpretation as entanglement entropy [Srednicki 93, Bombelli et al 86])

6 [Susskind 93] 2. Highly excited fundamental string states (Free) string microscopic entropy: S micro l s M But S BH GM 2?? Redshift factor at distance l s from horizon ( stretched horizon ) = l s /GM M loc GM 2 /l s S micro l s M loc GM 2

7 3. BPS black holes Saturating BPS bound for given charge Q: M = Z(Q) G Here Z(Q) = central charge; depends on vacuum moduli. Remains true with M M loc for any local static observer, even with varying moduli and metric! Unlike non-bps case, mass not renormalized between near and far! entropy = log # states of charge Q and energy E = Z(Q) G hor. Q 0 Need to count (BPS) D-brane states.

8 3. BPS black holes (continued) Use: 1. generically: degeneracy at finite coupling (good) index at finite coupling 2. index invariant under change of couplings and compute semiclassically in weakly coupled regime. S micro = S BH [Stroninger-Vafa 96, Maldacena-Strominger-Witten 97] [However all successful cases dual to D4-D0/M5-P/AdS 3 systems!]

9 4. (BPS) Fuzzballs [Mathur, Benna-Warner, Balasubramanian-Gimon-Levi, de Boer et al, Strominger et al, Skenderis-Taylor et al, X et al] There exist very intricate 4d multiparticle BPS bound states. Scaling solutions: asymptotically indistinguishable from black hole. Huge internal degeneracies. Partial quantitative successes, however so far not shown to be huge enough to reproduce full 4d S BH. Adding spherical D2-branes seems to ± reproduce S BH.

10 5. AdS 5 -CFT 4 Comparison N = 4 SU(N) YM field theory entropy S 0 computed at zero t Hooft coupling g 2 N and black hole entropy at large t Hooft coupling S : S = 3 4 S 0 Renormalization effect quite well understood by now.

11 6. Computer simulations [Anagnostopoulos-Hanada-Nishimura-Takeuchi 07] Monte Carlo simulations of U(N) susy QM with 16 supercharges, dual to (nonextremal) black hole with D0 charge N in IIA. Perhaps most convincing confirmation ever that string theory gets quantum gravity right!

12 Some recent developments in (non-fuzzball) microstate counting: [de Wit-Mohaupt-Lopes Cardoso-Kappeli 99-00]: Wald R 2 corrections to S BH match subleading corrections microscopic M5-P entropy. [Ooguri-Strominger-Vafa 04] All order refinement the OSV conjecture: Z BH Z top 2. [Dabholkar 04,Sen 05] Entropy of small black holes [Sen 05] Non-BPS extremal BH attractors in arbitrary higher derivative gravity; Entropy function formalism. [Gaiotto-Strominger-Yin 06,de Boer-Cheng-Dijkgraaf-Manschot-Verlinde 06] Relation AdS 3 -CFT 2 elliptic genus to OSV. [Gaiotto-Strominger-Yin 06] Exact M5 elliptic genera [Gaiotto Shih Strominger Yin Sen Dabholkar Murthy Narayan Banerjee Nampuri David Jatkar Srivastava Mukherjee Mukhi Nigam...] ((re-)re-)re-counting N=4 dyons [Denef-Moore 07] Partial proof version of OSV; wall crossing formulae [Sen 07-08,Cheng-Verlinde 07-08]: General N=4 wall crossing + alg [Kontsevitch-Soibelman,Gaiotto-Moore-Neitzke]: General N=2 wall crossing formula

13 BPS bound states and wall crossing for dummies Wall Crossing Simple derivation of index jumps for (1,N) splits! Denef '00-'03 Denef-Moore '07

14 Realizations of BPS states in string theory

15 Setting IIA on Calabi-Yau X 4d N = 2 supergravity +(h 1,1 + 1) gauge fields 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 )

16 BPS states and stability BPS bound for mass of particle with charge Γ = (p 0, p, q, q 0 ) in vacuum with complexified Kähler moduli t B + ij: where M M BPS = Z M p ( ) (Im t) 3 1/2 ( ) Z = p 0 t3 6 6 p t2 2 + q t q 0 + inst. corr. For generic t: Z(Γ 1 + Γ 2, t) < Z(Γ 1, t) + Z(Γ 2, t) BPS states absolutely stable.

17 Decay at marginal stability Stability argument fails when t such that arg Z(Γ 1, t) = arg Z(Γ 2, t) since then Z(1 + 2) = Z(1) + Z(2) : marginal stability. BPS states can disappear from spectrum when crossing walls of marginal stability. t Γ Γ 1 Γ 2 BPS particle splits in two BPS particles conserving different susies. Even index of BPS states jumps!

18 BPS states at g s 0 and V CY Localized at single point in noncompact space. Wrapped on holomorphic cycles. Bound states with lower dim branes: gauge flux: holomorphic vector bundles brane gas : ideal sheaves

19 BPS states at g s 0 near marginal stability Decay Γ Γ 1 + Γ 2 at marginal stability often invisible in IIA large volume geometrical D-brane picture. Stringy microscopic description [Kachru-McGreevy]: Γ 1 Γ 2 I 12 Γ 1 Γ 2 Light 1 2 open string modes φ i, i = 1,..., I 12 have D-term potential: V D i ( φ i 2 ξ) 2 M susy = CP I FI term ξ changes sign when crossing MS wall susy config. exists on one side, not on other: tachyon glue iff ξ > 0.

20 BPS states in 4d supergravity (g s Γ 1) Simplest possibility: spherically symmetric BPS black hole of charge Γ (p 0, p, q, q 0 ): ds 2 = e 2U(r) dt 2 + e 2U(r) d x 2 t Solutions attractors [Ferrara-Kallosh-Strominger]: Radial inward flow of moduli t(r) is gradient flow of log Z(Γ, t).

21 Existence of spherically symmetric BPS black holes Three possibilities [Moore]: 1. Gradient flow ends in minimum t = t (Γ) with Z(Γ, t ) 0. Regular black hole with horizon area A = 4π Z(Γ, t ) Flow ends in boundary point t = t 0 with Z(Γ, t 0 ) = 0. Zero area black hole, but still BPS solution (e.g. pure D6, D2-D0; note: regular after uplifting to 5d). 3. Flow ends in interior point t = t 0 with Z(Γ, t 0 ) = 0. No BPS black hole solution.

22 BPS black hole molecules More general BPS solutions exist: multi-centered bound states: ds 2 = e 2U( x) ( dt ω i dx i) 2 + e 2U( x) d x 2. Centers have nonparallel charges. Bound in the sense that positions are constrained by gravitational, scalar and electromagnetic forces. Stationary but with intrinsic spin from e.m. field

23 Explicit multicentered BPS solutions N-centered solutions characterized by harmonic function H( x) from 3d space into charge space: H( x) = N i=1 Γ i x x i + H with H determined by t x = and total charge Γ. Positions constrained by N j=1 Γ i, Γ j x i x j = 2 Im ( e iα Z(Γ i ) ) x = where Γ 1, Γ 2 = Γ m 1 Γe 2 Γe 1 Γm 2 and α = arg Z(Γ). All fields can be extracted completely explicitly from the entropy function S(Γ) on charge space, e.g. e 2U( x) = π S(H( x))

24 Decay at marginal stability 2-centered case: Γ 1 Γ 2 Equilibrium distance from position constraint: x 1 x 2 = Γ 1, Γ 2 Z 1 + Z 2 2 Im(Z 1 Z 2 ) x = When MS wall is crossed: RHS and then becomes negative: decay

25 Example: pure D4 = D6 D6 molecule Pure D4 with D4-charge P has Z P t2 2 P3 24. Z(t) = 0 at t i P No single centered solution. Instead: realized as bound state of single D6 with U(1) flux F = P/2 and anti-(single D6 with flux F = P/2): D6[P/2] -D6[-P/2] Stable for Im t > O(P). M-theory uplift: smooth bubbling geometry. [Benna-Warner,Cheng]

26 Transition between g s Γ 1 and g s Γ 1 pictures Mass squared lightest bosonic modes of open strings between Γ 1 and Γ 2 : M 2 /Ms 2 x 1 x α l 2 s = c(t) g 2 s + α On stable side of MS wall α < 0, so if g s gets sufficiently small, open strings become tachyonic and branes condense into single centered D-brane. [FD qqhh] single D-brane Quiver Higgs Quiver Coulomb two particles (sugra) 0 α 3/2 c c α Igs

27 The flow tree - BPS state correspondence Establishing existence of multicentered BPS configurations not easy: position constraints, S(H( x)) R + x,... However: Theorem/conjecture: Branches of multicentered configuration moduli spaces in 1-1 correspondence with attractor flow trees: Γ E' Γ E Γ E'' ms Much simpler to check & enumerate!

28 Flow tree decomposition of BPS Hilbert space Flow trees can also be given microscopic interpretations (decay sequences / tachyon gluing). Hilbert space of BPS states of charge Γ in background t has canonical decomposition in attractor flow tree sectors: H(Γ, t) =

29 Uplift to M-theory on AdS 3 S Flow trees which start at infinite CY volume can be realized as multicentered bound states in asymptotic AdS 3 S 2 : [de Boer-(Denef-)El Showk-Messamah-Van den Bleeken 08] relevant for (0, 4) CFT 2 s.

30 Wall crossing formulae

31 The BPS index Hilbert space of BPS states in 4d N = 2 theories: H(Γ, t) = ( 1 2, 0, 0) H (Γ, t) Index: Ω(Γ, t) = Tr H (Γ,t) ( 1) 2J 3 How does Ω change when t crosses MS wall?

32 Wall crossing formula for primitive splits J Γ 1 Γ 2 Near marginal stability wall Γ Γ 1 + Γ 2 (with Γ 1 and Γ 2 primitive), the decaying part of H (Γ, t) has following factorized form: H (Γ, t) = (J) H (Γ 1, t) H (Γ 2, t) with J = 1 2 ( Γ 1, Γ 2 1). Spin J factor: macroscopically from intrinsic angular momentum monopole-electron system ( 1/2 from relative c.o.m. spin) microscopically from quantizing open string tachyon moduli space M susy = CP 2J. Implies index jump Ω = ( ) 2J (2J + 1) Ω(Γ 1, t ms ) Ω(Γ 2, t ms ).

33 Indices from wall crossing: examples Pure D4 on P > 0 with pulled back flux S splits in D6 with flux P/2 + S and anti-(d6 with flux P/2 + S), so Ω = Γ 1, Γ 2 = P3 6 + c 2P 12 =: I P = χ(m P ). 2. D6-D2 brane gas bound state with D2 = U V, V > U, first splits off D6 with flux U, remainder next splits in D6 with flux V and anti-(d6 with flux U + V ), so Ω = Γ 1, Γ 2 + Γ 3 Γ 2, Γ 3 = I V I V U I U = χ(m U V )

34 Nonprimitive splits 1: halos Γ Γ 1 Ν Γ 2 Halo = bound state of one Γ 1 particle ( core ) with N Γ 2 particles. Wall crossing formula for index from generating function: N ( ) Ω(Γ 1 + NΓ 2, t) u N = Ω(Γ 1 ) 1 ( 1) Γ 1,Γ 2 Γ1,Γ 2 Ω(Γ 2,t) u Most general nonprimitive splits: Γ (MΓ 1, NΓ 2 )??

35 Example: D6-D0 bound states in large B-field For B Re t sufficiently large: D6 - N D0 BPS bound states exist as halos: D D Γ Z D6 D0 (u; t) := n Ω(D6+nD0, t) u n = k (1 ( u) k ) kχ(x ) (product is over halos of D0-particles of D0-charge k.)

36 D2-D0 halos and D6-DT-GV correspondence 10 Γ Γ Similarly: D2D0 halos around D6- cores. MS lines asymptoting to vertical lines B q D0 /q D2, J. only in suitable B limit identification: Z D6D2D0 = Z DT = Z GV Genus r = 0 part of Z GV counts halo states, genus r > 0 core states. Asymmetry under inversion D0-charge of r = 0 part due to different signs halo D0 charge stable for different signs B-field.

37 Side remark: scaling solutions multicentered scaling solutions without MS lines, asymptotically indistinguishable from single black hole E.g. multicenter configuration corresponding to three node closed loop quiver. Microscopic index: Ω = 0 ds e s L 1 a 1(s) L 1 b 1 (s) L1 c 1(s) + Polynomial in (a, b, c) in regime without scaling solutions, Exponential in regime with scaling solutions: Ω (abc) 1/3 2 a+b+c why?

38 The OSV conjecture

39 [Warning: 2 π i 1.] The OSV conjecture Defining Z osv (φ) q Ω(p, q) e φ q [Ooguri-Strominger-Vafa] conjectured: with identifications: Z osv (φ) Z top (g top, t) Z top (g top, t) g top = 1 φ 0 + i p 0, ta = φa + i p A φ 0 + i p 0. Inverting: Ω(p, q) dφ e φ q Z top 2 (p, φ). RHS in leading saddle point approx. = e S BH(p,q).

40 Deriving OSV for p 0 = 0: rough outline 1. Identify Z osv = lim β 0 Z D4 (β, C 1, C 3 ). 2. Use SL(2, Z) TST-duality to rewrite Z D4 at t = i as Fareytail/Rademacher series built on polar part Z D4 : Z osv = f (A, φ 0 ) Z D4 (A (φ0, φ)) A SL(2,Z) 3. Polar BPS states split: no attractor point, Ω (Γ ) = 0, so Ω(Γ, t) = ( ) Γ1,Γ2 1 Γ 1, Γ 2 Ω(Γ 1, t ms ) Ω(Γ 2, t ms ) Γ Γ 1 +Γ 2 4. At large P, SL(2, Z) element A = S : φ 0 1/φ 0, and splits into Γ 1 (Γ 2 ) = single (anti-)d6 with dilute D2-D0 gas dominate nonpolar part of Z D4 in Fareytail sum, provided φ 0 ( 1/g top ) not too large (and P large). 5. Dilute gasses fully factorize in suitable regime: Z osv Z D6 Z D6 Z DT Z DT = Z top 2.

41 Pictorial summary D4 S D6 D6 Z top Z top

42 Final result Index of nonpolar charge Γ = (0, P, Q, q 0 ): Ω(Γ; t = i ) = dφ µ(p, φ) e 2πq Λφ Λ e F ɛ (P,φ)+δF, where, with substitutions g 2π φ 0, t A 1 φ 0 (φ A + i PA 2 ): µ(p, φ) = 4π ɛ g 2 e K (g,t, t) F ɛ (P, φ) = F ɛ top(g, t) + F ɛ top(g, t) δf = O(e ɛgp3 ) with F ɛ top topological string free energy cut off by taking only DT invariants N DT (β, n) with β P < ɛp 3, n < ɛp 3, and K ɛ Kähler potential derived from this. Must take ɛ < O(P 1 ) for factorization and DT id., so error δf e gp2 e (Im t)2 /g

43 Range of validity Unless freaky cancelations of contributions to indices occur, we find restriction g > O(1) i.e. strong topological string coupling! Equivalently (as g saddle q 0 /P 3 ): q 0 > O(P 3 ). Technical reason: only when g > g crit O(1) are non-factorizable terms in fareytail series sufficiently suppressed (entropy overwhelms Boltzmann suppression phase transition). Not artifact of derivation, but related to entropy enigma.

44 The Entropy Enigma For Γ = Λ(0, P, Q, Q 0 ) in large Λ limit, and in background with Im t O(Λ), there always exists two centered D6-anti-D6 type black hole configuration such that S BH,2 := S BH,1 (Γ 1 ) + S BH,1 (Γ 2 ) Λ 3 while leading order OSV prediction is log Ω S BH,1 (Γ) Λ 2. S ~ Λ 2 S ~ Λ 3 [Denef-Moore 07]

45 The Entropy Enigma demystified M-theory uplift [de Boer-(Denef-)El Showk-Messamah-Van den Bleeken 08]: EE = Susy version of [Banks-Douglas-Horowitz-Martinec] thermodynamic instability of Schwarzschild-AdS to localization on sphere.

46 Does this mean OSV is wrong? Does entropy enigma imply that large Λ / weak g ( 1/Λ) OSV conjecture is wrong? No, physically one only really expects the conjecture to be valid at attractor point, i.e. finite t : Ω(Γ; t (Γ))? dφ e φ q Z top 2 (p, φ). [No troubling Λ 3 solutions there.] But: interpretation lost of counting large volume D-brane ground states; direct microscopic (D-brane) counting beyond reach at this point.

47 Wall crossing for non-dummies

48

49 Kontsevich-Soibelman formula Consider two charges Γ 1 and Γ 2 near a wall of marginal stability. Let, for positive m, n: Ω ± (m, n) := Ω(m Γ 1 + n Γ 2, t ± ) where t ± are moduli immediately on left/right of wall. Let k := Γ 1, Γ 2 and define the maps (symplectomorphisms) T m,n : (x, y) ( x(1 ( 1) mn x m y n ) n, y(1 ( 1) mn x m y n ) m). Then the KS wall crossing formula states m/n T kω+(m,n) m,n = m/n Example: k = 1: T 10 T 01 = T 01 T 11 T 10. T kω (m,n) m,n. KS reproduces and extends (1,n) w.c. formula to (m,n)!