BPS Black Holes Effective Actions and the Topological String ISM08 Indian Strings Meeting Pondicherry, 6-13 December 2008 Bernard de Wit Utrecht University
1: N=2 BPS black holes effective action attractor phenomena entropy function & free energy subleading & non-holomorphic corrections 2: Partition functions and OSV mixed partition function topological string versus the effective action 3: Non-holomorphic deformation equivalence classes 4: Summary / conclusion
N=2 BPS black holes N=2 supergravity: vector multiplet sector vector multiplets scalars X I (Wilsonian effective action) projectively defined: X I Y I Lagrangian encoded in a holomorphic homogeneous function BPS : attractor phenomena F (λy ) = λ 2 F (Y ) full supersymmetry enhancement at the horizon extremal versus extremal non-supersymmetric black holes We don t have to work in terms of the (complicated) effective actions!
Attractor equations (horizon behaviour) Y I Ȳ I = ip I F I F I = iq I magnetic charges electric charges Ferrara, Kallosh, Strominger, 1996 Cardoso, dw, Käppeli, Mohaupt, 2000 homogeneity: entropy and area are proportional to Q 2 R hor g s Q large Q l s and consistent with E/M duality duality: equivalence classes or invariances large black hole macroscopic black hole
BPS entropy function Σ(Y, Ȳ, p, q) = F(Y, Ȳ ) q I(Y I + Ȳ I ) + p I (F I + F I ) X + X I and F I + F I play the role of electro- and magnetostatic potentials at the horizon F(Y, Ȳ ) free energy δσ = 0 attractor equations πσ = S macro (p, q) subleading corrections?? det[imf IJ ] 0 entropy, like the area, scales quadratically in the charges
Higher-derivative interactions chiral class: Weyl background homogeneity: F (λy, λ 2 Υ) = λ 2 F (Y, Υ) F (Y ) F (Y, Υ) Weyl background attractor equations remain valid and Υ = 64 Υ dependence induces -terms in the action and subleading corrections in area and entropy R 2 F (Y, Υ) = F (0) (Y ) + g=1(y 0 ) 2 2g Υ g F (g) (t) subleading corrections t A = Y A /Y 0
Free energy : F(Y, Ȳ, Υ, Ῡ) = i ( Ȳ I F I Y I FI ) 2i ( ΥFΥ Ῡ F Υ ) example : F (Y, Υ) = 1 6 C ABC Y A Y B Y C Y 0 c 2A Y A 24 64 Y 0 Υ leads indeed to the microscopic result S macro = 2π Cardoso, dw, Käppeli, Mohaupt, 2006 Cardoso, dw, Mohaupt, 1998 1 6 ˆq 0 (C ABC p A p B p C + c 2A p A ) triple intersection number second Chern class c 2A subleading correction! Maldacena, Strominger, Witten, 1997 Vafa, 1997 membrane charges : ˆq 0 = q 0 1 2 CAB q A q B C AB = C ABC p C p 0 = 0 dictated by symmetry
area/entropy { Q 2{ } 1 + O(Υ/Q 2 ) Q { } Υ 1 + O(Υ/Q 2 ) R hor l s g s Q 1 R hor l s g s Q 1 large/macroscopic small/microscopic elementary string states Tested extensively for N=4 supersymmetric string compactifications (in N=2 formulation)
Problem : Non-holomorphic corrections So far: holomorphicity standard SG Lagrangians Wilsonian effective action (integrated out modes with cutoffs) integrating out massless modes leads to non-local terms holomorphicity is lost! non-holomorphic corrections are required: to realize certain symmetries Dixon, Kaplunovsky, Louis, 1991 background dependence of topological string The full non-local action is not known! Bershadsky, Cecotti, Ooguri, Vafa, 1994
Early example : N=4 supersymmetry with S-duality F = Y 1 Y 0 Y a η ab Y b Cardoso, dw, Mohaupt, 1999 + i [Υ log η 12 (S) + 256 π Ῡ log η12 ( S) + 12(Υ + Ῡ) log(s + S) 6] is = Y 1 Y 0 More general decomposition: F = Y 1 Y Y a η 0 ab Y b + 2iΩ ( ) ( ) Y I p I harmonic related to threshold correction real, homogeneous non-holomorphic required by S-duality Harvey, Moore, 1996 F I transforms as q I under duality rotations (monodromies) This determines the transformation of Ω The function is not invariant! F
Furthermore microscopic results 1/4 BPS states dyonic degeneracies d k (p, q) = Φ k (Ω) k = 10, 6, 4, 2, 1 ( ) ρ υ Ω = υ σ 3 cycle dω eiπ[ρ p2 +σ q 2 +(2υ 1)p q] Dijkgraaf, Verlinde, Verlinde, 1997 period matrix of g=2 Riemann surface Shih, Strominger, Yin, 2005 Jatkar, Sen, 2005 formally S-duality invariant Leading degeneracy for large charges : make saddle-point approximation on a leading divisor The result is identical as that obtained on the basis of : Σ(S, S, p, q) = q2 ip q(s S) + p 2 S 2 S + S + 4 Ω(S, S, Υ, Ῡ) Cardoso, dw, Käppeli, Mohaupt, 2004
Partition functions and OSV Y I = φi + ip I 2 where φ I p I electrostatic potentials (mixed ensemble) magnetic charges reduced entropy functions ] F E (p, φ) = 4 [Im F (Y, Ȳ, Υ, Ῡ) Ω(Y, Ȳ, Υ, Ῡ) Topological string: F (Y, Υ) = F (0) (Y ) + g=1(y 0 ) 2 2g Υ g F (g) (t) Σ = F E q I φ I Y I =(φ I +ip I )/2 (Y 0 ) 2 F (0) (t) Y 0 loop-counting parameter genus-g partition function of a twisted non-linear sigma model with CY target space t A = Y A /Y 0
Z BH (p, φ) = e F E Z top (p, φ) = e 2iF Z BH (p, φ) Z top (p, φ) 2 Strominger, Ooguri, Vafa, 2004 Topological string: Holomorphic anomaly Topological string coupling : Y 0 = g 1 top Duality invariant sections F (g) Bershadsky, Cecotti, Ooguri, Vafa, 1994 tf (g) 0 F (g) (t, t) F (g) captures certain string amplitudes Antoniadis, Gava, Narain, Taylor,1993 Note : identification with the effective action! Non-holomorphic extension? Cardoso, dw, Käppeli, Mohaupt, 2006
The mixed partition function Z BH (p, φ) = {q} d(p, q) e π q I φ I e π F E(p,φ) inverse Laplace transform : d(p, q) dφ e π (F E q I φ I ) = dφ e π Σ(φ,p,q) saddle-point approximation δ(f E q I φ I ) = 0 S macro = π Σ q I = F E φ I integrals ill-defined (contour, convergence) E/M duality problematic
improving : define a duality invariant canonical partition function Z(φ, χ) = d(p, q) e π[q I φ I p I χ I ] {p,q} defines a free energy (naturally formulated as a function of the electro- and magnetostatic potentials and χ ) φ inverse Laplace transform: d(p, q) dχ I dφ I Z(φ, χ) e π[ q I φ I +p I χ I ] over periodicity intervals (φ i, φ + i) (χ i, χ + i)
identify with the field-theoretic data: 2π H(φ/2,χ/2,Υ,Ῡ) Z(φ, χ) e Hesse potential : Legendre transform of Im[F ] with respect to (Y Ȳ )I 1 equal to as defined previously, including the -dependence 2 F Υ {p,q} d(p, q) e π[q I φ I p I χ I ] shifts 2π H(φ/2,χ/2,Υ,Ῡ) e
complex formulation: {p,q} d(p, q) e π[q I (Y +Ȳ )I p I ( ˆF + ˆ F )I ] inverse Laplace transform: d(p, q) shifts π F(Y,Ȳ,Υ,Ῡ) e d(y + Ȳ )I d( ˆF + ˆ F π Σ(Y,Ȳ,p,q) )I e dy dȳ Σ(Y,Ȳ,p,q) (Y, Ȳ ) eπ measure factor : implied by duality! [ ± (Y, Ȳ ) = det Im [2 F KL ± 2 F K L] Cardoso, dw, Käppeli, Mohaupt, 2006
Saddle-point approximation d(p, q) = (Y, Ȳ ) + (Y, Ȳ ) (Y, Ȳ ) + (Y, Ȳ ) attractor 1 mixed partition function : Z(p, φ) = {q} attractor d(p, q) e π q I φ I e S macro(p,q) shifts (semiclassical approximation) required by duality (p, φ) e π F E(p,φ) (and higher-order corrections!) modification of OSV predictive power is lost! These results have been confirmed Shih, Yin, 2005 in a variety of applications, e.g., Cardoso, dw, Käppeli, Mohaupt, 2006 Denef, Moore, 2007 Cardoso, David, de Wit, Mahapatra, 2008
A more subtle question : F (Y, Υ) = F (0) (Y ) + g=1(y 0 ) 2 2g Υ g F (g) (t) this same expansion is applied to (a) the topological string and (b) the effective action! But are they identical functions? NO! use duality arguments : effective action (and still agreement with string amplitudes?) F (g) the are NOT invariant the periods transform correctly under monodromies the duality transformations are Υ-dependent topological string F (g) F (0) the are INVARIANT sections the periods refer to the duality transformations are Υ-independent
difference has been confirmed: F (1) is still invariant for g 2 there are differences explicit evaluation and comparison of the non-holomorphic corrections for the FHSV model supports this conclusion. consistent with the reality of Ω Grimm, Klemm, Marino, Weiss, 2007 Cardoso, dw, Mahapatra, 2008 What then is the precise relation? Recall : amplitudes connected graphs 1PI graphs Then : compare with E/M duality properties of Lagrangian L Hamiltonian H Im[F (Y, Υ)] Hesse potential H Legendre transform Free energy F
Compare : electromagnetism L(E, B) (E H) (B D) under E/M duality D = L E H = L B transformations depend on the details of the Lagrangian (which is not invariant) depend on the details of the Lagrangian H(D, B) invariant under monodromies transformations do not depend on the details of the Lagrangian
Example : Born-Infeld Lagrangian L = g 2 det[η µν + g F µν ] + g 2 spherical ds 2 = dt 2 + dr 2 + r 2 (sin 2 θ dϕ 2 + dθ 2 ) symmetry F rt = e F ϕθ = p sin θ L red = dϕ dθ L [ = 4πr 2 g 2 1 g2 e 2 ] 1 + g 2 p 2 r 2 1 1 g δe = p e 2 1 + g 2 p 2 symmetry: 1 + g δp = e e 2 1 g 2 p 2 (suppress: 4π, r )
define electric charge Hamiltonian q = L red e [ ] H = g 2 1 + g2 (p 2 + q 2 ) 1 symmetry: δq = p δp = q Schrödinger, 1935 independent of the coupling constant g!
Non-holomorphic deformation F F + 2iΩ Special geometry : N=2 supersymmetric gauge theory encoded in function F (X) X I F (X) complex scalar fields and F I = ( ) X I X I period vector Ω F I A I,B J electric/magnetic duality (monodromies): ( ) ( ) ( ) ( ) X I XI U I J Z IJ X J = J F I F I V I W IJ F J integrable : Sp(2n, R) F ( X) 1 X I 2 FI ( X) = F (X) 1 2 XI F I (X) +
with non-holomorphic deformation : The starting point: monodromies so that X I X I = U I JX J + Z IJ F J (X, X) F I (X, X) F I ( X, X) = VI J F J (X, X) + W IJ X J X I X J SI J = U I J + Z IK F KJ As a first result we derive ( F IJ J F I ) : F IJ F IJ = (V I L ˆFLK + W IK ) [Ŝ 1 ] K J where ˆF IJ = K L F IJ F I K Z F LJ Ŝ I J = U I J + Z IK ˆFKJ Z IJ = [S 1 ] I K Z KJ X I X J = ZIK F K J
Assume: F I J = e iα F JI so that ˆF IJ = ˆF JI provided that F IJ = F JI In that case: FIJ = F JI so that both F I and FI can be integrated: F I = F X I One may also derive FI = F X I F I J F I J = [Ŝ 1 ] K I [ S 1 ] L J F K L = [S 1 ] K I [ Ŝ 1 ] L J F K L as well as similar formulae for higher derivatives! Furthermore, with external parameter dependence η : F (X, X; η ) = F (0) (X) + 2iΩ(X, X; η ) one derives η F ( X, X; η) = η F (X, X; η) i.e. transforms as a function!
Summary / conclusions F F + 2iΩ seems consistent with special geometry free energy F duality invariant BPS attractor equations with non-holomorphic terms E/M equivalence classes seem to be realized F transforms as a function confirmed by explicit results for FHSV and STU models prediction for measure factor in class of N=2 models the measure factor for the STU model Justin David s talk Precise relation effective action topological string remains open Many open questions!