Black Hole Partition Functions and Duality Gabriel Lopes Cardoso April 8, 2009 Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 1 / 14
Summary of Talk The OSV conjecture (2004) states that the microstates of any N = 2 , D = 4 BPS black hole are captured by the topological string: Z BH = | Z top | 2 (1) However: Duality invariance/covariance not manifest. Black hole degeneracies are sometimes captured by genus 2 Siegel modular forms. Complicated objects, do not lead to (1). Need to change the OSV relation into Z BH ∝ | Z top | 2 to make it compatible with duality invariance. Will encounter non-holomorphic deformation of special geometry. Relation LEEA ↔ topological string amplitudes more subtle than previously envisioned. Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 2 / 14
OSV Proposal Ooguri + Strominger + Vafa, hep-th/0405146 Let’s consider BPS black holes in four-dimensional N = 2 supergravity theories, dyonic, with electric/magnetic charges ( q , p ) , single-center Define a mixed black hole partition function Z BH in terms of black hole microstate degeneracies d ( p , q ) (a suitable index), � d ( p , q ) e π q φ d φ Z BH ( p , φ ) e − π q φ Z BH ( p , φ ) = d ( p , q ) = � ILPT − → q Here, φ are the electrostatic potentials. OSV proposal: Z BH ≡ e 4 π Im F top , with F top topological free energy. If true, � d φ e π [ 4 Im F top − q φ ] , d ( p , q ) = universal formula in terms of topological string data. Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 3 / 14
Duality Symmetries Weak topological string coupling g top : ∞ g 2 g − 2 F top ( g top , z A ) = F g ( z A ) � , holomorphic top g = 0 IIA: z A Kähler class moduli of Calabi-Yau threefold. F g ’s enter in the Wilsonian action as follows: metric on Kähler class moduli space is computed from F 0 higher F g ’s ( g ≥ 1) are coupling functions for higher-curvature terms proportional to the square of the Weyl tensor. N = 2 theory may have duality symmetries. Duality invariance requires the F g ’s ( g ≥ 1) to acquire non-holomorphic corrections: needed in the LEEA to make symmetries of the theory manifest; encoded in the holomorphic anomaly equations of the topological string. Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 4 / 14
This Talk Work at weak topological string coupling g top . Describe a method to include non-holomorphic corrections into the OSV proposal, necessary for duality covariance. Suggests consistent non-holomorphic deformation of special geometry. Departure from topological string. Use saddle-point arguments to infer measure factor in OSV integral. Confront with proposal for microstate degeneracy in a specific N = 2 model, the S-T-U model. A. Sen + C. Vafa, hep-th/9508064 , J. David, arXiv:0711.1971 Agreement! With Justin David, Bernard de Wit and Swapna Mahapatra, arXiv:0810.1233 Bernard de Wit and Swapna Mahapatra, arXiv:0808.2627 Bernard de Wit, Jürg Käppeli and Thomas Mohaupt, hep-th/0601108 . Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 5 / 14
BPS Black Holes BPS black holes in D = 4 , N = 2: extremal, supported by (VM) complex scalar fields Y I ( I = 0 , . . . , n ), charges ( p I , q I ) . Calabi-Yau compactifications: Wilsonian Lagrangian contains higher-curvature interactions ∝ Weyl 2 → encoded in holomorphic homogeneous function F ( Y , Υ) . Here Υ is the Weyl background. g = 0 ( Y 0 ) 2 − 2 g Υ g F g − F ( Y , Υ) = � ∞ Υ -expansion → topol. string Attractor mechanism: Ferrara, Kallosh, Strominger Y I → Y I Hor ( p , q ) at horizon , Υ → − 64 Attractor equations (in the presence of Weyl 2 ): Y I − ¯ I i p I F I = ∂ F ( Y , Υ) /∂ Y I ¯ Y = , magnetic , F I − ¯ F ¯ i q I F Υ = ∂ F ( Y , Υ) /∂ Υ = , electric , I Y I + ¯ Y ¯ I F I + ¯ F ¯ Electro/magnetostatic potentials: , I Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 6 / 14
Variational Principle Attractor equations can be obtained from a variational principle, based on a BPS entropy function Σ : Υ) − q I ( Y I + ¯ ¯ I ) + p I ( F I + ¯ Σ( Y , ¯ Y , p , q ) = F ( Y , ¯ Y , Υ , ¯ Y F ¯ I ) , where F is the free energy � ¯ I F I − Y I ¯ � F ( Y , ¯ Y , Υ , ¯ Υ) = − i Y F ¯ − 2 i Υ F Υ − ¯ F ¯ ¯ Υ¯ � � I Υ Stationary points: set Υ = − 64 i ( Y I − ¯ ¯ I − ip I ) δ ( F I + ¯ I − iq I ) δ ( Y I + ¯ ¯ I ) Y F ¯ I ) − i ( F I − ¯ F ¯ Y δ Σ = δ Σ = 0 ← → attractor equations At attractor point, get macroscopic (Wald’s) entropy: π Σ | attractor = S macro ( p , q ) So far, Wilsonian. Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 7 / 14
Duality Transformations: Entanglement with Υ ! Charges ( p I , q I ) undergo duality transformations. If these constitute symmetries of LEEA, macroscopic entropy is invariant under them. These leave Υ invariant, but act as symplectic Sp ( 2 n + 2 , Z ) transformations on the vector ( Y I , F I ( Y , Υ)) in the attractor equations. Entanglement with the Weyl background! Departure from topological string approach. Precise form of N = 2 LEEA not known − → cannot rely on an action principle to incorporate non-holomorphic corrections needed for duality invariance. Instead, demand: attractor equations retain their form; they follow from a variational principle based on free energy F , I F I − Y I ¯ � ¯ � F ( Y , ¯ Y , Υ , ¯ Υ) = − i Y ¯ F ¯ − 2 i Υ F Υ − ¯ Υ¯ F ¯ � � , I Υ but now based on a general function F ( Y , ¯ Y , Υ , ¯ Υ) , not necessarily holomorphic. Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 8 / 14
Non-Holomorphic Corrections Variation of F : with the attractor value Υ = − 64 i ( Y I − ¯ ¯ I ) δ ( F I + ¯ I ) δ ( Y I + ¯ ¯ I ) Y F ¯ I ) − i ( F I − ¯ F ¯ Y δ F = � � 2 Υ δ F Υ + Y I δ F I − F I δ Y I � � i + c . c . − Vanishing of second line: at least two solutions, namely F holomorphic, F ( λ Y , λ 2 Υ 2 ) = λ 2 F ( Y , Υ) , usual Wilsonian case F = 2 i Ω , Ω( λ Y , λ ¯ Y , λ 2 Υ , λ 2 ¯ Υ) = λ 2 Ω( Y , ¯ Y , Υ , ¯ Ω real, Υ) Without loss of generality: F = F ( 0 ) ( Y ) + 2 i Ω( Y , ¯ Y , Υ , ¯ Υ) When Ω is harmonic (i.e. Ω = holo + anti-holo), get back usual Wilsonian case. Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 9 / 14
Consistent Deformation of Special Geometry? Second option seems to be a consistent non-holomorphic deformation of special geometry, e.g.: under symplectic (duality) transformations, ( Y I , F I ) → ( ˜ Y I , ˜ F I ) ; can show that F F I = ∂ ˜ ˜ Y I ; ∂ ˜ ( L , F ) and ( ˜ F ) in same equivalence class; L , ˜ in addition, can show that F Υ − → F Υ , i.e. F Υ transforms as a scalar. It follows that F and Σ are invariant under duality transformations that define a symmetry. Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 10 / 14
Duality Invariant OSV Integral Consider the following duality invariant integral, expressed in terms of entropy function Σ , with the attractor value Υ = − 64, � � d ( Y I + ¯ ¯ I ) d ( F I + ¯ I ) e π Σ( Y , ¯ Y , p , q ) = Y ) e π Σ( Y , ¯ Y , p , q ) Y F ¯ dY d ¯ Y ∆ − ( Y , ¯ Duality covariance requires measure factor ∆ − = | det F JK − F J ¯ � � �� Im | K Evaluate integral in saddle-point approximation about attractor point: e π Σ | attractor = e S macro ( p , q ) for large charges Duality invariant. Expect saddle-point approximation to hold for dyonic black holes. Suggests to identify the above with d ( p , q ) , as in OSV. Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 11 / 14
Prediction for Mixed Partition Function On the other hand, when only integrating over Y I − ¯ Y ¯ I in saddle-point Y I = ( φ I + ip I ) / 2, approximation so that get modified OSV-type integral, � ∆ − ( p , φ ) e π [ F E ( p ,φ ) − q I φ I ] , d ( p , q ) = d φ � where Im F ( Y , ¯ Y , Υ , ¯ Υ) − Ω( Y , ¯ Y , Υ , ¯ � � F E = 4 Υ) | Y I =( φ I + ip I ) / 2 , Υ= − 64 Inverting yields prediction for N = 2 mixed black hole partition function, √ √ d ( p , q ) e π q I φ I = ∆ − e 4 π Ω nonholo e π F holo Z BH ( p , φ ) = ∆ − e π F E = � E q Test requires knowledge of microscopic degeneracies. Gabriel Lopes Cardoso (LMU) Black Hole Partition Functions and Duality GGI, April 8, 2009 12 / 14
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