THE 4D/5D CONNECTION BLACK HOLES and HIGHER-DERIVATIVE COUPLINGS

T I U THE 4D/5D CONNECTION BLACK HOLES and HIGHER-DERIVATIVE COUPLINGS Mathematics and Applications of Branes in String and M-Theory Bernard de Wit Newton Institute, Cambridge Nikhef Amsterdam 14 March 2012 Utrecht University S A R N O S S O L S T U L L I Æ I T I

Motivation: What is the relation between: microscopic/statistical entropy macroscopic/field-theoretic entropy microstate counting entropy S micro = ln d(q, p) supergravity: Noether surface charge Wald, 1993 first law of black hole mechanics (BH thermodynamics) A reliable comparison requires a (super)symmetric setting! Ideal testing ground: supergravities with 8 supercharges D=4 space-time dimensions with N=2 supersymmetry D=5 space-time dimensions with N=1 supersymmetry

To clarify previous results on 4D and 5D black hole solutions, to further the understanding between them, and to obtain new results. Higher-derivative terms (4D and 5D) Chern-Simons terms (5D) Bergshoeff, de Roo, dw, 1981 Cardoso, dw, Mohaupt, 1998 Hanaki, Ohashi, Tachikawa, 2007 dw, Katmadas, van Zalk, 2009, 2010 5D black holes (and black rings) Castro, Davis, Kraus, Larsen, 2007, 2008 dw, Katmadas, 2009 the 4D/5D connection Gaiotto, Strominger, Yin, 2005 Behrndt, Cardoso, Mahapatra, 2005 more recent developments Sen, 2011 Banerjee, dw, Katmadas, 2011

Methodology: superconformal multiplet calculus off-shell irreducible supermultiplets in superconformal gravity background extra superconformal gauge invariances gauge equivalence (compensating supermultiplets) 4D dw, van Holten, Van Proeyen, et al., 1980-85 5D Kugo, Ohashi, 2000 Bergshoeff, Vandoren, Van Proeyen, et al., 2001-04 Fujita, Ohashi, 2001 Hanaki, Ohashi, Tachikawa, 2006 6D/5D off-shell connection

BPS black holes in four space-time dimensions N=2 supergravity: vector multiplet sector (Wilsonian effective action) vector multiplets contain scalars X I projectively defined: X I Y I (residual scale invariance) Lagrangian encoded in a holomorphic homogeneous function F (λy ) = λ 2 F (Y ) with near-horizon geometry: AdS 2 S 2

Subleading corrections to Bekenstein-Hawking area law: extend with one extra complex field, originating from pure supergravity Υ F (Y ) F (Y, Υ) Weyl background homogeneity: F (λy, λ 2 Υ) = λ 2 F (Y, Υ) Υ -dependence leads to terms (R µνρ σ ) 2 in effective action BPS: supersymmetry at the horizon Y I Ȳ I = ip I magnetic charges F I F Ferrara, Kallosh, Strominger, 1996 I = iq I electric charges Cardoso, dw, Käppeli, Mohaupt, 2000 covariant under dualities! refers to the full function F! Furthermore Υ= 64 leads to subleading corrections

The Wilsonian effective action is not sufficient to realize all dualities. When integrating out the massless modes, so as to obtain the 1PI action, one encounters non-holomorphic corrections. For the BPS near-horizon region one has a natural infrared cut-off provided by the S 2 radius. In Minkowski space-time the integration over massless modes is problematic! It is possible to determine the short-distance corrections that depend logarithmically on the S 2 radius. Sen, 2011 Surprisingly enough these are consistent with the terms found from a study of degeneracies of BPS states of D-branes on compact Calabi-Yau manifolds. Denef, Moore, 2007 There is another issue related to the coupling to Gauss-Bonnet terms, which has been used to obtain the entropy for heterotic black holes. However, their supersymmetric form is not known. Sen, 2005

BPS black holes and rings in five space-time dimensions two different supersymmetric horizon topologies! S 3 (SPINNING) BLACK HOLE S 1 S 2 BLACK RING Breckenridge, Myers, Peet, Vafa, 1996 Elvang, Emparan, Mateos, Reall, 2004 with near-horizon geometry: AdS 2 S 3 or AdS 3 S 2 (this result does not depend on the details of the Lagrangian)

5D scalar fields: vector supermultiplets contain vector fields: σ I I W µ abelian field strengths F µν I = µ W ν I ν W µ I supergravity (Weyl) multiplet contains (auxiliary) tensor v 2 (T 01 ) 2 + (T 23 ) 2 T µν T ab T 01 T 23

supersymmetry + partial gauge choice σ I = constant (remain subject to residual (constant) scale transformations!) T µν conformal Killing-Yano tensor ds 2 = 1 16 v 2 D ρ T µν = 1 2 g ρ[µ ξ ν] ξ µ T µν = 0 Killing vector associated with the fifth dimension ψ ξ µ e 1 ε µνρστ T νρ T στ r 2 dt 2 + dr2 r 2 + dθ 2 + sin 2 θ dϕ 2 σ = 1 4 v 2 e g T 23 r dt T 01 cos θ dϕ AdS 2 S 2 + e 2g dψ + σ 2

Two distinct cases: T 01 = 0 SPINNING BLACK HOLE Breckenridge, Myers, Peet, Vafa, 1996 angular momentum T 23 T 01 T 01 = 0 BLACK RING Elvang, Emparan, Mateos, Reall, 2004 Additional horizon condition and magnetic charges F I µν = 4 σ I T µν Q µν = µ σ ν ν σ µ F θϕ p I = σi 4 v 2 T 23 Q θϕ p 0 = e g 4 v 2 T 01 attractor equations

Scale invariance (residue of conformal invariance) σ I, T ab, v, e g scale uniformly the metric is scale dependent Action in 5 space-time dimensions consists of two cubic invariants, each containing a Chern-Simons term: L C IJK ε µνρστ W µ I F νρ J F στ K L c I ε µνρστ W µ I R νρ ab R στ ab Hanaki, Ohashi, Tachikawa, 2006 dw,katmadas, 2009 The Chern-Simons terms cause non-trivial complications in the determination of entropy, electric charges and angular momenta!

An example: 5D electromagnetism with CS term L total = L inv (F µν, ρ F µν,ψ, µ ψ)+ε µνρστ A µ F νρ F στ The Noether potential associated with the abelian gauge symmetry takes the form Q µν gauge(φ, ξ) =2L µν F ξ 2 ρl ρ,µν F ξ +6e 1 ε µνρστ ξa ρ F στ where δl inv = L µν F δf µν + L ρ,µν F local gauge parameter δ( ρ F µν )+L ψ δψ + L µ ψ δ( µψ) ν Q µν gauge = J µ Noether =0 definition for δ ξ φ =0

Q gauge is a closed (d-2)-form for symmetric configurations! Electric charge is defined as q = Σ hor ε µν Q µν gauge(φ, ξ) bi-normal (This definition coincides with the definition based on the field equations! ) The electric charge now contains the integral over a 3-cycle of the CS term! This poses no difficulty for black holes for which the gauge fields are globally defined. However, the mixed CS term leads to the integral over a gravitational CS term, which is problematic. For black rings the gauge fields are not globally defined. Both CS terms are therefore problematic.

Entropy and angular momentum Entropy (based on first law of black hole mechanics S macro = π ε µν Q µν (ξ) Σ hor ξ µ µ = / t timelike Killing vector [µ ξ ν] =ε µν ; ξ µ =0 bi-normal Wald, 1993 Angular momenta J(ξ) = ε µν Q µν (ξ) Σ hor ξ µ periodic Killing vector

Evaluating the CS terms for black rings The correct evaluation of the CS term for the ring geometry yield Q CS I Q CS I C IJK W J F K C IJK a J p K Σ Hence, integer shifts of the Wilson line moduli induce a shift in the integrated CS term For concentric rings, one finds with 6 C IJK P J P K = 12 C IJK (a J + 1 2 pj ) i p K i P I = i p J i i The integrated CS terms are not additive! Hanaki, Ohashi, Tachikawa, 2007 Confirmed by explicit results for global solutions. Gauntlett, Gutowski, 2004 Additive charges take the following form (upon solving the Wilson line moduli in terms of the charges) q I 6 C IJK p J p K

5D black ring versus 4D black hole: S BR macro = 4π φ 0 C IJK p I p J p K + 14 c Ip I q I = 12 C IJK p J a K J ϕ = p I (q I 1 6 C IJp J ) J ψ J ϕ 1 24 CIJ (q I 6C IK p K )(q J 6C JL p L )= 2 φ 02 D = 5 C IJK p I p J p K + 14 c Ip I up to calibration S BH 4D = 2π φ 0 D IJK p I p J p K + 256 d I p I q I 4D = 6 φ 0 D IJK p J φ K ˆq 0 4D q 0 4D + 1 12 DIJ q I q J = 1 φ 02 D IJK p I p J p K + 256 d I p I D = 4

COMMENTS: Confirmation from near-horizon analysis in the presence of higher-derivative couplings. Partial results were already known (but somewhat disputed at the time). Bena, Kraus, etc The Wilson line moduli are defined up to integers. This implies that the electric charges and angular momenta are shifted under the large gauge transformations (spectral flow) induced by these integer shifts. Indeed, under a I a I + k I one finds, q I J ϕ J ψ q I 12 C IJK p J k K J ϕ 12 C IJK p I p J k K Bena, Kraus,Warner, Cheng, de Boer, etc J ψ q I k I 6 C IJK p I p J k K +6C IJK p I k J k K These transformations are in agreement with the corresponding 4D black holes where the above transformations correspond to a duality invariance! 4D/5D connection: difference resides in the contributions from the Chern-Simons term!

The spinning black hole S BH macro = π eg 4 v 2 CIJK σ I σ J σ K + 4 c I σ I T 23 2 q I = 6 eg 4 T 01 CIJK σ J σ K c I T 01 2 p 0 = e g 4 v 2 T 01 J ψ = T 23 e 2g T 01 2 choose scale invariant variables CIJK σ I σ J σ K 4 c I σ I T 01 2 φ I = σi 4T 01 φ 0 = e g T 23 4v 2 = p0 T 23 T 01

S BH macro = q I = J ψ = 4πp 0 p 02 C IJK φ I φ J φ K + 1 (φ 02 + p 02 ) 2 4 c Iφ I φ 02 6 p 0 C IJK φ J φ K 1 D = 5 φ 02 + p 02 16 c I 4φ 0 p 0 C IJK φ I φ J φ K 1 (φ 02 + p 02 ) 2 4 c Iφ I up to calibration relative factor 4 3 S BH 4D = q I 4D = q 0 4D = 2πp 0 p 02 D IJK φ I φ J φ K + 256 d I φ I φ 02 (φ 02 + p 02 ) 2 3 p0 D IJK φ J φ K 256 φ 02 + p 02 3 d I D = 4 2φ 0 p 0 D IJK φ I φ J φ K 256 d I φ I (φ 02 + p 02 ) 2 agrees!

Comments: Agrees with microstate counting for J ψ = 0 Two important differences with the literature! Vafa, 1997 Huang, Klemm, Mariño, Tavanfar, 2007 Castro, Davis, Kraus, Larsen, 2007,2008 The expression for J ψ is rather different! We have J ψ = q 0. The electric charges receives different corrections from the higher-order derivative couplings: 3 c I 3 φ02 c I identical when J ψ =0. p 02 It turns out that the discrepancy between 4D/5D results resides in the mixed Chern Simons term. Banerjee, dw, Katmadas, 2011 To exclude the possibility that contributions from other terms could be present one has to make a detailed comparison of 4D and 5D supergravity. the off-shell 4D/5D connection

The off-shell 4D/5D connection also enables one to understand why the field equations of the 5D and the 4D theory are different (in the bulk), as has been observed in the literature. Upon dimensional reduction the invariant Lagrangians with at most two derivatives lead to the corresponding 4D Lagrangians. There will be no qualitative differences. However, the higher-order derivative term in 5D leads upon reduction to three separate 4D invariant Lagrangians. One of these Lagrangians is known to give no contribution to the entropy and the electric charges of 4D BPS black holes, because of a non-renormalization theorem. dw, Katmadas, van Zalk, 2010

Off-shell dimensional reduction Weyl multiplet, vector multiplet, hypermultiplet First Kaluza-Klein ansatz with gauge choices: e A M = e µ a B µ φ 1, e M A = 0 φ 1 e a µ e ν a B ν, b M = 0 φ b µ 0 There are compensating transformations to preserve this form. Upon the reduction the 5D Weyl multiplet decomposes into the 4D Weyl multiplet and a Kaluza-Klein vector multiplet. Continue with the other fields and then consider the action of the five-dimensional supersymmetry transformations. In this case, there is no conflict between conformal invariance and dimensional reduction. Banerjee, dw, Katmadas, 2011

The supersymmetry transformations turn out to satisfy a uniform decomposition: δ Q () reduced Φ=δ 5D Q () 4D Φ+δ S ( η) 4D Φ+δ SU(2) ( Λ) 4D Φ+δ U(1) ( Λ 0 ) 4D Φ. The Kaluza-Klein vector multiplet has a real scalar field, rather than a complex one. Moreover, the R-symmetry remains restricted to SU(2) rather than extending to SU(2) U(1)! Explanation: It turns out the the result takes the form of a gauge-fixed version with respect to! U(1) compensating transformation

Weyl multiplet 5D 4D (reducible) φ = 2 X 0 B µ = W µ 0 KK vector multiplet missing pseudoscalar V Mi j = Vµi j = V µ j i 1 4 ε ik Y kj 0 X 0 2 W µ 0 V 5i j = 1 4 ε ik Y kj 0 X 0 2 T a5 = 1 12 ie µ D µ X 0 a X 0 D X µ 0 X 0 T ab = i 24 X 0 ε ij T ab ij X0 F ab (B) +h.c. gauge fixed formulation w.r.t. U(1)

vector multiplet 5D 4D σ = i X 0 (t t) W M = Wµ = W µ 1 2 (t + t) W µ 0 W 5 = 1 2 (t + t) Y ij = 1 2 Y ij + 1 4 (t + t) Y ij 0 t = X X 0

5D vector multiplet action 8π 2 L vvv = 3EC ABC σ A 1 2 D M σ B D M σ C + 1 4 F MN B F MNC Y ij B Y ijc 3 σ B F MN C T MN 1 8 i C ABCε MNP P QR W A M F B C NP F QR EC ABC σ A σ B σ C 1 8 R 4 D 39 2 T AB T AB 5D higher-derivative action 8π 2 L vww = 1 4 Ec AY A ij T CD R j CDk (V ) ε ki +Ec A σ A 1 64 R CD EF (M) R CD EF (M)+ 1 96 R MNj i (V ) R MN i j (V ) 1 128 iεmnpqr c A W M A R NP CD (M) R QRCD (M)+ 1 3 R NPj i (V ) R QRi j (V ) + 3 16 Ec A 10 σ A T BC F BC A R(M) DE BC T DE +

yields the following 4D actions: F (X, (T ab ij ε ij ) 2 )= 1 2 C ABC X A X B X C X 0 1 2048 corresponding to the chiral superspace density invariant c A X A X 0 (T ab ij ε ij ) 2 H(X, X) = 1 384 ic A X A X 0 ln X 0 X A X 0 ln X0 corresponding to a full superspace density invariant, described in terms of a Kähler potential. This class of actions does not contribute to the electric charges and the entropy of BPS black holes. dw, Katmadas, van Zalk, 2010

e 1 L = H IJ K L 1 4 + H IJ K +H I K Supersymmetric 4D Lagrangian arising from dimensional reduction of five-dimensional supergravity: F abi F ab J 1 2 Y ij I Y ijj F abk + F +ab L 1 2 Y K ij Y L ij +4 D a X I D b XK D a X J D b XL +2F ac J F + b c L 1 4 ηab Yij J Y L ij 4 D a X I D a X J D 2 XK F ab I F J ab 1 2 Y ij I Y Jij ) c X K + 1 8 F K ab +8 D a X I F J ab 4 c XI + 1 8 F + I ab Dc F + cb K 1 2 D c X K T ij cb ε ij Da X I Y J ij D a Y K ij +h.c. T ab ijε ij c X K + 1 8 F K ab T abij ε ij +4D 2 X I D 2 XK +8 D a F abi D c F +c b K D a Y ij I D a Y K ij + 1 4 T ab ij T cdij F ab I F +cd K + 1 6 R(ω, e)+2d Y ij I Y ij K +4F ac I F + bc K R(ω, e) a b ε ik Y ij I F +ab K R(V) ab j k +h.c. εij +h.c. T abij ε ij D c XK D c T ij ab F I ab +4T ij cb D a F I ab +8 R µν (ω, e) 1 3 gµν R(ω, e)+ 1 4 T µ b ij T νb ij +ir(a) µν g µν D D µ X I D ν XK

Surprise: there is one more higher-derivative coupling emerging from 5D! Some characteristic terms: 8π 2 L vww 1 384 ic At A 2 3 R abr ab + R(V) +i ab j R(V) +abj i 1 768 ic A(t A t A )(X 0 ) 1 ε ij T cdij R(M) ab cd F 0 ab +h.c. where R µν is the Ricci tensor Its structure is not yet known. Neither is it known whether this coupling is subject to some non-renormalization theorem. It will probably have some bearing on the supersymmetric Gauss-Bonnet invariant.

Conclusions/comments Off-shell dimensional reduction enables one to get a precise comparison, also in the presence of higher-order invariants, between complicated Lagrangians in 4 and 5 dimensions. The known higher-derivative invariant coupling in 5D gives rise to three independent higher-derivative couplings in 4D. The indication is that only one of them contributes to electric charges and entropy of BPS black holes. One is not yet explicitly known, and seems to be related to a supersymmetric coupling to the Gauss-Bonnet term. The difference in the near-horizon behavior between 4- and 5-dimensional solutions is exclusively related to non-trivial aspects of Chern-Simons terms. The off-shell dimensional reduction method can be used in many more situations. For instance, in performing the c-map of the topological string.