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Special Geometry, Black Holes, and Instantons

Thomas Mohaupt

Department of Mathematical SciencesUniversity of Liverpool

Inaugural Workshop on Black Holes in Supergravity andM/Superstring Theory, September 2010

Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 1 / 59

Introduction

Approach to special geometry (here: N = 2 vector multiplets) basedon a combination of old and new ideas

Special coordinates, homogeneity (conformal calculus), as in B.de Wit and A. Van Proeyen (1984).

More recent work in differential geometry (D. Freed, N. Hitchin, V.Cortés), see ‘Special complex manifolds’, D.V. Alekseevsky, V.Cortés and C. Devchand (1999).

In this talk:

Modifications: Euclidean supersymmetry and para-complexgeometry.

Generalizations: non-supersymmetric theories with target spacesencoded by potentials (not necessarily symmetric orhomogeneous spaces).


References

‘Euclidean special geometry’:

Rigid vector multiplets: V. Cortés, C. Mayer, T.M. and F.Saueressig, JHEP 03 (2004) 028.

Rigid hypermultiplets (bosonic sector, only): V. Cortés, C. Mayer,T.M. and F. Saueressig, JHEP 06 (2005) 025.

Local vector multiplets (bosonic sector, only): V. Cortés and T.M.,JHEP 07 (2009) 066.

Applications:

Multi-centered extremal black hole solutions: T.M. and K. Waite,JHEP 10 (2009) 058.

Black hole solutions and instantons. T.M. and K. Waite, inpreparation.

Single-centered non-extremal black hole solutions. T.M. and O.Vaughan, arXiv:1006.3439 and work in progress.


Part I

Special geometry of vector multiplets: modificationsand generalizations


Special real geometry

5d vector multiplets (M. Gunaydin, G. Sierra, P. Townsend (1984)).

Prelude: rigid case

gIJ(σ) =∂2V

∂σI∂σJ ,

Susy: Hesse potential V(σ) is a polynomial of degree three.

Hessian metric: flat, torsion-free connection ∇, such that ∇g iscompletely symmetric. Locally (∇-affine coordinates):

∂IgJK completely symmetric ⇔ ΓIJ|K completely symmetric .


Special real geometry

Local case: scalar manifold is a hypersurface

M = V(σ) = 1 ⊂ M

in Hessian manifold M with metric

gIJ(σ) =∂2V

∂σI∂σJ , V ≃ − log V(σ)

and where prepotential V(σ) is a homogeneous polynomial of degree3.

Generalization (loose Susy): allow V to be a homogeneous function ofarbitrary degree p. M is ‘conical Hessian’.


r-map (dimensional reduction)

Rigid case. Reduce

L ∼ −12

gIJ(σ)∂µσI∂µσJ − 14

gIJ(σ)F IµνF J|µν

over space/time:

L ∼ −12

gIJ(σ)(∂mσI∂mσJ ± ∂mbI∂mbJ) + · · ·

= −12

gIJ(X , X )∂mX I∂mX J + · · ·

where bI = AI∗, and

Space-like: X I = σI + ibI , X I = σI − ibI , i2 = −1Time-like: X I = σI + ebI , X I = σI − ebI , e2 = 1

e = para-complex unit (called ‘hyperbolic complex’ by G. Gibbons, M.Green and M. Perry (1995)).


Para-complex geometry

Almost para-complex structure J = tensor field of type (1, 1) such that

J2 = 1 , with ‘balanced’ eigenvalues .

Integrability, para-Hermitian, para-Kähler etc. can be definedanalogous to complex geometry.

Remark: para-Hermitian metrics are indefinite (split signature).

Additional references: several articles in V. Cortés (ed.), Handbook ofPseudo-Riemannian Geometry and Supersymmetry, EMS (2010).


Euclidean supersymmetry and para-complexstructures

Zumino (1977): Euclidean supersymmetry implies ‘non-compact chiraltransformation’ due to change of R-symmetry group with space-timesignature:

SU(2)R × U(1)R → SU(2)R × SO(1, 1)R

Generator of abelian factor = complex/para-complex structure onvector multiplet scalar manifold.

Special para-Kähler geometry (para-holomorphic prepotential).Rigid case: V. Cortés, C. Mayer, T.M., F. Saueressig (2004)Local case (bosonic sector, only) : V. Cortés, T.M. (2009).


Minkowksi or Euclidean?

δX I = i ǫ+ λI+ ,

δX I = i ǫ− λI− ,

δλiI+ = −1

4γmnF I

−mnǫi+ − i

2∂/X Iǫi

− − Y ij Iǫ+ j ,

δλiI− = −1

4γmnF I

+ mnǫi− − i

2∂/X Iǫi

+ − Y ij Iǫ− j ,

δAIm =

12

(

ǫ+γmλI− + ǫ−γmλI

+

)

,

δY ij I = −12

(

ǫ(i+∂/λ

j) I− + ǫ

(i−∂/λ

j) I+

)

.


Comments

i → e only for those i corresponding to the complex structure ofthe scalar manifold, e.g. σI + ibI → σI + ebI.

To obtain uniform expressions, need to use reality conditions forspinors which appy to both signatures. Here: symplecticMajorana.

‘Chiral projections’ for spinors, and ‘(anti-)selfduality projections’ ofgauge fields involve factors of e. Example:

F I±|mn =

12(F I

mn ± eF Imn)

Geometry: use para-complexified tangent space of scalarmanifold.

Can do with light cone coordinates X I = σI ± bI, but loose analogy.

Complex form.


r-maps with Susy

spatial reduction : special real → special Kähler.M. Gunaydin, G. Sierra, P. Townsend (1984), B. de Wit and A. VanProeyen (1992).

temporal : special real → special para-Kähler.V. Cortés and T.M., JHEP 07 (2009) 058, and work in progress.


r-maps without Susy

When dropping Susy, we loose the ‘special’ properties in 4d (flatconnection, prepotential), but retain a (para-)Kähler potential.

Hessian → (para-)Kähler (with isometries)

X I = σI + iǫbI, iǫ = i , e. Hesse potential → (para-)Kähler potential:

gIJ =∂2V(σ)

∂σI∂σJ = 4∂2V(X + X )

∂X I∂X J

D. Alekseevsky and V. Cortés, arXiv:0811.1658, T.M. and K. Waite,JHEP 10 (2009) 058, V. Cortés and T.M. work in progress.


Part II

Multi-centered extremal black holes


Some references

Generating stationary solutions via dimensional reduction over time.

G. Neugebauer und D. Kramer (1969)P. Breitenlohner, D. Maison and G. Gibbons (1988)

Branes and higher-dimensional theories:G. Clement and D. Gal’tsov PRD 54 (1996) 6136, D. Gal’tsov and O.A.Rychkov PRD 58 (1998) 122001,E. Cremmer, I. Lavrinenko, H. Lü, C. Pope, K. Stelle and T. Tran, NPB534 (1998) 40, . . .


Some more recent references

Many recent applications, including

M. Gunaydin, A. Neitzke, B. Pioline and A. Waldron, Phys. Rev. D 73(2006) 084019, JHEP 09 (2007) 056, A. Ceresole and G. Dall’Agata,JHEP 03 (2007) 110, G.L. Cardoso, A. Ceresole, G. Dall’Agata, J.M.Oberreuter and J. Perz, JHEP 10 (2007) 063, D. Gaiotto, W. Li and M.Padi, JHEP 12 (2007) 093, J. Perz, P. Smyth, T. Van Riet and B.Vercnocke, JHEP 03 (2009) 150, E. Bergshoeff, W. Chemissany, A.Ploegh, M. Trigiante and T. Van Riet, NPB 812 (2009) 343,Bossard’s talk, . . .

Most work assumes symmetric or homogeneous target spaces.


The 5d theory

Lagrangian

L ≃ 12

R(5) −34

(

aIJ(σ)∂µσI∂µσJ)

V=1− 1

4aIJ(σ)F I

µνF J|µν + · · ·

n gauge fields, n − 1 scalars valued in V(σ) = 1 ⊂ M. Scalar metricis ‘conical Hessian’:

aIJ(σ) = ∂2I,JV(σ) , V(σ) = −1

plog V(σ) ,

V(σ) homogeneous of degree p. (Susy: p = 3.) Goal: find static,purely electric solutions (black holes).


Dimensional reduction over time:

ds2(5) = −e2σ(dt + AKK )2 + e−σds2

(4)

Absorb KK scalar: σI → eσσI . Reduced Lagrangian

L ≃ 12

R(4) − aIJ(σ)(∂mσI∂mσJ − ∂mbI∂mbJ) + · · ·

2n independent real scalar fields σI , bI = AI0.

X I = σI + ebI: Hesse potential → Para-Kähler potential:

aIJ(σ) =∂2V(σ)

∂σI∂σJ = 4∂V(X + X )

∂X I∂X J

‘Special’ for p = 3. Need homogeneity (scaling properties. Not:homogeneous space).


4d equations of motion

1√

|h|∂m(

√

|h|aIJ(σ)∂mσJ) − 12

∂IaJK (∂mσI∂mσJ − ∂mbI∂mbJ) = 0

∂m(√

|h|aIJ(σ)∂mbJ) = 014

aIJ(σ)(∂mσI∂nσJ − ∂mbI∂nbJ) =

16

Rmn

Drastic simplification when imposing ‘extremality’ ∂mσI = ±∂mbI.Rmn = 0 ⇒ can take hmn = δmn.Remaining field equations:

∂m(aIJ(σ)∂mσJ) = 0 .


Extremality

What does ∂mσI = ±∂mbI mean?For ds2

(4) = δmndxmdxn, the lifted line element corresponds to anextremal black hole:

ds2(5) = −e2σdt2 + e−σds2

(4) , where epσ = V(σ).

(We will discuss regularity of eσ later. )Susy case (p = 3). Euclidean BPS condition for purely scalar fieldconfigurations. (Lifts to 5d BPS condition).When dualizing axions bI into tensors Bmn|I,

Hmnp|I = 3!∂[mBnp]I = 3!aIJǫmnpq∂qbJ

then the dual, positive definite Euclidean action

L ≃ 12

aIJ(σ)∂mσI∂mσJ +1

2 · 3!aIJ(σ)Hmnp|IH

mnpJ

has a genuine Bogomol’nyi bound saturated by

∂mσI = ±ǫmnpqaIJ∂nBpqJ

(Observation frequently used for axionic theories.)Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 20 / 59

Axions vs Tensors

Wormholes: S.B. Giddings and A. Strominger, NPB 306 (1988)890, C.P. Burgess and A. Kshirsagar, NPB 324 (1989) 157, S.Coleman and K. Lee, NPB 329 (1990) 387, . . .

type-IIB D-instanton: G.W. Gibbons, M.B. Green and M.J. Perry,PLB 370 (1996) 37, M.B. Green and M. Gutperle, NPB 498 (1997)195.

Hypermultiplet instantons: M. Gutperle and M. Spalinski, JHEP 06(2000) 037, NPB 598 (2001) 509, U. Theis and S. Vandoren,JHEP 09 (2002) 059, M. Davidse, U. Theis and S. Vandoren, NPB697 (2004) 48, . . .


Extremality (cont’d)

What does ∂mσI = ±∂mbI mean?

Geometrically, solution is parallel to the eigendirections of thepara-complex structure J.


Harmonic maps

Equations of motion of non-linear sigma model on E with target spaceN ⇔ Harmonic map

Φ : E → N .

Construct solutions using totally geodesic submanifolds S

Φ : E → S ⊂ N

Extremal solutions ⇔ E flat, S isotropic submanifold.

P. Breitenlohner, D. Maison and G. Gibbons (1988), . . .


Using para-complex geometry

N is para-Kähler, with para-complex structure J, S is defined byeigendirections of para-complex structure.

Eigendirections define a submanifold (J is integrable).

S is totally geodesic (J is parallel).

S is totally isotropic (J is an anti-isometry).

S is flat wrt pulled-back Levi-Civita connection of N (algebraicstructure of Riemann tensor). Therefore

Φ : E → S ⊂ N

is a harmonic map between flat manifolds and solutions can beexpressed in terms of harmonic functions.


Harmonic functions

Remaining field equations:

∂m(aIJ(σ)∂mσJ) = 0

Define ‘dual scalars’:

σI := ∂IV(σ) ⇒ ∂mσI = aIJ(σ)∂mσj

so that field equations become

∆σI = 0 .

Solution given by n harmonic functions HI(x)! Can be multi-centered.Solving σI = ∂IV = HI for the σI is equivalent to solving the black holeattractor equations.


Lifting to 5d

By lifting to 5d we recover (for p = 3) and generalize (for p 6= 3) theresults of A. Chamseddine and W. Sabra PLB 426 (1998) 36, PLB 460(1990) 63.

Attractor equations (‘global’ form), setting σI = eσhI:

σI = HI ⇒ e−σ ∂V(h)

∂hI = HI

determine 5d metric and scalars in terms of n harmonic functions.Asymptotics at a center located at r = 0:

HI ≈qI

r2 , e−σ ≈ Zr2

Attractor equations (fixed point):

Z∂V(h)

∂hI

∣

∣

∣

∣

∣

∗

= qI


Z generalizes the central charge: Z = 1p qIhI

∗.

Mass:

MADM =32

∮

d3Σme−σ∂mσ = σI(∞)qI ,

Entropy:

SBH =π2

2Z 3/2∗


Part III

Black holes and instantons


Instantons?

Expect correspondence between 0-branes (black holes/solitons) and(−1)-branes (instantons).I.p. Electric charge ≃ axionic charge, Mass (Tension) ≃ Action.

But can the Euclidean solutions we used to generate black holesolutions be regarded as instantons?Consistent saddle point approximation requires (i) finite positiveinstanton action, (ii) damped Gaussian around the saddle point.Problems: (i) Instanton action is zero, (ii) Action functional is indefinite(damping?)


Euclidean actions for axions

Should axions be ‘Wick rotated’, aI → bI = iaI?

Euclidean supersymmetry: B. Zumino PLB 69 (1977) 369, P. VanNieuwenhuizen and A. Waldron, PLB 389 (1996) 29, . . .

Wormholes: C.P. Burgess and A. Kshirsagar, NPB 324 (1989)157, S. Coleman and K. Lee, NPB 329 (1990) 387, . . .

D-instanton: G.W. Gibbons, M.B. Green and M.J. Perry, PLB 370(1996) 37, M.B. Green and M. Gutperle, NPB 498 (1997) 195, . . .

Complex actions and reality conditions: E. Bergshoeff, J. Hartong,A. Ploegh, J. Rosseel and D. Van den Bleecken, JHEP 07 (2007)067, T.M. and K. Waite, JHEP 10 (2009) 058 and to appear.


Indefinite Euclidean action

Indefinite Euclidean action (obtained by dimensional reduction overtime or modified Wick rotation)

L ≃ aIJ(σ)(∂mσI∂mσJ − ∂mbI∂mbJ)

Instanton candidate ∂mσI = ±∂mbI.

‘Instanton’ is a solution of the field equations.

‘Instanton’ lifts to ‘soliton’.

Problem: zero instanton action.

Problem: action functional is indefinite.


Definite Euclidean action

Definite Euclidean action (obtained by standard Wick rotation)

L ≃ aIJ(σ)(∂mσI∂mσJ + ∂maI∂maJ)

Instanton candidate ∂mσI = ±i∂maI.

Problem: ‘instanton’ is not a solution (Derrick’s theorem),

Problem: zero instanton action.

At least, this action functional is positive definite.


Amplitude calculation

Semi-classical evaluation of transition amplitude, adapting calculationsof S. Coleman and K. Lee, NPB 329 (1990) 387. Hypermultiplets: M.Chiodaroli and M. Gutperle, PRD 79 (2009) 085023.

Amplitude (schematically):

A ≃∫

Da e−Sdef[a]−Sbd[a]

Note boundary term.Consistent saddle point approximation:

a = ib∗ + a

Imaginary saddle point, real fluctuation.

A ≃ e−Sbd[ib∗]

∫

Da e−Sdef[a]

Analogy: one-dimensional real integral dominated by complex saddlepoint.


Observation: can re-write, using ‘rotated’ variable (a = ib):

A ≃∫

Db e−Sindef[b]−Sbd[b]

Consistent saddle point approximation:

a = ib∗ + a ⇔ b = b∗ + b = b∗ − i a

Real saddle point, imaginary fluctuation.

Same physical amplitude expressed in different variables.Consistency of saddle point approximation.

Can’t avoid to consider complex field configurations.


Instanton action and ADM mass

Investigate relation between mass of 0-brane (black hole) and action of(-1)-brane (instanton).Instanton action (boundary term)

Sinst = iaI∗(∞)qI = bI

∗(∞)qI .

Axionic shift symmetry broken to discrete subgroup aI → aI + 2πZ.ADM mass

MADM = σI∗(∞)qI

Extremality condition ∂mσI = ±∂mbI does not fix constant part of bI.Instanton action = ADM mass if σI(∞) = bI(∞) (up to discrete shifts).Global shift symmetry of axions vs local gauge symmetries of vectorfields.


Axions vs tensors, quantum

Similar observation when comparing instanton solutions of axionictheory and dual tensor field theory:

Saxioninst = iaI

∗(∞)qI = bI∗(∞)qI .

Stensorinst = σI

∗(∞)qI .

Quantum equivalence ‘up to zero modes’, C.P. Burgess and A.Kshirsagar, NPB 324 (1989) 157, . . .

Can relate physical quanitites, but need to be careful regarding zeromodes, boundary conditions.


Part IV

Non-extremal Black Holes


Some previous and related work

Higher-dimensional black holes: F. Tangherlini, Nuovo Cimento 77(1963) 636, R. Myers and R. Perry, Ann. Phys. 172 (1986) 304.

Non-extremal black holes in supergravity/string theory (entropy,U-duality, Hawking radiation, etc.): C. Callan and J. Maldacena,NPB 472 (1996) 591, G. Horowitz and A. Strominger, PRL 77(1996) 2368, M. Cvetic and D. Youm, PRD 54 (1996) 2612, K.Behrndt, M. Cvectic and W. Sabra, PRD 58 (1997) 084018, . . .

Non-extremal branes, instantons: G. Gibbons and K. Maeda NPB298 (1988), G. Horowitz and A. Strominger PRD 43 NPB 360(1991) 197, M.J. Duff, H. Lü and C. Pope, PLB 382 (1996) 73, M.Cvetic and A. Tseytlin, NPB 478 (1996) 181, E. Cremmer, I.Lavrinenko, H. Lü, C. Pope, K. Stelle and T. Tran, NPB 534 (1998)40, . . .


Recently: 1st order equations, integrability for symmetric targetsH. Lu, C. Pope and J. Vazquez-Poritz, NPB 709 (2003) 47, C.Miller, K. Schalm and E. Weinberg, PRD 76 044001, L.Andrianapoli, R. D’Auria and E. Orazi JHEP 11 (2007) 032, B.Janssen, P. Smyth, T. Van Riet and B. Vercnocke, JHEP 04 (2008)007, G. Cardoso and V. Grass, NPB 803 (2008) 209, J. Perz, P.Smyth, T. Van Riet and B. Vercnocke, JHEP 03 (2009) 150, W.Chemissany, J. Rosseel, M. Trigiante and T. Van Riet NPB 830(2010) 391, W. Chemissany, P. Fré, J. Rosseel, A. Sorin, M.Trigiante and T. Van Riet, arXiv:1007.3209.


Clues

5d version of Reissner-Nordström solution (isotropic coordinates):

ds2(5) = −W (ρ)

H2(ρ)dt2 +

H(ρ)

W 1/2(ρ)

[

dρ2

W 1/2(ρ)+ W 1/2(ρ)ρ2dΩ2

(3)

]

, where

H = 1 +qρ2 , W = 1 − 2c

ρ2

Solution is build up from harmonic function (on E = R4).

For extremal solutions, W = 1, dimensional reduction over time

ds2(5) = −e2σdt2 + e−σds2

(4)

results in a flat Euclidean 4d metric.

The non-extremal solution is obtained by ‘dressing’ the extremalsolution with the harmonic function W (ρ). (Similar results knownfor black branes.)


Equations of motion

Remember

1√

|h|∂m(

√

|h|aIJ(σ)∂mσJ) − 12

∂IaJK (∂mσI∂mσJ − ∂mbI∂mbJ) = 0

∂m(√

|h|aIJ(σ)∂mbJ) = 014

aIJ(σ)(∂mσI∂nσJ − ∂mbI∂nbJ) =

16

Rmn

Impose spherical symmetry:

ds2(4) = e6A(τ)dτ2 + e2A(τ)dΩ2

(3) ,

τ = radial coordinate. Solve Einstein equation (regularity).


Universality

The four-dimensional geometry is the time-reducedReissner-Nordstöm metric, irrespective of the matter sector:

ds2(4) =

c3

sinh3(2cτ)dτ2 +

csinh(2cτ)

dΩ2(3)

=dρ2

W (ρ)1/2+ W 1/2(ρ)ρ2dΩ2

(3) , W (ρ) = 1 − 2cρ2 = e−4cτ .

where

ρ2 =ce2cτ

sinh(2cτ)→c→0

12τ

.

τ → 0 ⇔ ρ → ∞: asymptotically flat.τ → ∞ ⇔ ρ →

√2c: outer horizon.

ρ → 0: inner horizon (extension!).Note: Lifted (5d) metric is real for all ρ.


Equations of motion

Remaining field equations:

ddτ

(aIJ(σ)σJ ) − 12∂IaJK (σJ σK − bJ bK ) = 0

ddτ

(aIJ(σ)bJ) = 0

aIJ(σ)(σJ σK − bJ bK ) = 4c2

Radial coordinate τ = affine parameter on geodesic curve C ⊂ N.Non-extremality parameter c related to (constant) length of tangentvectors.


4d solution

b-equations solved in terms of conserved charges:

aIJ(σ)bJ = qI = const.

Using dual coordinates, remaining equation is

σI +12

∂IaJK (σJ σK − qJqK ) = 0

Still complicated, but contraction with σI gives

aIJσI σJ = 4c2 .

Crucial identity: aIJσIσJ = 1. Parametrize our ignorance:

4c2σI = σI + XI , σIXI = 0

Are there solutions where XI = 0?


Solutions with XI = 0

The equation obtained by setting XI = 0,

σI = 4c2σI

is solved by

σI = AI cosh(2cτ) +1

2cBI sinh(2cτ) →c→0 AI + BIτ .

Re-substitution: constraints on the integration constants AI, BI .

Can always recover 5d non-extremal Reissner-Nordström withconstant 5d scalars.

For models with diagonal metric and connection, obtain black holesolution with general scalar profile.


Example

Para-version of dilaton/axion system, (σI = e−2φI).

Hesse potential:V ≃ − log(σ1σ2 · · · σp)

Manifold:

N =

(

SL(2,R)

SO(1, 1)

)p

Then

σI = AI cosh(2cτ) +12c

BI sinh(2cτ)

is the general solution.

p = 3: supersymmetric STU model.General p: ‘Seed solution’ for symmetric target spaces. P.Breitenlohner, D. Maison and G. Gibbons, CMP 120 (1988) 295, . . .


Extend solution to inner horizon

Replace affine parameter τ by standard radial coordinate ρ.

σI = AI cosh(2cτ) +12c

BI sinh(2cτ) =HI(ρ)

W 1/2(ρ),

where

HI(ρ) = AI +QI

ρ2 , W (ρ) = 1 − 2cρ2

Integration constants: AI = σI(ρ → ∞),

QI =BI − 2cAI

2=

electric charges qI , for c = 0‘dressed’ charges, for c 6= 0

Solution works indeed by (twofold) ‘dressing’!


Lifting to 5d

Metric ds2(5) = −e2σdt2 + e−σds2

(4)

ds2(5) = − W

(H1H2 · · ·Hp)2/pdt2 +

(H1H2 · · ·Hp)1/p

W 1/2ds2

(4)

= − W(H1H2 · · ·Hp)2/p

dt2 + (H1H2 · · ·Hp)1/p[W−1dρ2 + ρ2dΩ2

(3)]

p = 3: recover non-extremal type-IIB D5-D1-pp solution of C. Callanand J. Maldacena, NPB 472 (1996) 591.

For H1 ∝ H2 ∝ · · · ∝ Hp ∝ H, recover non-extremalReissner-Nordström solution:

ds2(5) = −W

H2 dt2 + H[W−1dρ2 + ρ2dΩ2(3)]


5d scalars/dressed attractors

Same expression for 5d scalars as in non-extremal case

hI =(H1 · · ·Hp)1/p

HI

Inner horizon limit:

hI →ρ→0(Q1 · · ·Qp)1/p

QI.

‘Dressed attractor formula’.

QI =BI − 2cAI

2→c→0

BI

2= qI .

Not a true attractor for c 6= 0, since QI depend on AI ∼ hI(∞).Similar formula at outer horizon, with differently dressed charges.What does this tell us?


Black holes vs geodesics

A generic geodesic has 4n parameters: σI(∞), σI(∞), bI(∞), bI(∞).

Black hole solutions only have 2n:

σI(∞) , σI(∞) or σIHorizon ∼

qI ∼ bI(∞) , for c = 0QI(σ

J (∞), qJ) , for c 6= 0

Interpretation: Regularity of lifted solution ⇔ fine tuning of asymptoticsof geodesic (relating σI and bI) ⇔ (dressed) attractors, geodesicequations reduces to gradient flow equation.

Extremal limit:

σI = ±bI ⇒ σI = ∂IW , W = qIσI

Non-extremal case:

σI = ∂IW ⇒ W(σ(τ)) = −V(σ(τ)) + 4c2τ + W0

Can find W(σ) explicitly for STU model.Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 50 / 59

Integrability

pI := aIJ σJ = σI are the canonical momenta associated to σI .

qI := aIJ bJ = electric charges = canonical momanta associated to bI.

For extremal black holes, σI = ±bI, we have manifest integrability forpara-Kähler target spaces without assuming them to be homogeneousor symmetric spaces, because pI and qI are conserved.

Symmetric spaces: geodesic equations fully integrable (Liouville andHamilton/Jacobi), regular black hole solutions ≃ vanishingHamiltonians. (W.Chemissany, P. Fré, J. Rosseel, A.S. Sorin, M.Trigiante and T. Van Riet, arXiv:1007.3209).

What to expect for (non-homogeneous) para-Kähler targets?


Brief Summary

Modifications and generalizations of special geometry can beobtained systematically and have interesting applications.

Para-complex geometry is a useful tool, i.p. in combination withtemporal reduction and harmonic maps.

Can go beyond homogeneous/symmetric target spaces usingpotentials (integrability properties), special coordinates (definedintrinsically in terms of special data, like flat connections) andhomogeneity properties (homothetic Killing vector fields).


Outlook

Non-extremal black holes, gradient flow equations, integrability.

(generalized) hypermultiplet geometries, c-map

4d/3d reduction, inclusion of magnetic charges. (For purelyelectric charges, 5d/4d reduction is straightforward to extend tod + 1/d .)


Appendix


Complexification

Basic dilaton/axion system: X = e−2φ + iǫa, iǫ = i , e.

L ∼ ∂mX∂mXRe(X )2 ∼ −(∂mφ∂mφ ± 1

4e4φ∂ma∂ma)

Two real forms of one complex-Riemannian space:

SL(2,R)

SO(2)⊂ SL(2,C)

GL(1,C)⊃ SL(2,R)

SO(1, 1)

Change of real form via intermediate complexification is useful forhandling dualities, constructing solutions. Maximal 10d supergravity:E. Bergshoeff, J. Hartong, A. Ploegh, J. Rosseel and D. Van denBleecken, JHEP 07 (2007) 067Not restricted to symmetric spaces (assuming analyticity/integrabilityproperties): T.M. and K. Waite, JHEP 10 (2009) 058 and to appear.


Complexifying the complex

Complexify a complex field:

σI + ibI → (σI + j σI) + j(bI + jβI)

Note the second complex structure/unit j .Since ij = ji and i2 = −1 = j2, we get a para-complex structure for free:e := ij , with e2 = 1, and a real form carrying a para-complex structure:

σI + ijβI = σI + eβI .

Analytical continuation from complex to para-complex.


Comment: non-BPS extremal solutions

If aIJ has discrete isometries

aKLRKI RL

J = aIJ

then ∂mσI = ±∂mbI can be replaced by

∂mσI = RIJ∂mbJ

A. Ceresole and G. Dall’Agata, JHEP 03 (2007) 110, G. LopesCardoso, A. Ceresole, G. Dall’Agata, J. Oberreuter and J. Perz, JHEP10 (2007) 063.Such ‘non-BPS extremal’ solutions mix eigendirections of J, isometryguarantees existence of totally geodesic submanifold.


Comment on the role of discrete isometries

aKLRKI RL

J = aIJ

allowing relaxed ansatz

∂mσI = RIJ∂mbJ

For general (non-symmetric) target spaces such isometriesappear somewhat non-generic. But there are examples: P. Kauraand A. Misra, Fortsch. Phys. 54 (2006) 1109, S. Belluci, S.Ferrara, S. Marrani and A. Yeranyan, Riv. Nuovo Cimento 29N5(2006) 1.

For symmetric spaces there are many isometries beyond thegeneric ones we assume. Huge literature on non-BPS extremalblack holes, see review S. Belluci, S. Ferrara, M. Gunaydin and A.Marrani, arXiv:0905.3451.


BPS instantons of 4d Euclidean STU models

Stringy interpretation:

Instanton action ∼ g−2: wrapped five-brane (space-timeinstanton)!

ds2StrFrame = −dt2 + (dy1)2 + · · · + (dy5)2

+H(x)((dx1)2 + · · · + (dx4)2)

e2(Φ−Φ∞) = H(x) , dB = ⋆4dH(x)

is the 0 + 4 → 1 + 9 lift of the basic SL(2,R)/SO(1, 1) solutionwith X = S = e−2φ + eb = 1

g2 + eb.

Same solution, X = T , U geometrical modulus.Instanton action ∼ g0: must be world-sheet instanton.


special geometry, black holes, and instantonsmedia/blackholes_supergravity/... · special geometry,...

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