FIRST ORDER FORMALISM FOR

NON-SUPERSYMMETRIC MULTI

BLACK HOLE CONFIGURATIONS

A . S h c h e r b a k o vLNF INFN Frascati

(Italy)

i n c o l l a b o r a t i o n w i t h A . Ye r a n y a n

S u p e r s y m m e t r y i n I n t e g r a b l e S y s t e m s - S I S ' 1 2

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Purpose

In the framework of N=2 D=4 supergravity, construct the first order equation formalism governing the dynamics of the graviton, scalar and electromagnetic fields in the background of extremal black hole(s)

1. multiple black hole configuration2. supersymmetric and non-supersymmetric3. rotating black holes

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Why equations and not solutions?

The main goal – to find a solution.

The equations of motion are coupled non-linear differential equations of the second order.

The known solutions are just particular ones.

Why not to rewrite the equations of motion in an easier-to-solve manner?

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Results

EquationsTwo possible cases

A.

B.

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Setup

Einstein gravity coupled to electromagnetic fields

in a stationary background

With N=2 D=4 SUSY, the σ-model metric Gaā and couplings μΛΣ and νΛΣ are expressed in terms of a holomorphic prepotential F=F(z).

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Reduction to three dimensions

Reduction is performed in Kaluza-Klein manner

metric

vector-potentialThree dimensional vector potentials a and w can be dualized in scalars

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If the three dimensional space is flat, the equations of motion read

with an additional constraint

These equations contain the following objects

Equations of motion

divergenless

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Black hole potential

In the case of a single non-rotating black hole

tensorial black hole potential reduces to a singlet

For N=2 D=4 SUGRA

where

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Black hole potential

Single non rotating BH General case

rotation & Maxwell

Recall

hints to introduce

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Equations of motion (summary)

The equations of motion has the following form

with the constraint

where

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Present state of art

supersymmetric, single center supersymmetric, multi center

non-supersymmetric, single center non-supersymmetric, multi center

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Supersymmetric single o multi-center

Single center

Natural splitting

Entropy

Multi center S.Ferrara, G.Gibbons, R.Kallosh ‘97 F.Denef ‘00

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Non supersymmetric single center

Analogous description for non-BPS black holes

Entropy

A.CeresoleG. Dall’Agata ‘07

S.Bellucci, S.Ferrara,A.Marrani, A.Yeranyan ‘08

Example of a fake superpotential

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Constructing the first order equations

General form of the first order equations

plus other equations (if any).The algebraic constraint

imposes a relation

What functions W, Pi and li are equal to?

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As a starting point, let us consider the spatial infinity and the supersymmetric flow.

Wi and Pi are defined by ADM mass M, NUT charge N and scalar charges πAt spatial infinity

Phase restoration

Constructing flow-defining functions

G.Bossard’11

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Now let us generalize the consideration for the whole space:

Constructing flow-defining functions

To pass to a non-supersymmetric solution, “charge flipping” is needed.

G.Bossard’11D0

D2D2D2D4D4D4

D6

Toy example:

1. Composite 2. Almost BPS

A.Yeranyan ‘12

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Composite

Full set of equations

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Almost BPS

Full set of equations

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Properties

We showed that solutions1. Rasheed-Larsen black holes2. magnetic/electric multi-black hole satisfy the corresponding equations of motion.Let us stress that all these solutions are particular ones and not general.Appearance of the phases demonstrates how the concept of “flat directions” gets generalized for multi-black hole configurations.

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THANK YOU!

- I think you should be more explicit here in step two…