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FIRST ORDER FORMALISM FOR NON-SUPERSYMMETRIC MULTI BLACK HOLE CONFIGURATIONS. A.Shcherbakov LNF INFN Frascati (Italy). in collaboration with A.Yeranyan. Supersymmetry in Integrable Systems - SIS'12. Purpose.

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first order formalism for non supersymmetric multi black hole configurations

FIRST ORDER FORMALISM FORNON-SUPERSYMMETRIC MULTI BLACK HOLE CONFIGURATIONS

A.Shcherbakov

LNF INFN Frascati (Italy)

in collaboration with A.Yeranyan

Supersymmetry in Integrable Systems - SIS'12

purpose
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)

multiple black hole configuration

supersymmetric and non-supersymmetric

rotating black holes

Supersymmetry in Integrable Systems 2012

why equations and not solutions
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?

Supersymmetry in Integrable Systems 2012

results
Results

Equations

Two possible cases

Supersymmetry in Integrable Systems 2012

setup
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 νΛΣare expressed in terms of a holomorphic prepotential F=F(z).

Supersymmetry in Integrable Systems 2012

reduction to three dimensions
Reduction to three dimensions

Reduction is performed in Kaluza-Klein manner

metric

vector-potential

Three dimensional vector potentials a and w can be dualized in scalars

Supersymmetry in Integrable Systems 2012

equations of motion
Equations of motion

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

with an additional constraint

These equations contain the following objects

divergenless

Supersymmetry in Integrable Systems 2012

black hole potential
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

Supersymmetry in Integrable Systems 2012

black hole potential1
Black hole potential

Single non rotating BH General case

Recall

rotation & Maxwell

hints to introduce

Supersymmetry in Integrable Systems 2012

equations of motion summary
Equations of motion (summary)

The equations of motion has the following form

with the constraint

where

Supersymmetry in Integrable Systems 2012

present state of art
Present state of art

supersymmetric, single center supersymmetric, multi center

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

Supersymmetry in Integrable Systems 2012

supersymmetric single o multi center
Supersymmetric single o multi-center

Single center

Natural splitting

Entropy

Multi center

S.Ferrara, G.Gibbons, R.Kallosh ‘97

F.Denef ‘00

Supersymmetry in Integrable Systems 2012

non supersymmetric single center
Non supersymmetric single center

Analogous description for non-BPS black holes

Entropy

A.Ceresole

G. Dall’Agata ‘07

Example of a fake superpotential

S.Bellucci, S.Ferrara,

A.Marrani, A.Yeranyan ‘08

Supersymmetry in Integrable Systems 2012

constructing the first order equations
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 liare equal to?

Supersymmetry in Integrable Systems 2012

constructing flow defining functions
Constructing flow-defining functions

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 π

At spatial infinity

Phase restoration

G.Bossard’11

Supersymmetry in Integrable Systems 2012

constructing flow defining functions1
Constructing flow-defining functions

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

G.Bossard’11

1. Composite

D0

D4

2. Almost BPS

D4

D4

D6

A.Yeranyan ‘12

Toy example:

Now let us generalize the consideration for the whole space:

D2

D2

D2

Supersymmetry in Integrable Systems 2012

composite
Composite

Full set of equations

Supersymmetry in Integrable Systems 2012

almost bps
Almost BPS

Full set of equations

Supersymmetry in Integrable Systems 2012

properties
Properties

We showed that solutions

Rasheed-Larsen black holes

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.

Supersymmetry in Integrable Systems 2012

thank you
THANK YOU!

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

Supersymmetry in Integrable Systems 2012