Non-commutative D-Brane World, Non-commutative D-Brane World, Black Holes & Extra Dimensions Black Holes & Extra Dimensions Supriya K. Kar

Dept.of Physics & Astrophysics

University of Delhi New Delhi, INDIA

Based on a research in progress

Some related research in:

[1] Journal of High Energy Physics 10 (2007) 052 & Physical Review D74 (2007) 126002

[2] Physical Review D74 (2006) 066003 & Int. Journal of Mod. Physics A21 (2006) 6087 (with Sumit Majumdar)

Particle Physics, Astrophysics and Quantum Field Theory: 75 Yrs. since Solvay, 27-29 Nov '08 @ Institute of Advanced Studies, Nanyang Technological University, Singapore

THEME:-THEME:-.......................................................

• Some aspects of Quantum Gravity in String Some aspects of Quantum Gravity in String Theory:Theory:

• ( inspired thoughts........ )

• Dimension of our space-timeDimension of our space-time • ( are there extra dimensions ? )

• Equivalence Principle in quantum gravityEquivalence Principle in quantum gravity

• Black Hole Geometries on D-Brane WorldBlack Hole Geometries on D-Brane World• ( inspired by noncommutativity on its world-volume )

• Microscopic Black HolesMicroscopic Black Holes• ( possibility of laboratory black holes )

• Construction of de Sitter VacuaConstruction of de Sitter Vacua

Motivation:

Black Holes in GTR Event horizons enclose Point singularity

Quantum-mechanical Singular sources are Smeared out

String theory Non-commutativity on a D-brane at short distances

Non-commutative analogue of Schwarzschild black holeLimits the mass to a non-zero minimum ~ Lnc ~ Lpl

But Lnc could possibly >> Lpl, if extra dimensions exist

Possibility of Laboratory (micro) black holes (mass)BH ~ few TeV’s

NC on a D-brane world in String Theory

Non-linear EM-Field on the D-brane

Potential candidate to address the quantum aspects of gravity

Einstein gravity decouples ………..

the effective gravity on a D-brane may be governed by the non-linear EM-Field

Dynamics of a curved D-brane may be seen to be governed by an appropriate potential in the moduli

space of scalars in a string theory

Notion of a curved D-brane world in Quantum Gravity

PPLLAANN

[1] [1] Non-commutative D-Brane World in String Non-commutative D-Brane World in String TheoryTheory

[2] (Anti) de Sitter Black Hole Geometries[2] (Anti) de Sitter Black Hole Geometries

[3] Black Hole Horizon As Attractor[3] Black Hole Horizon As Attractor

[4] Extra Dimensions,[4] Extra Dimensions,

**((Hegedorn)Hegedorn) phase transitions & tunnelingphase transitions & tunneling

Curved D-Brane FormalismCurved D-Brane Formalism

Boundary Dynamics of Open String Notion of a D-BraneNotion of a D-Brane

U(1) Gauge theory + Gravity (back reaction)

Motion of D-Brane along the Cigar Geometry (string bulk)

Induces (scale dependent) Curvature on its Brane World

Non-commutative scaling on a D5-Non-commutative scaling on a D5-brane worldbrane world((schematic)schematic)

Classical Gravity

NC-NC-scalingscaling

Q G

[ X , Z ] = i (theta) -> [ X , Z ] = i (theta) -> non-commutativitynon-commutativity

XX transverse 4D (ordinary) space coordinates &transverse 4D (ordinary) space coordinates &

ZZ Longitudinal 2D (ordinary) space coordinatesLongitudinal 2D (ordinary) space coordinates

Seiberg-Witten MapSeiberg-Witten Map ………………………….

U(1) U(1) U(1) U(1)ncnc

( g, b( g, b22, F, F22 ) ) ( G, F ( G, Fncnc ) )

G(g,b) G(g,b) Modified metric Modified metric

• bb22 Global mode Global mode NC geometry on a D-brane NC geometry on a D-brane • Non-linear electric field: ENon-linear electric field: Enlnl = (b + E) = (b + E)

• NC (theta)NC (theta) term constraints space-time term constraints space-time dimensionsdimensions

““extra dimensions”extra dimensions” in the formulation in the formulation

D3-Brane Dynamics: (

Minkowski inequlity: |E| =|B|

Non-linear electric field:

Curved Brane Dynamics in String Curved Brane Dynamics in String TheoryTheoryD3-brane + D=10 type IIB string on K3 X T2.

In a static gauge:

For a stable minima in V4 Lmn= Const.

: Const. as Lmn fixed point on EH

RELEVENT DYNAMICS (RELEVENT DYNAMICS (USING NON-COMMUTATIVE USING NON-COMMUTATIVE

SCALINGSCALING))

: potential between moduli & F2’s

: electric charge on D3-brane

NC-scaling vacuum field configurations for some of the fields:

Curved D3-brane effective action:

Axially Symmetric (Anti) de Sitter Black Axially Symmetric (Anti) de Sitter Black HoleHole

Constant scalar moduli EM on the brane only

: to 2nd order in GN

: k (+1,0,-1) constant curvature geometry at the event horizon

: C1,2,3,4 & Ceff (light) mass terms

ADM Mass:

Extremal Extremal GeometriesGeometries…………………………………………

…..Axially symmetric Spherically symmetric Black Holes . Geometries are independent of GN

Generic Black Geometries ( arbitrary Generic Black Geometries ( arbitrary moduli )moduli )

• Non trivial potential in the moduli space non-vanishing Enl on D-brane & EM-field in string bulk

de Sitter Charged Black Hole to O(Gde Sitter Charged Black Hole to O(GNN):):……………………….

Anti de Sitter Charged Black Hole to O(GAnti de Sitter Charged Black Hole to O(GNN):):

Shrinking SShrinking S22 Emerging 2D Black Emerging 2D Black HolesHoles……….

Moduli ~ EM-fields V2 & event horizon acts as an attractorCharges force the horizon radius to shrink to zero in g 0 limit

dS2

AdS2

dSdS22 AdS AdS22,,

when k when k - - kk

Motion of D-brane Motion of D-brane Variation of V Variation of V22 in moduli in moduli spacespace

Classical Classical Planckian regime ( Hagedorn Phase ) Planckian regime ( Hagedorn Phase )

Semi-classical BH’s (Hawking radiation) Semi-classical BH’s (Hawking radiation) Extremal BH’s Extremal BH’s

Decoupling of EDecoupling of Enlnl (Hagendorn transition) (Hagendorn transition) Near extremal dS Near extremal dS BH’s BH’s

For large r (near extremal) For large r (near extremal)

Topology Topology interchanges:interchanges:

Extra Dimensions……………………………………………………

……………..Decoupling Decoupling gravity + gauge non-linearitygravity + gauge non-linearity near horizon near horizon geometry governs a typical monopole black hole solution: geometry governs a typical monopole black hole solution:

reduced massreduced mass

Gravitational potential generated by the reduced mass

underlying gravity in 5D (ordinary geometry) !

“ “3 Large Extra Dimensions” to the 2D Monopole Black Hole3 Large Extra Dimensions” to the 2D Monopole Black Hole

OROR-------->>

plausible scenario !plausible scenario !

space-time dimension ~ scale dependent schematic:

Concluding remarks:

1. Dynamics of a curved D3-brane world, inspired by a its non-commutative world volume geometry, is explicitly investigated in a string theory.

2. D3-brane is seen to be governed by a potential in the moduli space of scalars.

3. Axially symmetric & spherically symmetric AdS and dS extremal black hole geometries due to the non-linear EM-field are obtained in the gravity decoupling limit.

4. A plausible scenario leading to a tunneling between dS2 to AdS2 vacua is highlighted under the Hagedorn transition in presence of a B-field in string theory.

5. Hint for large extra dimensions in the formalism.

16

hijhkl∂k¯hαβ ∂l¯hγδ − 2∂ihαβ ∂j¯hγδ

ǫαγǫβδ = (8 exπ)

6. Generalization to a non-commutative D5-brane world is discussed.

7. Non-commutative scaling on the D5-brane world may be used to decouple the 2D quantum gravity in longitudinal space from the 4D classical (ordinary) geometry along the transverse directions.

8. Microscopic black holes in 2D may be obtained due to the non-linear EM-field. However, the stability of these black holes need to be investigated.

9. Emerging 4D extremal macroscopic black hole geometries may be obtained in the gravity decoupling limit on the D5-brane world.

10. The ADM mass and the electric charge of the black holes may be seen to receive corrections due to the non-commutative parameter onthe D5-brane world.

11. Plausible scenario leading to a tunneling between dS4 to AdS4 inthe formalism may enlighten us with the dS world.

Thanks.