1. Dept.of Physics & Astrophysics
Non-commutative D-Brane World, 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)
[2] Physical Review D74 (2006) & Int. Journal of Mod. Physics A21 (2006) 6087 (with Sumit Majumdar)‏
Particle Physics, Astrophysics and Quantum Field Theory: 75 Yrs. since Solvay, Nov '08 @ Institute of Advanced Studies, Nanyang Technological University, Singapore 1 1

2. THEME:
Some aspects of Quantum Gravity in String Theory:
( inspired thoughts )
Dimension of our space-time
( are there extra dimensions ? )‏
Equivalence Principle in quantum gravity
Black Hole Geometries on D-Brane World
( inspired by noncommutativity on its world-volume )‏
Microscopic Black Holes
( possibility of laboratory black holes )
Construction of de Sitter Vacua

3. Motivation: Point singularity Smeared out
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 hole
Limits 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 3

4. 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 4

5. PLAN [1] Non-commutative D-Brane World in String Theory
[2] (Anti) de Sitter Black Hole Geometries
[3] Black Hole Horizon As Attractor
[4] Extra Dimensions,
*(Hegedorn) phase transitions & tunneling 5 5

6. Curved D-Brane Formalism
Boundary Dynamics of Open String  Notion 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 6 6

7. Non-commutative scaling on a D5-brane world (schematic)‏
Classical Gravity
NC-scaling
4D World
D5-brane
Q G
[ X , Z ] = i (theta) -> non-commutativity
X transverse 4D (ordinary) space coordinates &
Z Longitudinal 2D (ordinary) space coordinates 7 7

8. Seiberg-Witten Map ………………………….
U(1)   U(1)nc
( g, b2, F2 )   ( G, Fnc )‏
G(g,b)  Modified metric
b2  Global mode  NC geometry on a D-brane
Non-linear electric field: Enl = (b + E)‏
NC (theta) term constraints space-time dimensions
 “extra dimensions” in the formulation 8 8

9. D3-Brane Dynamics: (  Minkowski inequlity: |E| =|B|
Non-linear electric field: 9 9

10. Curved Brane Dynamics in String Theory
D3-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 10 10

11. RELEVENT DYNAMICS (USING NON-COMMUTATIVE SCALING)
: 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: 11 11

12. Axially Symmetric (Anti) de Sitter Black Hole
Constant scalar moduli  EM on the brane only
ADM Mass:
: to 2nd order in GN
: k (+1,0,-1) constant curvature geometry at the event horizon
: C1,2,3,4 & Ceff  (light) mass terms 12 12

13. Extremal Geometries……………………………………………..
Axially symmetric  Spherically symmetric Black Holes  Geometries are independent of GN 13 13

14. Generic Black Geometries ( arbitrary moduli )‏
Non trivial potential in the moduli space
 non-vanishing Enl on D-brane & EM-field in string bulk 14 14

15. de Sitter Charged Black Hole to O(GN):……………………….
Anti de Sitter Charged Black Hole to O(GN): 15 15

16. Shrinking S2  Emerging 2D Black Holes……….
Moduli ~ EM-fields  V2 & event horizon acts as an attractor
Charges force the horizon radius to shrink to zero in g  0 limit
dS2
AdS2
dS2 AdS2,
when k  - k 16 16

17. Motion of D-brane  Variation of V2 in moduli space
Classical  Planckian regime ( Hagedorn Phase )‏
Semi-classical BH’s (Hawking radiation)  Extremal BH’s
Decoupling of Enl (Hagendorn transition)  Near extremal dS BH’s
For large r (near extremal)
Topology interchanges: 17 17

18. Extra Dimensions…………………………………………………………………..
Decoupling gravity + gauge non-linearity  near horizon geometry governs a typical monopole black hole solution:
reduced mass
Gravitational potential generated by the reduced mass
 underlying gravity in 5D (ordinary geometry) !
“3 Large Extra Dimensions” to the 2D Monopole Black Hole
----> OR
plausible scenario ! 18 18

19. ( x2 , x3 ) S 2 AdS ( Ordinary ) 5 ( t , x1, x4 , x5 , x6 , x7 )
space-time dimension ~ scale dependent
schematic:
( x2 , x3 ) S 2
AdS
( Ordinary ) 5
( t , x1, x4 , x5 , x6 , x7 ) 19 19

20. 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)‏ 20

21. 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 on
the D5-brane world.
11. Plausible scenario leading to a tunneling between dS4 to AdS4 in
the formalism may enlighten us with the dS world.
Thanks. 21