Noncommutative DBrane World Black Holes Extra Dimensions Supriya

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 D 74 (2007) 126002 [2] Physical Review D 74 (2006) 066003 & Int. Journal of Mod. Physics A 21 (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 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

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 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 Te. V’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

PL AN [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

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

Non-commutative scaling on a D 5 -brane world (schematic) NC-scaling Classical Gravity QG [ X , Z ] = i (theta) -> non-commutativity X transverse 4 D (ordinary) space coordinates & Z Longitudinal 2 D (ordinary) space coordinates

Seiberg-Witten Map ……………. U(1)nc ( g, b 2, F 2 ) ( G, Fnc ) G(g, b) Modified metric • b 2 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

D 3 -Brane Dynamics: ( Minkowski inequlity: |E| =|B| Non-linear electric field:

Curved Brane Dynamics in String Theory D 3 -brane + D=10 type IIB string on K 3 X T 2. In a static gauge: For a stable minima in V 4 Lmn= Const. : Const. as Lmn fixed point on EH

RELEVENT DYNAMICS (USING NON-COMMUTATIVE SCALING) : potential between moduli & F 2’s : electric charge on D 3 -brane NC-scaling vacuum field configurations for some of the fields: Curved D 3 -brane effective action:

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

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

Generic Black Geometries ( arbitrary 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(GN): ……………. ): Anti de Sitter Charged Black Hole to O(GN):

Shrinking S 2 Emerging 2 D Black Holes………. Holes Moduli ~ EM-fields V 2 & event horizon acts as an attractor Charges force the horizon radius to shrink to zero in g 0 limit d. S 2 Ad. S 2, when k - k

Motion of D-brane Variation of V 2 in moduli space Classical Planckian regime ( Hagedorn Phase ) Semi-classical BH’s (Hawking radiation) Extremal BH’s Decoupling of Enl (Hagendorn transition) Near extremal d. S BH’s For large r (near extremal) Topology interchanges:

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 5 D (ordinary geometry) ! “ 3 Large Extra Dimensions” to the 2 D Monopole Black Hole ----> OR plausible scenario !

space-time dimension ~ scale dependent schematic:

Concluding remarks: 1. Dynamics of a curved D 3 -brane world, inspired by a its non -commutative world volume geometry, is explicitly investigated in a string theory. 2. D 3 -brane is seen to be governed by a potential in the moduli space of scalars. 3. Axially symmetric & spherically symmetric Ad. S and d. S extremal black hole geometries due to the non-linear EMfield are obtained in the gravity decoupling limit. 4. A plausible scenario leading to a tunneling between d. S 2 to Ad. S 2 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.

6. Generalization to a non-commutative D 5 -brane world is discussed. 7. Non-commutative scaling on the D 5 -brane world may be used to decouple the 2 D quantum gravity in longitudinal space from the 4 D classical (ordinary) geometry along the transverse directions. 8. Microscopic black holes in 2 D may be obtained due to the non-linear EM-field. However, the stability of these black holes need to be investigated. 9. Emerging 4 D extremal macroscopic black hole geometries may be obtained in the gravity decoupling limit on the D 5 -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 D 5 -brane world. 11. Plausible scenario leading to a tunneling between d. S 4 to Ad. S 4 in the formalism may enlighten us with the d. S world. Thanks.