Updates on K-strings from the Supersymmetric D-brane Perspective

Transcription

1 Updates on K-strings from the Supersymmetric D-brane Perspective Kory Stiffler Indiana University Northwest Miami Conference 2015: 100 Years of GR 12/21/2015 1/54

2 Outline What is a k-string? Gauge/Gravity Duality Supersymmetric Probe D-branes Bosonic Modes: Lüscher term Fermionic Modes:? Conclusions 2/54

3 What is a k-string? 3/54

4 Lüscher s Fundamental String (Lüscher, 1981) quark-antiquark pair tied together by a string Forms color flux tube of energy: π ( d 2) 1 E = T L + β + O 2 24L L T: tension d: space-time dimension L: quark-antiquark separation 4/54

5 SU(N) k-strings: Tension Energy: α 1 Ek = Tk L+ + β + O 2, k = l m L L Tension vanishes when k=0,n (N-ality k) T k kπ N sin sine law N N k k casimir law N Lüscher string k-string: mul>ple Lüscher strings, d <<L, k= l -m Exhibited by models in la;ce gauge theory analy>c Hamiltonian methods string theory using gauge/gravity duali>es 5/54

6 N-ality k N=3 k=3-string 6/54

7 N-ality k N=3 k=3-string Baryon & anti-baryon form 7/54

8 N-ality k N=3 k=3-string Baryon & anti-baryon form Tension vanishes Behavior called 3-ality 3 Generally: N-ality k 8/54

9 SU(N) k-strings: Lüscher Term Energy: α 1 Ek = Tk L+ + β + O 2, k = l m L L Lattice calculations ( , ): α String theory: Dp-brane probes of Supergravity ( , , ): Lüscher string k-string: mul>ple Lüscher strings, d <<L, k= l -m These match for 9/54

10 Gauge/Gravity Duality k-strings = D-branes 10/54

11 Gauge/Gravity Duality Closed strings = Supergravity(SUGRA) Open strings + Dp-branes = gauge theory Dp-branes: membranes on which strings can end p = spatial dimension of the brane. Open strings attached to D2-branes 11/54

12 Gauge/Gravity Duality Open strings + D-branes = gauge theory Gauge flux flows between string to D-brane: D-brane itself IS the color flux tube 12/54

13 Gauge/Gravity Duality D-branes = k-strings? Gauge flux flows between string and D-brane: D-brane ~ SU(N) k-strings 13/54

14 Gauge/Gravity Duality Open strings + D-branes = U(N) gauge theory 1 D-brane = U(1) gauge theory 1 distinct way to connect string to the brane Dual to gauge theory with adjoint representation of dimension 1 => U(1) 3 stacked D-branes => U(3) gauge theory The D-branes are stacked in the sense that they are infinitesimally close together. 3 2 =9 distinct ways to connect strings to the branes Dual to gauge theory with adjoint representation of dimension 3 2 =9 => U(3) 14/54

15 Gauge/Gravity Duality Open strings + D-branes = U(N) gauge theory N stacked D-branes => U(N) gauge theory N 2 distinct ways to connect strings to the branes Dual to gauge theory with adjoint representation of dimension N 2 => U(N) 15/54

16 Gauge/Gravity Duality Open to Closed strings: D-branes as a Supergravity Source N stacked D-branes = U(N) gauge theory 16/54

17 Gauge/Gravity Duality Open to Closed strings: D-branes as a Supergravity Source N stacked D-branes = U(N) gauge theory Naturally couple to Ramond-Ramond (R-R) flux (closed strings) Ex: 17/54

18 Gauge/Gravity Duality Open to Closed strings: D-branes as a Supergravity Source N stacked D-branes = U(N) gauge theory Naturally couple to Ramond-Ramond (R-R) flux (closed strings) Closed strings = supergravity (low energy effective theory) 18/54

19 Gauge/Gravity Duality Supergravity instructions for a k-string 1) SUGRA dual to supersymmetric SU(N) theory 2) Probe with D-brane wrapped with flux 3) Calculate for several years Result: The Probe brane exhibits behavior of a k-string ~ 19/54

20 What has been done SUGRA Background Dimension of dual gauge theory Klebanov-Strassler (KS) d=3+1 p=3 Cvetic, Gibbons, Lu, and Pope (CGLP) d=2+1 p=4 Dp-brane probe Maldacena-Nunez (MN) d=3+1 p=3, p=5 Maldacena-Nastase (MNa) d=2+1 p=3, p=5 Dp-brane bosonic Energy: T k kπ N sin sine law N N k k casimir law N α 1 Ek = Tk L+ + β + O 2, k = l m L L 20/54

21 Supersymmetric Probe Branes 21/54

22 Building a k-string 1) MN/MNa SUGRA background: dual to SU(N) theory ρ 22/54

23 Building a k-string 1) MN/MNa SUGRA background: dual to SU(N) theory ρ Dilaton: 23/54

24 Building a k-string 1) MN/MNa SUGRA background: dual to SU(N) theory ρ Dilaton: 24/54

25 Building a k-string 1) MN/MNa SUGRA background: dual to SU(N) theory ρ Dilaton: 25/54

26 Building a k-string 1) MN/MNa SUGRA background: dual to SU(N) theory ρ 26/54

27 Building a k-string 1) MN/MNa SUGRA background: dual to SU(N) theory ρ 27/54

28 Building a k-string 1) MN SUGRA background: dual to SU(N) in 3+1 X=(t,x,y,z,ρ,θ 1,θ 2,φ 1,φ 2,ψ) 28/54

29 Building a k-string 1) MNa SUGRA background: dual to SU(N) in 2+1 X=(t,x,y,ψ 1,ρ,θ 1,θ 2,φ 1,φ 2,ψ 2 ) 29/54

30 Building a k-string 2) Probe with Dp-brane wrapped with flux: Born-Infeld Action X(ξ),Θ(ξ) 30/54

31 Energy 3) Probe brane energy ~ k-string energy X(ξ),Θ(ξ) ~ E k = T k L + α L + β +O 1 L 2 31/54

32 Classical Energy 3) Probe brane energy ~ k-string energy X(ξ),Θ(ξ) ~ E k = T k L + α L + β +O 1 L 2 32/54

33 Classical Energy 3) Probe brane energy ~ k-string energy ~ X(ξ) E k = T k L + α L + β +O 1 L 2 33/54

34 Classical Energy 3) Probe brane energy ~ k-string energy p = 3 X=(t,x,0,0,0,θ,θ,φ,φ,ψ 0 ) E k = T k L + α L + β +O 1 L 2 34/54

35 Classical Energy 3) Probe brane energy ~ k-string energy E k = T k L + α L + β +O 1 L 2 35/54

36 Classical Energy 3) Probe brane energy ~ k-string energy E k = T k L + α L + β +O 1 L 2 36/54

37 Classical Energy 3) Probe brane energy ~ k-string energy E k = T k L + α L + β +O 1 L 2 37/54

38 Classical Energy 3) Probe brane energy ~ k-string energy ~ X=(t,x,0,0,0,θ,θ,φ,φ,ψ 0 ) D3-brane probe of MN/MNa Background: E k = T k L + α L + β +O 1 L 2 T k N sin kπ N 38/54

39 Bosonic One-Loop Energy 3) Probe brane energy ~ k-string energy ~ X+δx E k = T k L + α L + β +O 1 L 2 Luscher term 39/54

40 Bosonic One-Loop Energy 3) Probe brane energy ~ k-string energy X+δx 40/54

41 Bosonic One-Loop Energy 3) Probe brane energy ~ k-string energy X+δx Regularize 41/54

42 Bosonic One-Loop Energy 3) Probe brane energy ~ k-string energy ~ X+δx D3-brane one-loop bosonic energy in MN/MNa k-string energy E k = T k L + α L + β +O 1 L 2 42/54

43 Fermionic One-Loop Energy 2) Probe with Dp-brane wrapped with flux: Born-Infeld Action X,Θ =? 43/54

44 Fermionic One-Loop Energy 2) Probe with Dp-brane wrapped with flux: Born-Infeld Action X,Θ 44/54

45 Fermionic One-Loop Energy 2) Probe with Dp-brane wrapped with flux: Born-Infeld Action X,Θ 45/54

46 Fermionic One-Loop Energy 2) Probe with Dp-brane wrapped with flux: Born-Infeld Action X,Θ 46/54

47 Fermionic One-Loop Energy 2) Probe with Dp-brane wrapped with flux: Born-Infeld Action X,Θ 47/54

48 Fermionic One-Loop Energy 3) Probe brane fermionic energy ~? ~??? X,Θ =? 48/54

49 Fermionic One-Loop Energy 3) Probe brane fermionic energy not yet calculated in MN/ MNa. For Klebanov-Strassler background found: =? 49/54

50 Fermionic One-Loop Energy 3) Probe brane fermionic energy not yet calculated in MN/ MNa. For Klebanov-Strassler background found: =? 50/54

51 Fermionic One-Loop Energy 3) Probe brane fermionic energy not yet calculated. =? 51/54

52 Fermionic One-Loop Energy 3) Probe brane fermionic energy not yet calculated. =? 52/54

53 Fermionic One-Loop Energy 3) Probe brane fermionic energy ~? ~??? X,Θ =? 53/54

54 Summary K-strings: color flux tubes Branes Source supergravity Endowed with gauge theories k-strings/d-branes: gauge/gravity duals Classical solution: tension Bosonic fluctuations Regularize Lüscher term universality Fermionic fluctuations Regularize?? 54/54