Holographic k-string Tensions in Higher Representations and Lüscher Term Universality

Transcription

1 Holographic k-string Tensions in Higher Representations and Lüscher Term Universality Kory Stiffler Based on: B. Button*, S. J. Lee*, L. Pando Zayas**, V. G. J. Rodgers*, and K. Stiffler***, Holographic k- string Tensions in Higher Representations and Lüscher Term Universality, Phys. Rev. D 87, (2013), arxiv: [hep-th]. The University of Iowa* The University of Michigan** Indiana University Northwest and the University of Maryland***

2 Outline What is a k-string? Branes, gauge theory, and supergravity k-strings/d-branes: gauge/gravity duals Conformal Case: Fundamental WL: fundamental string Symmetric WL: D3-brane Antisymmetric WL: D5-brane Our results (confining cases) fundamental WL: fundamental string Symmetric WL: larger brane Antisymmetric WL: smaller brane Lüscher term universality Conclusions 2/42

3 What is a k-string? 3/42

4 Lüscher s Fundamental String (Lüscher, 1981) Simple model of quarkantiquark 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 4/42

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 5/42

6 SU(N) k-strings: Lüscher Term Energy: α 1 Ek = Tk L+ + β + O 2, k = l m L L Lattice calculations in d=2+1(teper, 2008): d 2 π α = 4 = 24 6 Lüscher string k- string: mul?ple Lüscher strings, d <<L, k= l -m Supergravity(SUGRA) dual models in d = 2+1, d=3+1(our work): π ( d + p 3) α = 24 6/42

7 N-ality k N=3 K=3-string 7/42

8 N-ality k N=3 K=3-string Baryon and anti-baryon form 8/42

9 N-ality k N=3 K=3-string Baryon and anti-baryon form Tension vanishes Behavior called 3-ality 3 Generally: N-ality k 9/42

10 Branes, Gauge Theory, and Supergravity 10/42

11 Branes, Gauge Theory, and Supergravity Closed strings = Supergravity Open strings + D-branes = gauge theory 11/42

12 Branes, Gauge Theory, and Supergravity Open strings + D-branes = gauge theory (N=1) Gauge flux flows between string to D-brane: 1 d- brane => N=1 gauge theory D-brane perspective: D- brane itself IS the color flux tube (N=1) 12/42

13 Branes, Gauge Theory, and Supergravity Open strings + D-branes = gauge theory (N=3) N=3 D-branes = N=3 colors 9 types of strings => 9 gluons in a U(3) gauge theory when the D-branes are parallel and `squashed together so they are effectively on top of each other 13/42

14 Branes, Gauge Theory, and Supergravity N stacked D-branes = gauge theory U(N) N parallel and infinitesimally close D- branes (stacked D- branes) => U(N) gauge theory permeating their world volume 14/42

15 Branes, Gauge Theory, and Supergravity Open to Closed strings: D-branes as a Supergravity Source N stacked D-branes: a U(N) gauge theory 15/42

16 Branes, Gauge Theory, and Supergravity Open to Closed strings: D-branes as a Supergravity Source N stacked D-branes: a U(N) gauge theory Naturally couple to Ramond- Ramond (R-R )flux (closed strings) 16/42

17 Branes, Gauge Theory, and Supergravity Open to Closed strings: D-branes as a Supergravity Source N stacked D-branes: a U(N) gauge theory Naturally couple to Ramond- Ramond (R-R )flux (closed strings) Closed strings generate supergravity (low energy effective theory) 17/42

18 Branes, Gauge Theory, and Supergravity Open to Closed strings: D-branes as a Supergravity Source N stacked D-branes: a U(N) gauge theory Naturally couple to Ramond- Ramond (R-R )flux (closed strings) Closed strings generate supergravity (low energy effective theory) Low energy effective picture: stacked branes with R-R flux comprise a heavy object that generates supergravity and is also endowed with gauge theory => gauge/gravity duality 18/42

19 Branes, Gauge Theory, and Supergravity Probes of Supergravity: Dual Gauge Theory Objects String probes: dual to various fundamental Lüscher string situations and analogous Wilson loops in conformal field theories 19/42

20 Branes, Gauge Theory, and Supergravity Probes of Supergravity: Dual Gauge Theory Objects Brane probes: Wrapped with flux => dual to k-strings 20/42

21 Branes, Gauge Theory, and Supergravity Probes of Supergravity: Dual Gauge Theory Objects 21/42

22 k-strings/d-branes: gauge/gravity duals 22/42

23 Conformal Case Drukker, Fiol; Gomis, Passerini; Faraggi, Mueck, Pando Zayas Conformal field theory (CFT), no running couplings, so no confining structure such as the k- string Wilson loop (WL) in the CFT => analogous to k- strings in a confining theory different supergravity objects are dual to different WL representations Take AdS 5 x S 5 generated by N D3-branes in type IIB string theory: dual to 4D N=4 SU(N) gauge theory (CFT) Fundamental WL: fundamental string Symmetric WL: D3-brane endowed with electric flux Antisymmetric WL: D5-brane endowed with electric flux Various combinations of D3 and D5-branes: mixed representations These have been rigorously worked out 23/42

24 Conformal Case Drukker, Fiol; Gomis, Passerini; Faraggi, Mueck, Pando Zayas Conformal field theory (CFT), no running couplings, so no confining structure such as the k- string Wilson loop (WL) in the CFT => analogous to k- strings in a confining theory different supergravity objects are dual to different WL representations This suggests the following: Symmetric WL: D3-brane endowed with electric flux embedded in a D3-brane sourced background Antisymmetric WL: D5-brane endowed with electric flux embedded in a D3-brane sourced background 24/42

25 Conformal Case Drukker, Fiol; Gomis, Passerini; Faraggi, Mueck, Pando Zayas Conformal field theory (CFT), no running couplings, so no confining structure such as the k- string Wilson loop (WL) in the CFT => analogous to k- strings in a confining theory different supergravity objects are dual to different WL representations Implies the following: Symmetric WL: D5-brane endowed with electric flux embedded in a D5-brane sourced background Antisymmetric WL: D3-brane endowed with electric flux embedded in a D5-brane sourced background 25/42

26 Confining Case 26/42

27 k-strings from Supergravity Duals (Herzog and Klebanov, 2002) k-string is a probe Dp-brane with U(1) gauge flux F = da = Ftxdt dx + Fθφdθ dφ, embedded in a classical SUGRA sourced by F dc H db n =,,. n 3 = 2 Φ Sources are functions of Bosonic SUGRA coordinates, X m, which become fields in a field theory, along with bosonic U(1) fields A, and fermionic fields : S = µ d ς e g + + µ C F + S p + 1 Φ det ( F ) p p ab ab p q 2 f q F Θ ( ) ( ) = B2 + 2 πα ' F, Sf = S f ς, ς Θ Θ = 0 classically, g is pullback of SUGRA, G,onto Dp-brane: g = G X ς ab µν ab µν a b X ς µ ν 27/42

28 k-strings from Supergravity Duals (Herzog and Klebanov, 2002) k-string is a probe Dp-brane with U(1) gauge flux F = da = Ftxdt dx + Fθφdθ dφ, embedded in a classical SUGRA sourced by F dc H db n =,,. n 3 = 2 Φ Sources are functions of Bosonic SUGRA coordinates, X m, which become fields in a field theory, along with bosonic U(1) fields A, and fermionic fields : Θ S = µ d ς e g + + µ C F + S p + 1 Φ det ( F ) p p ab ab p q 2 f q Gauge/gravity Correspondence: minimized, classical Hamiltonian of this action gives us the k-string tension: H = TL 28/42 min k

29 k-string Tensions from Holographic Methods Tension is same for D5-brane probes of either Maldacena-Nunez (MN) or Maldacena_Nastase backgrounds. MN: dual to 3+1 SU(N) gauge theory MNa: dual to 2+1 SU(N) gauge theory Looks similar to Casimir formula: T k k N k N Casimir law 29/42

30 k-string Tensions in d=2+1 Maldacena-Nastase background (MNa) 30/42

31 k-string Tensions in d=3+1 Maldacena-Nunez background (MN) 31/42

32 k-string Tensions in d=2+1 Representation Analysis Maldacena-Nastase backgorund (MNa): D3-brane probe is antisymmetric rep D5-brane probe is symmetric rep Cvetic, Gibbons, Lu, and Pope backgorund (CGLP) D4-brane probe is antisymmetric rep 32/42

33 Confining vs. conformal Conformal: clear picture, known what to expect Symmetric WL: D5-brane probe of D5-brane sourced background or D3-brane probe of D3-brane sourced background Antisymmetric WL: D5-brane probe of D3-brane sourced background or D3-brane probe of D5-brane sourced background Confining: not known what to expect, have to compare results and see: Seems the smaller brane is the anti-symmetric rep, the larger is the symmetric rep (e.g. D3 in MNa is symmetric rep., D5 in MNa is antisymmetric rep.) 33/42

34 Lüscher Term Universality 34/42

35 Lüscher Term from Kory Stiffler k-strings Supergravity Duals (Pando Zayas, Rodgers, Stiffler, et al., 2008-present) Fluctuate about classical solution: X = X + δx, A= A + δa, Θ= 0+ δθ S = S X X A A Θ Θ = S + S + S + (0) (1) (2) [,,,,, ]... Invoke gauge/gravity correspondence: e ie 1 T gauge = Z 2 Z SUGRA = DX DADΘ DΘe i S 2 2 GaugeTheory 0 0 (0) (2) = S + + S + Supergravity /42

36 Lüscher Term from Kory Stiffler k-strings Supergravity Duals (Pando Zayas, Rodgers, Stiffler, et al., 2008-present) Fluctuate about classical solution: X = X + δx, A= A + δa, Θ= 0+ δθ 0 0 S = S X X A A Θ Θ = S + S + S + (0) (1) (2) [,,,,, ]... (0) (2) = S + + S MN: d+p-3 = 5 massless modes: (d = 3, p=5) 36/42

37 Lüscher Term from Kory Stiffler k-strings Supergravity Duals (Pando Zayas, Rodgers, Stiffler, et al., 2008-present) Fluctuate about classical solution: X = X + δx, A= A + δa, Θ= 0+ δθ 0 0 S = S X X A A Θ Θ = S + S + S + (0) (1) (2) [,,,,, ]... (0) (2) = S + + S MNa: d+p-3 = 6 massless modes: (d = 4, p=5) 37/42

38 Lüscher Term from Kory Stiffler k-strings Supergravity Duals (Pando Zayas, Rodgers, Stiffler, et al., 2008-present) Fluctuate about classical solution: S = S X X A A Θ Θ = S + S + S + Invoke gauge/gravity correspondence: e ie 1 T gauge = Z 2 Z SUGRA = DX DADΘ DΘe i S 2 2 GaugeTheory One loop Energy: X = X + δx, A= A + δa, Θ= 0+ δθ 0 0 (0) (1) (2) [,,,,, ]... E 1 (0) (2) = S + + S + Supergravity 0... α π( d+ p 3) = + β, α = L 24 38/42

39 Lüscher Term from Different Models Lattice calculations in d=2+1(teper, 2008): d 2 π α = 4 = 24 6 SUGRA duals(stiffler et. al., 2008,2009): π ( d + p 3) α = 24 k-string dual (CGLP background) D4-brane π d = 2+1 = 3, p = 4: α = 6 k-string dual (MNa background) D3-brane: d = 2+1 = 3, p = 3: α = π 8 39/42

40 Lüscher Term Universality Lattice calculations in d=2+1(teper, 2008): d 2 π α = 4 = 24 6 SUGRA duals(stiffler et. al., 2008,2009): Same formula for all supergravity models A judicious choice of d and p in the supergravity model will yield the Lüscher term of the lattice calculation: p = 3d 5 π ( d + p 3) α = 24 Are the SUGRA models in a Larger Universality class of which the Lattice models are a subset? 40/42

41 Summary K-strings: color flux tubes Branes Source supergravity Endowed with gauge theories k-strings/d-branes: gauge/gravity duals Conformal Case: Fundamental WL: fundamental string Symmetric WL: D3-brane in D3-brane background, D5-brane in D5-brane background Antisymmetric WL: D5-brane in D3-brane background, D3-brane in D5-brane background Our results (confining cases) fundamental WL: Symmetric WL: larger brane Antisymmetric WL: smaller brane Lüscher term universality: SUGRA models lattice models? 41/42

42 Thank You You the audience Co-authors: L. Pando Zayas, V.G.J. Rodgers, B. Button, S.J. Lee The Curtrights, the other organizers, and the U Lago Mar Resort Broward County Transit (routes 1 and 40) Adi Armoni, Parameswaran Nair, Michigan Center for Theoretical Physics, Aspen Center for Physics, National Science Foundation Grants Nos (Aspen) and (University of Iowa), Department of Energy grant DE-FG02-95ER40899 (University of Michigan), endowment of the John S. Toll Professorship, the University Maryland Center for String & Particle Theory, National Science Foundation Grant PHY , Indiana University Northwest 42/42