Vortex solutions on membranes

Transcription

1 Vortex solutions on membranes Shinsuke Kawai SKKU, Suwon APCTP YongPyong Winter School, February 2010

2 Outline 1 AdS/CFT AdS/CMT The ABJM model 2 Non-relativistic mass-deformed ABJM Relativistic mass-deformed ABJM The non-relativistic limit Non-relativistic SUSY 3 BPS vortex solutions The BPS equations The exact vortex solutions Examples Preserved supersymmetry 4 Summary and comments

3 What is string theory?

4 What is string theory? The theory of strong interaction

5 What is string theory? The theory of strong interaction

6 What is string theory? The theory of strong interaction The theory of everything

7 What is string theory? The theory of strong interaction The theory of everything AdS/CFT: Logical completion of QFT (incl. QCD, Condensed Matter Systems...)

8 AdS/CFT AdS/CFT correspondence (Some) strongly coupled field theories have gravity duals [Maldacena] Conformal group SO(d, 2) AdS d+1 (Poincaré M µν, P µ, Dilatation D, SCG K µ ) Poincaré coordinates are ds 2 = ρ 2 η µν dx µ dx ν + dρ2 ρ 2 Dilatation is D : x µ λx µ, ρ ρ/λ Dictionary: [Gubser, Klebanov, Polyakov] [Witten] e R φ 0O = Z AdS [φ 0 ] e S[φ0] Boundary conditions states Variation of boundary conditions correlation functions

9 AdS/CFT Radial coordinate energy scale ( holographic renormalization ) A realisation of holographic principle Canonical example: AdS 5 S 5 in Type II B string theory N = 4 SYM in 4 dim SO(4, 2) conf SO(6) R : isometries of the spacetime

10 AdS/CFT λ = gym 2 N: t Hooft coupling 3 versions of AdS/CFT weak: large λ strong: any finite λ but N and g s = g 2 YM 0 (exact in α but for small g s only) strongest: any g s and N (α and g s )

11 AdS/CMT Condensed matter applications of AdS/CFT Application to QCD relatively successful why not Condensed Matter Theory? AdS/CFT in laboratory Strongly correlated systems near criticality in low dimensions Rich examples Good control A new computational tool in CMT Beyond perturbative field theory and lattice A new arena of string theory Typically, non-relativistic ( (q q c ) νz, ξ (q q c ) ν ) D : t λ z t, x i λx i NR AdS/CFT duality: limited technology a new challenge

12 AdS/CMT Examples of condensed matter systems Superconductivity Superfluidity Quantum Hall systems AdS/CFT involves large N SYM. The AdS is realised by some other fields CMT is assumed to be a probe on a background geometry Supersymmetry broken by finite T (?) Phenomenological approach be maximally optimistic and use as a computational tool Questions Phase transition, solitonic excitation (brane excitations) Especially, vortices play important roles in above examples ABJM: M-theory on AdS 4 S 7 /Z k CFT on M2-branes

13 AdS/CMT Superfluidity in He 4 He phase diagram (left) and 3 He phase diagram (right) Source: Low Temperature Laboratory, TKK, Finland

14 AdS/CMT Vortices in 3 He superfluidity A closer look at the low temperature 3 He phase diagram (left) and two types of vortices in the superfluid B phase (right) Source: Low Temperature Laboratory, TKK, Finland

15 AdS/CMT Our focus: vortex solutions on the CFT side. Solitonic solutions in ABJM Abelian vortices (Jackiw-Lee-Weinberg type) and domain walls [Arai, Montonen, Sasaki 2008] Non-abelian vortices [Kim, Kim, Kwon, Nakajima 2009] [Auzzi, Kumar 2009] Abelian, non-relativistic vortices [Kawai, Sasaki 2009]

16 The ABJM model The ABJM model The ABJM model [Aharony, Bergman, Jafferis, Maldacena (2008)] ( ) SABJM bos = d 3 x L bos kin + L CS VD bos VF bos, L bos h kin = Tr (D µzâ) (D µ ZÂ) + (D µwǎ) (D µ i WǍ), L CS = V bos D = V bos F = k 4π ǫµνλ Tr ha µ νa λ + 2i 3 AµAνA λ  µ ν  λ 2i  µ  ν  λ i, 3 4π 2 h Z k 2 Tr ˆB Z ˆB Z ZÂZ ˆB Z ˆB ˇB W WˇB Z + ZÂWˇB W ˇB 2 + W ˇB WˇB W Ǎ Ǎ W WˇB W ˇB ˆB Z Z ˆB W Ǎ Ǎ + W Z ˆB Z ˆB 2 i, 16π 2 h ǫâĉ k 2 Tr ǫˇbď 2 WˇB ZĈWĎ + ǫǎč ǫˆb ˆD Z ˆB ˆD 2 i WČZ. A µ, µ U(N) U(N) gauge fields, and ZÂ, W Ǎ ( = 1, 2, Ǎ = 3, 4) complex scalars in U(N) U(N) bi-fundamental (N, N) rep

17 The ABJM model Chern-Simons-matter theory on 2+1 dimensions X1,...,X8 2+1 dim (ZÂ, W Ǎ ) e 2πi k (ZÂ, W Ǎ ) M-theory on AdS 4 S 7 /Z k N,k, λ = N k fixed ( t Hooft limit): IIA on AdS 4 CP 3

18 The ABJM model We consider: The ABJM model (Chern-Simons-matter theory) massive deformation Mass-deformed ABJM non-relativistic limit NR, mass-deformed ABJM solving BPS eqns BPS vortex solutions

19 Vortex solutions on membranes The ABJM model A bigger picture: ABJM CSm-th Massive ABJM NR Massive ABJM NR BPS vortices M-theory on AdS 4 S 7 /Z k Fuzzy spheres NR gravity dual of NR BPS vortices?

20 Non-relativistic mass-deformed ABJM Relativistic mass-deformed ABJM Relativistic mass-deformed ABJM Before taking the NR limit we take massive deformation [Hosomichi, Lee 3, Park 2008], [Gomis, Rodríguez-Gómez, Van Raamsdonk, Verlinde 2008] introducing a scale that is necessary for the solitonic solutions Maximally supersymmetric (N = 6) massive deformation + 4πm k SO(8) R SU(2) SU(2) U(1) Z 2 The scalars acquire equal masses The change [ of (the bos. part of) the Lanrangian: δl = Tr m 2 Z Â ZÂ m 2 W ǍWǍ ( (ZÂZ Â )2 (W ǍWǍ) 2 (Z ZÂ) 2 + (WǍW Ǎ) 2)] Â

21 Non-relativistic mass-deformed ABJM The non-relativistic limit The non-relativistic limit Now we consider the NR limit [Nakayama, Sakaguchi, Yoshida 2009], [Lee ] 1 Write down the action with ( c and explicitly ) 2 Decompose: Z = 2m e i mc2 t zâ + e i mc2 t ẑ  etc. (zâ, ẑ  are non-relativistic scalar fields) Keep the particle DoF (zâ,w Ǎ ) & drop the antiparticles 3 Send c, m and look at the leading orders The resulting (bos. part of the) Lagrangian: L NR,bos ABJM = k c 4π ǫµνλ Tr " i +Tr 2» A µ νa λ + 2i z  Dtz + D tzâ z  3 AµAνA λ  µ  ν  λ 2i  µ  ν  λ 3 «2 2m D izâd i z  + i wǎdtw Ǎ + Dtw Ǎ wǎ 2 2 2m D iw Ǎ Di wǎ + π 2 j 2ff # km (zâz Ǎ )2 (z  zâ) 2 (w Ǎ wǎ) 2 + (wǎw Ǎ ).

22 Non-relativistic mass-deformed ABJM The non-relativistic limit The equations of motion The scalar part: nonlinear Schrödinger equations i D tzâ = 2 2m D2 i zâ 2π 2 km (z ˆB z ˆB zâ zâz ˆB z ˆB ), i D tw Ǎ 2 2 2π = 2m D2 i w Ǎ + (w ˇB km wˇb w Ǎ Ǎ w w ˇB wˇb ). The gauge field part: Gauss-law constraints The fermionic part: i D tψ A + 2mc 2 δâ A ψ Â i cd ψ +A = 0, i D tψ +A + 2mc 2 δǎ A ψ +Ǎ i cd+ψ A = 0. (due to these NR Dirac eqns 1/2 of the fermionic DoF drop)

23 Non-relativistic mass-deformed ABJM Non-relativistic SUSY Non-relativistic SUSY Apply the same procedure of NR limit to SUSY trfn rules non-relativistic SUSY (super Schrödinger symmetry) 14 supercharges: 10 kinematical SUSY: ( ω + ˆB, ω ǍˇB, ω ) ±ÂˇB 2 dynamical SUSY: ( ω ˆB, ω ) +ǍˇB 2 conformal SUSY: ( ) ξâˆb, ξǎˇb SUSY parameters defined by ω AB = ǫ i Γ i AB (i = 1, 2,..., 6), ω = 1 2 «ω + ω +, ω i ω + i ω AB + ± = ( ω ±AB) = 1 2 ǫabcd ω ±CD Conformal SUSY Special conformal charge Dynamical SUSY, S = i[k, Q D ]

24 BPS vortex solutions The BPS equations The BPS equations The Hamiltonian density (Noether charge of the time translation) " 2 H = Tr 2m Diz m D iw Ǎ 2 π 2 km j (zâz  )2 (z  zâ) 2 (w Ǎ wǎ) 2 + (wǎw Ǎ ) 2ff #. Using D ± D 1 ±id 2 and Bogomol nyi completion = BPS bound: E = d 2 x H = [ d 2 2 x Tr 2m saturated by BPS equations D zâ D zâ = 0, D + w Ǎ = 0, Recall: zâ and w Ǎ are matrix-valued (N N) D + w Ǎ 2] 0 2m

25 BPS vortex solutions The BPS equations The fuzzy 3-sphere ansatz [Gomis, Rodríguez-Gómez, Van Raamsdonk, Verlinde 2008] zâ(x) = ψ z (x)s I, w Ǎ(x) = ψ w (x)s I, A i (x) = a i (x)s I S I, Â i (x) = a i (x)s I SI. Here, ψ z, ψ w, a i C, (S 1 ) mn = m 1δ mn, (S 2 ) mn = N mδ m+1,n S I = S J S J SI S I S J SJ, S I = S S J S I J S J SJ S I, Tr S I S I = Tr S S I = N(N 1). I Then the BPS eqns = the Jackiw-Pi vortex eqns [Jackiw and Pi 1990] (D 1 id 2 )ψ z (x) = 0, (D 1 + id 2 )ψ w (x) = 0 (D i i + ia i ) J-P eqns allow exact solutions

26 BPS vortex solutions The exact vortex solutions The exact vortex solutions Finding solutions: Set w Ǎ = 0 & solve for zâ, A i and Âi (BPS I), or set zâ = 0 & solve for w Ǎ, A i and Âi (BPS II) The radius of the fuzzy S 3 (in the case of BPS I): R 2 = 2 [ ] Tr NT ZÂZ = N 1 ψ z 2 M2 Â T M2 m, T M2 : the tension of an M2-brane

27 BPS vortex solutions The exact vortex solutions The BPS I solutions 1 Changing the variables ψ z (x) = e iθ(x) ρ 1 2 (x), (θ, ρ R), the BPS eqns become a i (x) = i θ ǫ ij j ln ρ. 2 Using the Gauss law constraint = Liouville equation 2 lnρ = 4π k ρ, solved by ρ(x) = k 2π 2 ln ( 1 + f (z) 2) (f (z): a holomorphic function of z = x 1 + ix 2 ) 3 θ is fixed by regularity of ψ z at z = 0

28 BPS vortex solutions Examples Examples Examples of BPS I solutions Choose a profile: f (z) = ( z 0 ) n, z n Z, z0 : a complex const; yielding ρ(x) = k 4n 2 2π r 0 2 r 2(n 1) r0» 1 + r r0 2n 2, θ = (n 1) arg z = (n 1) arctan(x 2 /x 1 ) These are non-topological vortices since ψ z 0 as z 0.8 Ρ x x Ρ x Ρ x2 0 x x Figure: ψ z 2 shown for f (z) = 1 z, 1 z 2 and 1 z(z 1), with k = 1

29 BPS vortex solutions Examples Examples of BPS II solutions Setting zâ = 0 we find similar solutions for w Ǎ : ψ w (x) = e iθ(x) ρ 1 2 (x), ρ(x) = k 2π 2 ln ( 1 + f (z) 2), θ = (n 1)arctan(x 2 /x 1 ).

30 BPS vortex solutions Preserved supersymmetry Preserved supersymmetry Check the SUSY trfns against the BPS eqns (in the BPS-I case) w Ǎ = 0, D zâ = 0. The fermion transformation rules: δ K ψ = ω + +ˆB z ˆB, δk ψ = +ω Ǎ ǍˆB z ˆB, δ D ψ = i + 2m ω ˆB D+z ˆB, δd ψ = 0, Ǎ δ S ψ ξ + ˆB z ˆB, δs ψ 0. Ǎ Hence δψ = 0 = ω + ˆB = ω ˆB = ω ǍˆB = ξ ˆB = 0 This means that the BPS-I solutions break 5 kinematical, 1 dynamical and 1 conformal SUSYs (i.e. exactly 1/2).

31 BPS vortex solutions Preserved supersymmetry The BPS II case is similar. Summarising the results, Type of Kinematical Dynamical Conformal SUSY ω + ¡B ω + ˆB ω ǍˇB ω ǍˆB ω ˆB ω + ǍˇB ξâˆb ξǎˇb BPS I BPS II Table: : preserved, : broken

32 BPS vortex solutions Preserved supersymmetry Summary of our solutions We find exact solutions of abelian vortices by solving BPS equations in the non-relativistic ABJM model: Jackiw-Pi combined with fuzzy 3-sphere These solutions preserve half of the super Schrödinger symmetry Any relevance in real physics? more realistic, parity broken models with external fields desirable.

33 BPS vortex solutions Preserved supersymmetry Vortex solutions in AdS Vortex line Pure AdS [Dehghani Ghezelbash Mann, 2001] AdS-Sch [Dehghani Ghezelbash Mann, 2001] with boundary magnetic field [Albash Johnson 2009] [Montull Pomarol Silva 2009] [Maeda Natuume Okamura 2009] with vanishing magnetic field on the boundary [Keränen Keski-Vakkuri Nowling Yogendran 2009]

34 Summary and comments Unsorted list of problems Non-relativistic AdS/CFT exists at all? In dim, spontaneous breaking of continuous symmetry is not possible at finite temperature (Mermin-Wagner). However, such a phase transition is found in holographic superconductor. How do we interpret? Large N artefact? Berezinskii-Kosterlitz-Thouless transition in AdS?