Seminar given at NTHU/NTS Journal lub, 010 Sep. 1 Holographic Anyons in the ABJM theory Shoichi Kawamoto (NTNU, Taiwan) with Feng-Li Lin (NTNU) Based on JHEP 100:059(010)
Motivation AdS/FT correspondence provides a useful tool to probe strong coupling dynamics of (gauge) field theory. Recently, this is applied to the analysis of the problems of condensed matter theory. For example, holographic superconductors. Today, we construct another tool for this application (not application itself, today)
AdS/FT correspondence Best nown case: D-branes and N=4 super Yang-Mills theory N D-branes 10D SUGRA urved space (Blac -brane) Flat space decouple Maldacena limit (N, α 0) Gauge theory (open string) z= z=0 : AdS boundary N=4 Supersymmetric Yang-Mills String (SUGRA) in AdS 5 * S 5 (Strong coupling regime: λ=g YM N= ) (wea curvature: L 4 =λα = ) orrespondence: Symmetry, States, correlation functions,...
Various AdS/FT correspondence There are several variants of AdS/FT correspondence (or rather gauge/gravity ). Symmetry Superconformal 4D N=4 super YM (AdS 5 ), D N=6 S (AdS 4 ) Less SUSY No SUSY, conformal Not SUSY, nonconformal orbifolding, etc. finite temperature (AdS-BH), holographic QD, phenomenological (not from string theory) holographic superconductor more reliable correspondence, field theory side is nown
Multiple M and AdS 4 It is nown that M-brane bacground also admits AdS 4 description in a near-horizon limit. However, no superconformal theory on M has been nown for long time. Recently, there are much progress on multiple M theory: BLG model and ABJM model. Both are superconformal hern-simons theory in +1 dimenions. uriosity in +1D: Anyonic states
What is Anyon? Anyon is a particles that obeys fractional statistics between bosons and fermions onsider the (adiabatic) exchange of the two identical particles: relative configuration space of these two particles in (d+1)-dim. d R Z { 0} d 1 \ RP
Introduction to Anyons B π 1 A d 1 ( RP ) = Z 1 RP d d A B ( ) Trivial Path One exchange (noncontractable) Exchange twice (contractable) ± 1 phase is allowed (boson and fermion) A B π 1 Trivial Path One exchange (noncontractable) Exchange twice (noncontractable) ( RP ) = Z inθ e ± 1 RP ( d = phase for n winding. ) Anyon!! A B
Anyons and hern-simons theory Abelian hern-simons Lagrangean with the source term: L = ε 4 π μνρ A μ ν A ρ A μ J μ For point charge distribution J = ( ρ,0,0) ρ = q δ x x ) ( i ρ q Equation of motion: B = = δ ( x xi ) hern-simons term attach a magnetic flux to each particle. B q
Anyon phase as AB phase onsider that a particle is adiabatically going around the other particle. Due to the magnetic flux, the particle gets the Aharanov-Bohm phase as, exp ( ) = iq iq A exp Exchange of these two particle is understood as half of this procedure. Anyon phase is given by θ = φ AB = q
Holographic realization of Anyon Motivation: Anyon is not just a mathematical construct, but appears in a real physical system lie Fractional Quantum Hall System. So... Question: an we realize anyons holographically? Setup: Want to have +1D hern-simons theory and its dual ABJM: +1D N=6 hern-simons-matter theory and its dual description of AdS 4 * P is well studied. A good starting point!
Plan 1.Introduction What is Anyon?.Gravity Side IIA Sugra on AdS space.field Theory Side N=6 Superconformal S 4.onclusion
Gravity Side IIA Supergravity on AdS 4 P
AdS 4 P bacground We start with the gravity side. The bacground geometry we wor with is AdS 4 times P spacetime F 4 F +1D boundary This space-time has RR four-form and two-form fluxes. ( dω Kahler form of P ) metric The curvature radius is R = 5/ π λ N λ = and IIA picture is valid for
Strategy We are constructing an analogue of the source-charge pair. Encircling will pic up a phase Source: D0-brane together with the bacground 4-form, it generates H = db harge: fundamental string B charged under NSNS -form field
Supergravity equations of motion (A part of) Type IIA supergravity with D0-brane sources + + + = Φ Φ Φ 4 4 4 4 10 ~ * ~ * * 4 1 F F B F F e F F e H H e S κ 0 μ 0 D 1 1 4 4 ~ H F F = 0 ) ( ~ * * 0 0 10 4 = + Φ Φ x D F e H F e d δ μ κ 1 equation of motion with the bacground four-form field F 4, D0-brane induces the NS-NS -form field B. linearized equation for induced fluctuation by D0 n n n F F F + ) (0
D0 - F1 pair AdS 4 Σ (we discuss this end later) We consider a fundamental string going around D0-brane. What we need to calculate is A B Σ An approximation: Minimal distance between D0 and F1 is long enough to neglect the position of P. (Originally considered by Hartnoll for D1-F1 in AdS 5 *S 5 and M in AdS 7 *S 4 )
AB phase from D0 source average over P liniearized eom Δφ F1D0 = π N F is massive form structure does not match ΔF F δ ( x)
D on P 1 and a survived phase The phase we obtained is subleading in large N... D-brane wrapped on P 1 is a baryon-vertex. (Witten) D-brane WZ term: F1 1 F πα ' A t P t To cancel this induced electric charge, we need to attach fundamental strings Then we exchange D (with F1) and D0, D on P 1 we get the phase in total: A t Δφ = π = N π λ
Anyon as D0-D bound state So far, what we have calculated is an AB phase. Anyon is equivalent particles that gets fractional phase under exchange. an we construct BPS one? θ 1 (allan-maldacena, Gibbons, Druer-Fiol) We can construct BPS D + F1 as a spie D-brane. F1 flux (SK, Lin) However, turning on magnetic fields (D0-brane charge), BPS condition breas down. No BPS anyons.
Summary for Gravity part D or D6 details of D4-D6 D0 or D4 F1 strings attached to D or D6 baryon pic up phase when they go around D0 or D4 source. Δφ D D0 = πλ 1 Δφ πλ D6 D4 = The other combinations will not have a trivial phase. Note: These phases are long-range effects, valid only when the source and F1s are separated enough. Otherwise, phases get modified and disappear. We can construct anyons by D0-D- F1 or D4-D6-N F1 bound state, though they are not BPS.
Field Theory Side N=6 hern-simons-matter theory
N=6 hern-simons-matter theory Field theory side by ABJM is Supersymmetric hern-simons-matter theory: L + = LS + Lin + Lψψ L 6 L S = Tr AdA 4π A ~ ~ AdA 4π ~ A U(N) * U(N) hern-simons-matter theory in +1 dimensions N=6 supersymmetry and SU(4)*U(1) symmetry Symmetric under parity transformation We mainly consider the operator made of ( ), : bosons ( N, N; 4) + 1 ( N, N; 4) 1 They are and of ( U ( N), U ( N) ; SU (4) R ) U (1) B
D0 and hiral primary operator The chiral primary operator that is dual to D0-brane is identified as (ABJM) But they are not gauge invariant. To mae it gauge invariant, we attach a monopole operator m. Magnetic flux (Dirac string) It transforms as ( SymN, SymN ) Going around this string, fields get gauge transformation, i φ N e,e i φ N (φ : polar angle around the loop) more
D baryon There are fundamental strings attached to D on P 1. Q The other end points need to end on the boundary, and give fields in the fundamental representation. Assume: They are bound state of N of first U(N). Q They are the dual of the fundamental strings, Wilson lines with copies of the fundamental representation of the first U(N).
AB phase in the field theory side Now D-brane baryon (Q ) is going around and Dirac string. Q Q e πi N Q Then the AB phase for this configuration is = e πi N Δφ = N Q = λ 1 Matching with the gravity side result!! Note: D6 baryon is N bound state of anti-q, and gets no phase.
Summary for Field Theory part or det() The state corresponding to D0 or D4, we need to attach magnetic flux ( t Hooft operator) to mae them gauge invariant. Q or Q N Extra state from D or D6 baryons get the phase when going around the flux Δφ Q = πλ 1 Δ = Q N det( ) φ πλ details of det() operator The other combinations will not have a trivial phase.
onclusion and Discussion We calculated the anyonic phase arising from Dp-Dq-F1 configuration. D0-D- F1 and D4-D6-N F1 pair lead anyonic phase. We may construct anyons by bound states of above combinations. We also calculated the same phase for (probably) corresponding field theory configuration. These phases are matched to D-brane results. Future directions... Other brane configurations. More realistic setup??
Than you!
AdS 4 P bacgound bac
The ABJM action = S ~ ~ ~ 4 4 Tr A AdA A AdA π π L L 6 L L L L + + + = ψψ in S Aharony-Bergman-Jafferis-Maldacena s N=6 Superconformal hern-simons-matter action ( ) I I I D i D ψ γ ψ μ μ μ + = L in Tr ( ) ( ) [ ] 1 1 1 1 ] [ ] [ six Tr 8 I I I I I I I I I I I I L + = π
omment on D6-D4 pair Fluctuation by D4 brane source can also be considered, and the linear perturbation is onsider D4 wrapping on J J, where J is proportional two Kahler two form on P. 1 Δφ = π
The phase from D4-D6 pair We then now consider D6 Baryon. Due to the magnetic six-form on P, F 6 = *F4 we have F 6 A t N A t Then N anti-fundamental strings are attached to it. Thus we get the phase for D4-D6 pair, N ΔφD4 D6 = π = πλ Note: D-D4 pair and D6-D0 pair have trivial phase! bac to the summary
The operator corresponding to D4 det( ) arrying (N, -N) U(1) b charge. dual to D4 on P (ABJM, Par) Need to cancel U(1) charge. Attach U(1) Wilson line or t Hooft operator (equivalent) det() To cancel N U(1) charge, t Hooft operator causes the gauge transformation i φ e,e i φ D baryon cannot detect it. monopole operator
The phase from det()-d6 Baryon N N i N N i N Q e Q e Q π π = The AB phase for this pair is λ φ = = Δ N We consider the similar baryonic state from D6-brane. N strings need to attach it and the direction is opposite to D ones. Assume we have N Q (N bound state of anti-fundamental) bac to the summary
Monopole operator (I) In +1D, t Hooft loop becomes a local operator, t Hooft operator. (localized magnetic flux) θ It generates a gauge transformation of the center T h e ihθ 1 1 1 h = diag,,, 1 N N N (Par) With hern-simons action, this is equivalent to the insertion of Wilson line SS SS δss Dg Th e Dg e = We + SS For SU(N) hern-simons of level, this Wilson line is in SymN representation and then can be used to mae gauge invariant. (Itzhai) bac
Monopole operator (added) Z N in the previous page is not the center for each U(N) in ABJM. We may define through U(1) subgroup of U(N)? (Kapustin-Witten, Klebanov et al, Imamura,...) It generates a gauge transformation around it: (localized magnetic flux) iαθ e label for artan For nonabelian group U(N), we can consider U(1) r artan rotations i A μ = λ i dθ Fact: There is U(N) hern-simons action, this monopole belongs to the representation i whose highest weight is given by λ i λ = ( 1,0,,0) SymN representation Not sure that it generates the desired gauge rotation.
Monopole operator (II) Diagonal U(1) part ~ A D A + A μ = couples only through A B F D μ μ d α = Dualize dual photon B A iα e is a monopole operator. This changes the flux F D 1 unit magnetic flux It transforms under U(1) B ( N, N ) Thus taing 1/N unit of anti-flux, we can mae gauge invariant. note: Since no field is charged under U(1) D, flux quantization condition can be relaxed. However, it may be problematic for the moduli space structure. (Par) return