AdS4/CFT3 correspondence and Chern-Simons gauge theories Jaemo Park (Postech ) Yong Pyong 2010. 02.25 TexPoint fonts used in EMF. Read the TexPoint manual.

AdS4/CFT3 correspondence and Chern-Simons gauge theories

Jaemo Park (Postech ) Yong Pyong

2010. 02.25

Outline

1. Motivation and some history2. N=4 theories via N=1 Formalism3. N=2 formalism and M2 Theories for gener

ic toric singulraities5. Monopole solutions of ABJM and nonpertu

rbative tests of M2 theory6. BPS spectrum and monopole operators7. Future directions

Construction of N=4 theoriesStart from N=1 theories to look for N=4 theories which has SO(4) R sym-

metry. [Gaitto, Witten]Lagrangians for the gauge multiplets

L = ktr(A ^dA +23A ^A ^A ¡ ¹ÂÂ)

For matter multipletsQ = q+ µª ¡ µµF

matter kinetic term is given by

¡12

Zd2µ(DQ)2 = ¡

12D¹ qD ¹ q+ ª (i° ¢D)¸ + F 2 + ª Âq

Integrating out  gives q2ª 2 which is not invariant under SU(2) £ SU(2) ´SO(4) in general. Add quartic homogeneous superpotentials W(Q), classicallyconformal, to make SO(4) invaraint.

The gauge algebras together with fermionic matters should form the super-group algebra to have SO(4) R symmetry. If we consider just hypermultiplets,possible supergroups are U(M jN ) or Osp(M jN ) and we have Chern-Simonsgauge theories of U(M ) £ U(N ) or O(M ) £ Sp(N ) and one bifundamental hy-per. Includingboth hyper and twisted hypers, wehavelinear (or circular) quivertype theories with alternating sequences of hypers and twised hypers satisfyingsupergroup constraints. [Hosomichi, Lee, Lee, Lee, Park]

The gauge group U(M1) £ U(N1) £ U(M2) £ U(N2) ¢¢¢O(M1) £ Sp(N1) £ O(M2) £ Sp(N2) ¢¢¢Under SO(4)R ´ SU(2)L £ SU(2)R

Hyper qA®(2;1);ª A

_®(1;2)Twistedhyper ~qA

_®(1;2); ~ª ®(2;1)

Possible configurations

Moduli space of (U(M ) £ U(M ))n with level (k;¡ k;k;¡ k;¢¢¢):Symmetric product of (C2=Zn)2=Zk

Previous constructions essentially exhaust all possible theories with N ¸ 4.Especially for n = 1 we have C4=Zk

(Aharony, Bergman, J a®eris, Maldacena : ABJ M)This is (conjecturally) dual to gravity theory on AdS4 £ (S7=Zk).This have weak coupling dual of string theory on AdS4 £ CP 3 for large k.

N=2(4 supercharges) formalism

1. gauge multiplets; (dimensional reduction of 4d)

A¹ ;Â;D ! A¹ ;¾;Â;D

L = § akatr(Aa ^dAa +23(Aa ^Aa ^Aa) ¡ ¹ÂaÂa + Da¾a)

2. Matter theories

L = Tr ¡Z

d4µ§ abXyabexp(¡ Va)X abexp(Va) +

Zd2µW(X ab) + c:c:

(X ab bifundamental or adjoint)3. vacuaa. F-term conditions;

@X abW = 0

b. D-term conditions ;¹ a(X ) = ka¾a

c.¾aX ab = X ab¾b

4. Abelian case (with § aka = 0)¾a = ¾so that ¹ a(X ) = 4ka¾we have G-2 ~¹ i (X ) = 0 (G; number of gauge group)and remaining U(1) decides the value of ¾and gives the discrete modding

Zk with k = gcd(ka).5. Non-abelian case; Generically symmetric product of the abelian moduli

space for U(N ) £ U(N ) £ ¢¢¢

4d-3d correspondenceGiven 4D CFT theory ! 3D theory with the same superpotentialmoduli space is non-compact Calabi-Yau four-fold (Membrane tilings)[Hanany, Za®aroni; Martelli, Sparks]Lots of N=2 theories are obtainedBut generic ones are not of this typePrime example C(Q111) coset of SU(2)3=U(1)2 one of Sasaki-Einstein(Analogue of the conifold C(T1;1);SU(2)2=U(1))Metric

ds2 = (dª + § 3i=1cosµi dÁi )2 +

12§ 3

i=1(dµ2i + sin2µi dÁ2

i )

Proposed theory has 6 chiral ¯elds » SU(2)3

W = C1A1B1C2A2B2 ¡ C1A1B2C2A2B1

One can compute the moduli space using the standard technique for toricgeometry (linear sigma model)

For k = (1;1;¡ 1;¡ 1) we indeed have C(Q111)

Evidences of AdS4/CFT3

• Moduli space• index computation (Bhattacharaya, Minwalla N=6; Choi, Lee, Song N=5,4, Kim N=6)• Graviton exchange with nonzero angular momentum• Matching BPS operators and spectrum

Monopole solutions in ABJM andnonperturbative test of M2 brane theory

U(2) £ U(2) theoryEuclideanized CS term contains iSaddle point con¯gurations aregenerally complexWe obtain the monople solution as the usual BPS monopole solutions via

complexi¯ed gauge transformation (Asymptotic vev is real)

D©= 0 ¤F = ¡ D(©¹©)

Important fact: Gaussian constraint relates the magnetic °ux with electriccharge

Monopolesolutions mediate the jump of the magenetic °ux and the chargeThis leads to the gauge invariant amplitude (cf. A² eck, Harvey, Palla,

Semeno®)Monopole vertex is of the form ( z1

z2)kmexp(im¾) for m monopole sector (¾

is the dual photon ¯eld)©= (z1;z2)

From the zero mode structure we expect the prefactor j@zj4

¹ 3

For M2 brane located at ~z and ~w in C4

the probe action gives v4 term

Spr obe =Z

d3xv4(f k;0(~z) + § 1m=0f k;m(~z; ~w)(eim¾+ e¡ im¾))

where f k;m is a Fourier coe±ceint of the distant functionFk(¾) = § f k;meim¾

which gives v4 qk m

¹ 3 eim¾ak;m(q)

• Monopole amplitude matches with the graviton exchange with nonzero angular momentum

(Analogue of Polchinski, Pouliot of membrane scattering in matrix theory)• It is interesting to work out the prefactor as well. (cf. Baek, Hyun, Jang, Yi)

BPS operators

• In Chern-Simons matter theory, this involves the monopole operator

• These operators in field theory could describe nonperturbative objects (e.g. giant gravitons, D2-D0 bound states)

),( r

• D4 brane ~ dibaryon oprator det (due to subtle quantization condition)

1

Conclusions and Future Directions

• Construct various CS gauge theories which can describe M2 brane theories and provide several evidences for them

• Should provide more evidences for AdS4/CFT3• Algorithm of constructing N=2 theories for M2 branes in an arbitrary CY4 singularities• Wilson loop evaluation• Beyond the chiral rings for N=2 theories• Integrability structures?