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# Families of conformal ﬁxed points of N = 2 Chern-Simons-Matter Theories

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### Families of conformal fixed points of N = 2 Chern-Simons-Matter Theories

Chi-Ming Chang

Department of Physics, Harvard University

Based on the work with Xi Yin,arXiv:1002.0568 [hep-th]

and previous work by Gaiotto and Yin 07’

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the worldvolume theory of M2-branes (ABJM model:

N =6 U(N) ×U(N)CSM theory describes N M2-branes living onR8/Z2)

the conformal field theory part of the AdS4/CFT3

correspondence (for example, ABJM model = the M-theory on AdS4×S7/Zk, or more general,N =2 CSM theories can be dual to M-theory on AdS4×M7, M7being the base of a Calabi-Yau 4-fold cone.)

[Aharony, Bergman, Jafferis and Maldacena ’08, Martelli and Sparks ’08

& ’09]

the conformal fixed point in condensed matter systems (for example, Z2spin liquid can be described by the double Chern-Simons theory (N =0 U(1) ×U(1)CSM theory))

[Xu and Sachdev ’09]

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### Outline

1 TheN =2 Chern-Simons-Matter theories Lagrangian

Nonrenormalization theorems The moduli spaceM

2 The Zamolodchikov metric onM

The chiral primary operators and Zamolodchikov metric The Zamolodchikov metric is Kähler

3 Diagram calculations

A 4-loop check of the nonrenormalization theorems Calculate the two-loop Zamolodchikov metric

4 Conclusion

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N =1 in 4d ⇒ N =2 in 3d . V : theN =2 vector superfield Φi: theN =2 chiral superfield (i =1, . . . ,M is the flavor index.)

The superspace Lagrangian for nonabelianN =2 Chern-Simons-Matter theory

L = Z

d4θK+

Z

d2θW +c.c.



K = k

Z 1 0

dt Tr[V Dα(etVDαetv)] + ΦieVΦi (1)

k is the Chern-Simons coupling. W can be any gauge invariant holomorphic function onΦi.

For simplicity, we assume G=U(N),Φi in the adjoint representation and W =0 or 14P αijklTr(ΦiΦjΦkΦl).

[Zupnik and Pak ’88, Ivanov ’91]

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### The N = 2 CSM: Lagrangian

Some supervertices given by the Lagrangian K = k

2π Z 1

0

dt Tr[V Dα(etVDαetv)] + ΦieVΦi

W =0 or 1 4

ijklTr(ΦiΦjΦkΦl)

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Φi Φl

Φj Φk

Φi Φ¯i

V

F-term vertex D-term vertex

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L = Z

d4θK+ Z

d2θW

K = k

Z 1 0

dt Tr[V Dα(e−tVDαetv)] + ΦieVΦi (3)

If W =0, we have U(M)flavor symmetry. Since there is no anomaly for continuous global symmetry in three

dimensions, this U(M)symmetry holds in the quantum theory. The U(M)symmetry and the holomorphy forbid any effective superpotential being generated.

The Chern-Simons level k is quantized to be integer valued in order for the path integral to be invariant under large gauge transformation. Any quantum correction to k at 2-loop or higher order will be suppressed by 1/k , which in general cannot be integer valued. So k is not

renormalized beyond a possible 1-loop shift.

[Avdeev, Grigorev, Kazakov and Kondrasuk ’92 & ’93]

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### The N = 2 CSM: nonrenormalization theorems

L = Z

d4θK+ Z

d2θW

K = k

Z 1 0

dt Tr[V Dα(etVDαetv)] + ΦieVΦi (4)

The Kahlar potential can be renormalized, but any corrections to it are either irrelevant in the IR or can be absorbed by a rescaling ofΦi.

In conclusion, the theory with W =0 is exactly marginal.

[Avdeev, Grigorev, Kazakov and Kondrasuk ’92 & ’93]

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However, W =0 is an unstable fixed point under the RG-flow if we turn on a general quartic superpotential W =14P αijklTr(ΦiΦjΦkΦl). By a two-loop computation

Tr(ΦiΦjΦkΦl)has dimension 2−b0/2k2+ O(k4) <2, where b0=8(M+1)N2. It is a chiral primary so the dimension∆ =4JΦ.

Further, the beta function forαijkl up to two-loop order takes the form

µdαijkl

dµ = (4JΦ−2)αijkl+ 1

2B(irαr jkl)+ higher loop (5) where(ijkl)stands for cyclic symmetrization.

[Gaiotto and Yin ’07]

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### The N = 2 CSM: The moduli space M

µdαijkl

dµ = (4JΦ−2)αijkl+ 1

2B(irαr jkl)+ higher loop (6) Bij = 12N2αiklmαjklm comes from the two-loop wave function renormalization.

The IR fixed points, up to the global flavor symmetry U(M), is parameterized by the quotient space

M0=



αijkl : 1

2N2αiklmαjklm =4π2(2−4JΦij



/U(M) (7) Denote by V the linear vector space of allαijkl’s. Then M0=V//U(M)is the standard simplectic quotient.

[Gaiotto and Yin ’07]

What happen to theM0if we include higher-loop corrections? A new nonrenormalization theorem!

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We consider the Wilsonian effective action, of the form SCSN =2(V) +

Z d3x

Z

d4θK(Φi, Φi,V) +

Z

d3xd2θf(k)X

αijklTr(ΦiΦjΦkΦl) +c.c.

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The nonrenormalization theorem for K is the same as for the case W =0.

By promotingαijkl to dynamical chiral superfields, the effective superpotential is holomorphic inαijkl, Φi and invariant under R-symmetry and the spurious U(M)flavor symmetry (which cannot be anomalous in 3d). This implies W =f(k,P αijklTr(ΦiΦjΦkΦl)). Furthermore, by assigning an R-charge 2 toαijkl, 1 toθ, and 0 toΦi’s, we conclude that the effective superpotential must be linear inαijkl, and the superpotential coefficient can only be renormalized by the Chern-Simons coupling 1/k .

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### The N = 2 CSM: The moduli space M

After canonically normalized the Kahlar potential, the quantum correction toαijkl amounts to an anomalous dimension for the operatorTr(ΦiΦjΦkΦl)together with a wave function renormalization. The beta function forαijkl

takes the form µijkl

= (4JΦ(k) −2)αijkl+ 1

2B(irαr jkl) (9) for some Bij(α, α,k) = 12N2αiklmαjklm+ O(1k, α3).

Therefore, the exact (to all-loop order) moduli space takes the form

M =n

αijkl : Bij =4π2(2−4JΦ) δijo

/U(M) (10) Mis generally a deformation of the two-loop moduli space M0=V//U(M)by the ’t Hooft couplingλ =N/k .

Especially, the dimensions ofMandM0are the same.

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The entire moduli spaceMcan exist in the perturbative regime and its dimension (∼M4) can be very large if the flavor number M is large, in contrast with the moduli space found in[Leigh and Strassler ’95]which interplaying the weak and strong regimes and is low dimensional (usually it is only one-dimensional).

Recently, the paper by[Green, Komargodski, Seiberg, Tachikawa and Wecht]shows that the similar moduli space exists in the four dimensionalN =1 theories, and the Leigh-Strassler’s moduli space is a subspace of it.

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### Chiral primary operators and Zamolodchikov metric

The moduli spaceMis naturally equipped with a Zamolodchikov metric. [Zamolodchikov ’86]

We may write the W 6=0 IR fixed point locus as

µij(α, α,k) =r(k)δji (11) Hereµij is proportional to Bij by normalizing the quadratic terms to be N2αimnpα¯jmnpij is usually called the moment map of the symplectic quotient). So, r(k) =h(k)(12J(k)), where h(k)is a normalization factor which is not important for our purpose. The tangent directionsδαijkl =cijkl are determined by

cmnpq

∂αmnpq

µij(α, α,k) =0, ∀i,j. (12) We define the quartic operators corresponding to tangent directionsOc =P

cmnpqTr(φmφnφpφq). They are chiral primary operators since they cannot be written as Tr(φi∂W

∂φi).

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The two-point functions of a quartic chiral primary and an anti-chiral primary take the form

hOc(x)Oc0(0)i = g(c, ¯c0)

|x|4 . (13) where the coefficient g(c, ¯c0)is the Zamolodchikov metric.

We expect the generic CSM theory to have a holographic dual, which may or may not have a gravity limit at strong ’t Hooft coupling. If there is a gravity dual say of the form M-theory on AdS4×M7, M7being the base of a Calabi-Yau 4-fold cone, then the analog of the manifoldMat strong ’t Hooft coupling would be the moduli space of this CY 4-fold cone. The geometry ofMat strong ’t Hooft coupling is difficult to understand from the field theory perspective.

[Martelli and Sparks ’08 & ’09, Hanany and Zaffaroni ’08, Franco, Hanany, Park and Rodriguez-Gomez ’08]

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### Chiral primary operators and Zamolodchikov metric

There is another (mathematically) natural metric onM.

SinceM =V//U(M)is defined by a symplectic quotient, by definition, given the U(M)action as a vector field on V

vij = αimnp

∂αjmnp

− ¯αjmnp

∂ ¯αimnp, (14) we have dµij = ιv

ijω, whereιstands for the contraction with a vector field. The symplectic formω induces a Kähler metric onM.

There is no reason for these two metrics to be the same, and indeed we will find out that they differ by a factor of two at the two-loop order.

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We argue that the Zamolodchikov metric is Kähler. The variation of the metric along a tangent direction

corresponding to a chiral primaryOc1 is δc1g(c2, ¯c3) = |x|4

Oc2(x)

Z

d3yQ2· Oc1(y)



O¯c3(0)

(15) The statement that g is Kähler amounts to

δc1g(c2, ¯c3) = δc2g(c1, ¯c3), for chiral primaries c1,c2,c3. Let us study the correlation function

F(c1,c2, ¯c3;x,y,z) = hOc1(x)h

Q2· Oc2(y)i

Oc3(z)i (16) Q2· Oc2 is a primary with respect to the bosonic conformal algebra. Apart from potential contact terms, the spatial dependence of the three-point function of the primaries as above are fixed by conformal symmetry, to be

|x−y|3|y−z|3|x −z|1.

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### The Zamolodchikov metric is Kähler

On the other hand, by Ward identity, we can move Q2from acting onOc2(y)to acting onOc1(x), and conclude that F(c1,c2, ¯c3;x,y,z) =F(c2,c1, ¯c3;y,x,z). This is

inconsistent with the naive spatial dependence determined by conformal symmetry, which implies that F must vanish up to contact terms.

On dimensional grounds we expect F to take the form F(c1,c2, ¯c3;x,y,z) = f(c1,c2, ¯c3)

|x −z|4 δ3(x −y). (17) where f(c1,c2, ¯c3)is symmetric in c1and c2by the Ward identity argument above. The closure of the Kähler form associated with g(c, ¯c0)then follows.

We will check the existence of the contact term by explicitly computing F(c1,c2, ¯c3;x,y,z) in perturbation theory.

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The holomorphy argument applies only to the Wilsonian effective action but not the 1PI effective action.

In the 1PI effective action, a priori, there are terms that could potentially contribute to the beta function ofαijkl in the form

βijkl = βijkl2loop+ C

k2α(ij mnα¯mnpqαpqkl)+ · · · (18) where we exhibited one possible 4-loop contribution. C is a constant coefficient that generally depends on M and N.

Such a 4-loop contribution cannot be absorbed into the wave function renormalization of the matter fields. If C is nonzero, the family of two-loop fixed pointsM0will further flow to a submanifold of lower dimension (i.e.

dimM <dimM0). But the dimensionality of the moduli space should not depend on which RG we use, so we expect C =0.

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### A 4-loop check of the nonrenormalization theorems

βijkl = βijkl2−loop+ C

k2α(ij mnα¯mnpqαpqkl)+ · · · (19) We provide a check on C=0 based on a 4-loop

calculation in the planar and large M limit. The term in the 1PI effective action that corresponds to the previous term in the beta function is

(c1M+c2)4π2

k2 N4αijmnα¯mnpqαpqklTr



ΦiΦjΦkΦl

(20) where c1 and c2are constants. We will show that c1=0.

In a supergraph that contributes to (20), the F-term

vertices are contracted according to the following structure:

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The D-term supervertices can be attached to the above graph to form a 4-loop diagram that contribute to the beta function. For example,

Because of the internal chiral superfield loop, in the large M limit, the last supergraph dominates the first two.

In the limit of large M, there are only three types of planar 4-loop supergraphs, given by

(a) (b) (c)

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### A 4-loop check of the nonrenormalization theorems

We computed 20 ordinary Feynman diagrams in component fields:

and show that

a=c = 1

256π2, b= − 1 128π2 c1 = 1

2(a+b+c) =0.

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The leading (zero-loop) contribution to the Zamolodchikov metric is simply given by the free correlator,

=⇒ g(0)(c, ¯c0) =X

cmnpq¯c0mnpq. (22) The moduli spaceMat two-loop order has the moment map

=⇒ µij =N2αimnpα¯jmnp (23) corresponding to the symplectic form

ω(0)=dαmnpqdα¯mnpq. (24) The Zamolodchikov metric onMat the leading order is the one induced from the symplectic quotient by the flavor symmetry U(M).

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### The two-loop Zamolodchikov metric

Generally, the two-loop correction to the two-point function of the quartic chiral primaries takes the form

g(c, ¯c0) =f(k)cmnpqc¯0mnpq+a2N2cijmnα¯mnpqαpqrs¯c0rsij +a3N2cimnpα¯mnpqαqrst¯c0rsti,

(25) where f(k) =1+ O(1/k2), a2,a3are constants. cijkl is constrained to be tangent to the moduli space, in particular, cimnpα¯mnpq = O(c/k3)from the two-loop constraints. So we can ignore a3in the expression for g(c, ¯c0).

a2can be evaluate by the supergraph

=⇒ a2= −161.

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With the 4-loop contributions taken into account,µij takes the form

µij =N2αimnpα¯jmnp +a1N4

αimnkα¯mnpqαpqrsα¯rskj + αkmniα¯mnpqαpqrsα¯rsjk + higher order.

(26) All other 4-loop corrections to Bij will be proportional to the first term in (26) and are subleading in 1/k2; they can be absorbed by a rescaling ofµij. The constant a1is simply given by the 4-loop wave function renormalization.

=⇒ a1= −1281 .

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### The two-loop Zamolodchikov metric

In summary, the Zamolodchikov metric with the next-to-leading correction included takes the form

g(c, ¯c0) =f(k)cmnpqc¯0mnpqN2

16cijmnα¯mnpqαpqrs¯c0rsij + · · · whereasαijkl’s are constrained by

µij =N2αimnpα¯jmnp

N4 128



αimnkα¯mnpqαpqrsα¯rskj + αkmniα¯mnpqαpqrsα¯rsjk + · · ·

=r(k)δij

Using dµij = ιv

ijω, we get ω =N2dαmnpqdα¯mnpqN4

32αrsijα¯pqijmnpqdα¯mnrs + · · · . g(c, ¯c0)is not the same as the natural symplectic metric on the quotient spaceM =V//U(M)defined using the symplectic formω.

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We also provide a check on the existence of such contact term by explicitly computing F(c1,c2, ¯c3;x,y,z) in

perturbation theory. To leading nontrivial order, this is computed by the same diagrams as interpreted as a three point function rather than a two-point function.

It is given by

F(c1,c2, ¯c3;x,y,z) ∼(c1)ijkl(c2)mnpq¯c3ijmnα¯klpq 1

|x−z|2|y −z|2 Z

d3w 1

|x−w|2|y −w|4 + higher order.

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### The existence of the contact term

The integration over w naively gives zero by analytic continuation in the exponents of the propagators. If we first integrate the integrand multiplied by a generic function of x over x , and then integrate over w , we see that

Z

d3w 1

|x−w|2|y−w|4 = −π

3(x −y). (28) This gives the contact term in F(c1,c2, ¯c3;x,y,z).

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We found a conformal moduli space ofN =2 Chern-Simons-Matter theories, which can exist in perturbative regime and in general has large dimension.

We checked the existence of the moduli space by doing a 4-loop Feynman diagram calculation.

We calculate the Zamolodchikov metric on the moduli space, and argued that it is Kähler.

The moduli spaceMand the Zamolodchikov metric g(c, ¯c0)at strong ’t Hooft coupling limit?

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