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

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]

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 2π

Z 1 0

dt Tr[V D^α(e^−tVDαe^tv)] + Φ_ie^VΦ_i (1)

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

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

[Zupnik and Pak ’88, Ivanov ’91]

The N = 2 CSM: Lagrangian

Some supervertices given by the Lagrangian K = k

2π Z 1

0

dt Tr[V D^α(e^−tVD_αe^tv)] + Φie^VΦi

W =0 or 1 4

Xα_ijklTr(Φ_iΦ_jΦ_kΦ_l)

(2)

Φⁱ Φ^l

Φ^j Φ^k

Φⁱ Φ¯ⁱ

V

F-term vertex D-term vertex

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L = Z

d⁴θK+ Z

d²θW

K = k 2π

Z 1 0

dt Tr[V D^α(e^−tVDαe^tv)] + Φ_ie^VΦ_i (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]

The N = 2 CSM: nonrenormalization theorems

L = Z

d⁴θK+ Z

d²θW

K = k 2π

Z 1 0

dt Tr[V D^α(e^−tVD_αe^tv)] + Φ_ie^VΦ_i (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(Φ_iΦ_jΦ_kΦ_l). By a two-loop computation

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

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

µdα_ijkl

dµ = (4J_Φ−2)αijkl+ 1

4π²B_(i^rα_{r jkl)}+ higher loop (5) where(ijkl)stands for cyclic symmetrization.

[Gaiotto and Yin ’07]

The N = 2 CSM: The moduli space M

µdαijkl

dµ = (4J_Φ−2)α_ijkl+ 1

4π²B_(i^rα_{r jkl)}+ higher loop (6) B_i^j = ¹₂N²α_iklmα^jklm 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

2N²αiklmα^jklm =4π²(2−4J_Φ)δ_i^j



/U(M) (7) Denote by V the linear vector space of allα_ijkl’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 S_CS^{N =2}(V) +

Z d³x

Z

d⁴θK(Φ_i, Φ_i,V) +

Z

d³xd²θf(k)X

αijklTr(ΦiΦjΦkΦl) +c.c.

(8)

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 inα_ijkl, Φ_i and invariant under R-symmetry and the spurious U(M)flavor symmetry (which cannot be anomalous in 3d). This implies W =f(k,P α_ijklTr(Φ_iΦ_jΦ_kΦ_l)). Furthermore, by assigning an R-charge 2 toα_ijkl, 1 toθ, and 0 toΦ_i’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 .

The N = 2 CSM: The moduli space M

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

takes the form µdα_ijkl

dµ = (4J_Φ(k) −2)αijkl+ 1

4π²B_(i^rα_{r jkl)} (9) for some B_i^j(α, α,k) = ¹₂N²α_iklmα^jklm+ O(¹_k, α³).

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

M =n

α_ijkl : B_i^j =4π²(2−4J_Φ) δ_i^jo

/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.

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)δ^j_i (11) Hereµ_i^j is proportional to B_i^j by normalizing the quadratic terms to be N²αimnpα¯^jmnp (µij is usually called the moment map of the symplectic quotient). So, r(k) =h(k)(¹₂−J(k)), where h(k)is a normalization factor which is not important for our purpose. The tangent directionsδα_ijkl =c_ijkl are determined by

c_mnpq ∂

∂αmnpq

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

c_mnpqTr(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)O_c0(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]

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

v_i^j = α_imnp ∂

∂αjmnp

− ¯α^jmnp ∂

∂ ¯α^imnp, (14) we have dµ_i^j = ι_v

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 δ_c1g(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.

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 = β_ijkl^2−loop+ C

k²α_{(ij mn}α¯^mnpqα_pqkl)+ · · · (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.

A 4-loop check of the nonrenormalization theorems

βijkl = β_ijkl^2−loop+ C

k²α_{(ij mn}α¯^mnpqα_pqkl)+ · · · (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 N⁴α_ijmnα¯^mnpqα_pqklTr



ΦⁱΦ^jΦ^kΦ^l

(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)

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.

(21)

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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

c_mnpq¯c^0mnpq. (22) The moduli spaceMat two-loop order has the moment map

=⇒ µ_i^j =N²α_imnpα¯^jmnp (23) corresponding to the symplectic form

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

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+a₂N²c_ijmnα¯^mnpqαpqrs¯c^0rsij +a₃N²c_imnpα¯^mnpqαqrst¯c^0rsti,

(25) where f(k) =1+ O(1/k²), a₂,a₃are constants. c_ijkl is constrained to be tangent to the moduli space, in particular, c_imnpα¯^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

µ_i^j =N²α_imnpα¯^jmnp +a₁N⁴

α_imnkα¯^mnpqαpqrsα¯^rskj + α_kmniα¯^mnpqαpqrsα¯^rsjk + higher order.

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

=⇒ a₁= −₁₂₈¹ .

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¯^0mnpq−N²

16c_ijmnα¯^mnpqαpqrs¯c^0rsij + · · · whereasαijkl’s are constrained by

µ_i^j =N²α_imnpα¯^jmnp

− N⁴ 128



α_imnkα¯^mnpqαpqrsα¯^rskj + α_kmniα¯^mnpqαpqrsα¯^rsjk + · · ·

=r(k)δij

Using dµ_i^j = ι_v

ijω, we get ω =N²dαmnpq∧dα¯^mnpq−N⁴

32α_rsijα¯^pqijdαmnpq∧dα¯^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¯c₃^ijmnα¯^klpq 1

|x−z|²|y −z|² Z

d³w 1

|x−w|²|y −w|⁴ + higher order.

(27)

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

4δ³(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?