### 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’

1 / 28

the worldvolume theory of M2-branes (ABJM model:

N =*6 U*(N) ×*U*(N)*CSM theory describes N M2-branes*
living onR^{8}/Z2)

*the conformal field theory part of the AdS*_{4}/CFT_{3}

correspondence (for example, ABJM model = the M-theory
*on AdS*_{4}×*S*^{7}/Z*k*, or more general,N =2 CSM theories
*can be dual to M-theory on AdS*_{4}×*M*_{7}*, M*_{7}being 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, Z*_{2}spin 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

3 / 28

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

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

L = Z

*d*^{4}θK+

Z

*d*^{2}θW +*c.c.*

*K* = *k*
2π

Z 1 0

*dt* Tr[V D^{α}(e^{−}^{tV}*D*α*e** ^{tv}*)] + Φ

_{i}*e*

*Φ*

^{V}*(1)*

_{i}*k is the Chern-Simons coupling. W can be any gauge*
invariant holomorphic function onΦ*i*.

*For simplicity, we assume G*=*U(N),*Φ* _{i}* in the adjoint

*representation and W*=0 or

^{1}

_{4}P α

*ijkl*Tr(Φ

*i*Φ

*j*Φ

*k*Φ

*l*).

[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^{−}^{tV}*D*_{α}*e** ^{tv}*)] + Φ

*i*

*e*

*Φ*

^{V}*i*

*W* =0 or 1
4

Xα* _{ijkl}*Tr(Φ

*Φ*

_{i}*Φ*

_{j}*Φ*

_{k}*)*

_{l}(2)

Φ* ^{i}*
Φ

^{l}Φ* ^{j}*
Φ

^{k}Φ* ^{i}*
Φ¯

^{i}*V*

F-term vertex D-term vertex

5 / 28

L = Z

*d*^{4}θK+
Z

*d*^{2}θW

*K* = *k*
2π

Z 1 0

*dt* Tr[V D^{α}(e^{−tV}*D*α*e** ^{tv}*)] + Φ

_{i}*e*

*Φ*

^{V}*(3)*

_{i}*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

*d*^{4}θK+
Z

*d*^{2}θW

*K* = *k*
2π

Z 1 0

*dt* Tr[V D^{α}(e^{−}^{tV}*D*_{α}*e** ^{tv}*)] + Φ

_{i}*e*

*Φ*

^{V}*(4)*

_{i}The Kahlar potential can be renormalized, but any
corrections to it are either irrelevant in the IR or can be
absorbed by a rescaling ofΦ*i*.

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

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

7 / 28

*However, W* =0 is an unstable fixed point under the
RG-flow if we turn on a general quartic superpotential
*W* =^{1}_{4}P α*ijkl*Tr(Φ* _{i}*Φ

*Φ*

_{j}*Φ*

_{k}*). By a two-loop computation*

_{l}Tr(Φ* _{i}*Φ

*Φ*

_{j}*Φ*

_{k}*)has dimension 2−*

_{l}*b*

_{0}/2k

^{2}+ O(k

^{−}

^{4}) <2,

*where b*

_{0}=8(M+1)N

^{2}. It is a chiral primary so the dimension∆ =

*4J*

_{Φ}.

Further, the beta function forα*ijkl* up to two-loop order
takes the form

µ*d*α_{ijkl}

*d*µ = (4J_{Φ}−2)α*ijkl*+ 1

4π^{2}*B*_{(i}* ^{r}*α

*+ higher loop (5) where(ijkl)stands for cyclic symmetrization.*

_{r jkl)}[Gaiotto and Yin ’07]

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

µ*d*α*ijkl*

*d*µ = (4J_{Φ}−2)α* _{ijkl}*+ 1

4π^{2}*B*_{(i}* ^{r}*α

*+ higher loop (6)*

_{r jkl)}*B*

_{i}*=*

^{j}^{1}

_{2}

*N*

^{2}α

*α*

_{iklm}*comes from the two-loop wave function renormalization.*

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

M0=

α*ijkl* : 1

2*N*^{2}α*iklm*α* ^{jklm}* =4π

^{2}(2−

*4J*

_{Φ})δ

_{i}

^{j}

/U(M)
(7)
*Denote by V the linear vector space of all*α* _{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!

9 / 28

We consider the Wilsonian effective action, of the form
*S*_{CS}^{N =2}(V) +

Z
*d*^{3}*x*

Z

*d*^{4}θK(Φ* _{i}*, Φ

*,*

_{i}*V*) +

Z

*d*^{3}*xd*^{2}θf(k)X

α*ijkl*Tr(Φ*i*Φ*j*Φ*k*Φ*l*) +*c.c.*

(8)

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

By promotingα*ijkl* to dynamical chiral superfields, the
effective superpotential is holomorphic inα* _{ijkl}*, Φ

*and*

_{i}*invariant under R-symmetry and the spurious U*(M)flavor symmetry (which cannot be anomalous in 3d). This implies

*W*=

*f*(k,P α

*Tr(Φ*

_{ijkl}*Φ*

_{i}*Φ*

_{j}*Φ*

_{k}*)). Furthermore, by assigning an R-charge 2 toα*

_{l}*, 1 toθ, and 0 toΦ*

_{ijkl}*’s, we conclude that the effective superpotential must be linear inα*

_{i}*ijkl*, 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

*Φ*

_{i}*Φ*

_{j}*Φ*

_{k}*)together with a wave function renormalization. The beta function forα*

_{l}*ijkl*

takes the form
µ*dα*_{ijkl}

*dµ* = (4J_{Φ}(k) −2)α*ijkl*+ 1

4π^{2}*B*_{(i}* ^{r}*α

*(9)*

_{r jkl)}*for some B*

_{i}*(α, α,*

^{j}*k) =*

^{1}

_{2}

*N*

^{2}α

*α*

_{iklm}*+ O(*

^{jklm}^{1}

*, α*

_{k}^{3}).

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

M =n

α* _{ijkl}* :

*B*

_{i}*=4π*

^{j}^{2}(2−

*4J*

_{Φ}) δ

_{i}*o*

^{j}/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.

11 / 28

The entire moduli spaceMcan exist in the perturbative
regime and its dimension (∼*M*^{4}) 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

µ*i**j*(α, α,*k*) =*r*(k)δ^{j}* _{i}* (11)
Hereµ

_{i}

^{j}*is proportional to B*

_{i}*by normalizing the quadratic*

^{j}*terms to be N*

^{2}α

*imnp*α¯

*(µ*

^{jmnp}*i*

*j*is usually called the moment

*map of the symplectic quotient). So, r*(k) =

*h(k*)(

^{1}

_{2}−

*J(k*)),

*where h(k*)is a normalization factor which is not important for our purpose. The tangent directionsδα

*=*

_{ijkl}*c*

*are determined by*

_{ijkl}*c** _{mnpq}* ∂

∂α*mnpq*

µ_{i}* ^{j}*(α, α,

*k) =*0, ∀i,

*j.*(12) We define the quartic operators corresponding to tangent directionsO

*c*=P

*c** _{mnpq}*Tr(φ

*m*φ

*n*φ

*p*φ

*q*). They are chiral primary operators since they cannot be written as Tr(φ

*i*∂W

∂φ*i*).

13 / 28

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

hO*c*(x)O* _{c}*0(0)i =

*g*(c, ¯

*c*

^{0})

|x|^{4} . (13)
*where the coefficient g(c, ¯c*^{0})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 AdS*_{4}×*M*_{7}*, M*_{7}being 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}∂α*jmnp*

− ¯α* ^{jmnp}* ∂

∂ ¯α* ^{imnp}*, (14)

*we have d*µ

_{i}*= ι*

^{j}

_{v}*i**j*ω, 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.

15 / 28

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

corresponding to a chiral primaryO*c*1 is
δ_{c}_{1}*g(c*_{2}, ¯*c*_{3}) = |x|^{4}

O*c*2(x)

Z

*d*^{3}*yQ*^{2}· O*c*1(y)

O¯*c*_{3}(0)

(15)
*The statement that g is Kähler amounts to*

δ*c*_{1}*g(c*_{2}, ¯*c*_{3}) = δ*c*_{2}*g(c*_{1}, ¯*c*_{3}), for chiral primaries c_{1},*c*_{2},*c*_{3}.
Let us study the correlation function

*F*(c1,*c*_{2}, ¯*c*_{3};*x,y,z*) = hO*c*1(x)h

*Q*^{2}· O*c*2(y)i

O*c*3(z)i (16)
*Q*^{2}· O*c*2 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 Q*^{2}from
acting onO*c*2(y)to acting onO*c*1(x), and conclude that
*F*(c1,*c*_{2}, ¯*c*_{3};*x*,*y*,*z) =F*(c2,*c*_{1}, ¯*c*_{3};*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*(c_{1},*c*_{2}, ¯*c*_{3};*x,y,z*) = *f*(c_{1},*c*_{2}, ¯*c*_{3})

|x −*z|*^{4} δ^{3}(x −*y*). (17)
*where f*(c_{1},*c*_{2}, ¯*c*_{3})*is symmetric in c*_{1}*and c*_{2}by the Ward
identity argument above. The closure of the Kähler form
*associated with g(c, ¯c*^{0})then follows.

We will check the existence of the contact term by explicitly
*computing F*(c1,*c*_{2}, ¯*c*_{3};*x*,*y*,*z)* in perturbation theory.

17 / 28

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 ofα*ijkl* in
the form

β*ijkl* = β_{ijkl}^{2}^{−}* ^{loop}*+

*C*

*k*^{2}α_{(ij mn}α¯* ^{mnpq}*α

_{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

β*ijkl* = β_{ijkl}^{2−loop}+ *C*

*k*^{2}α_{(ij mn}α¯* ^{mnpq}*α

_{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

(c_{1}*M*+*c*_{2})4π^{2}

*k*^{2} *N*^{4}α* _{ijmn}*α¯

*α*

^{mnpq}*Tr*

_{pqkl}

Φ* ^{i}*Φ

*Φ*

^{j}*Φ*

^{k}

^{l}(20)
*where c*_{1} *and c*_{2}*are constants. We will show that c*_{1}=0.

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

vertices are contracted according to the following structure:

19 / 28

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

256π^{2}, *b*= − 1
128π^{2}
*c*_{1} = 1

2(a+*b*+*c) =*0.

(21)

21 / 28

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

=⇒ *g*^{(0)}(c, ¯*c*^{0}) =X

*c** _{mnpq}*¯

*c*

^{0}

*. (22) The moduli spaceMat two-loop order has the moment map*

^{mnpq}=⇒ µ_{i}* ^{j}* =

*N*

^{2}α

*α¯*

_{imnp}*(23) corresponding to the symplectic form*

^{jmnp}ω^{(0)}=*d*α*mnpq*∧*d*α¯* ^{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, ¯c*^{0}) =*f(k)c**mnpq**c*¯^{0}* ^{mnpq}*+

*a*

_{2}

*N*

^{2}

*c*

*α¯*

_{ijmn}*α*

^{mnpq}*pqrs*¯

*c*

^{0}

*+*

^{rsij}*a*

_{3}

*N*

^{2}

*c*

*α¯*

_{imnp}*α*

^{mnpq}*qrst*¯

*c*

^{0rsti},

(25)
*where f*(k) =1+ O(1/k^{2}), a_{2},*a*_{3}*are constants. c** _{ijkl}* is
constrained to be tangent to the moduli space, in particular,

*c*

*α¯*

_{imnp}*= O(c/k*

^{mnpq}^{3})from the two-loop constraints. So

*we can ignore a*

_{3}

*in the expression for g*(c, ¯

*c*

^{0}).

*a*_{2}can be evaluate by the supergraph

=⇒ *a*_{2}= −_{16}^{1}.

23 / 28

With the 4-loop contributions taken into account,µ*i**j* takes
the form

µ_{i}* ^{j}* =N

^{2}α

*α¯*

_{imnp}*+*

^{jmnp}*a*

_{1}

*N*

^{4}

α* _{imnk}*α¯

*α*

^{mnpq}*pqrs*α¯

*+ α*

^{rskj}*α¯*

_{kmni}*α*

^{mnpq}*pqrs*α¯

*+ higher order.*

^{rsjk}(26)
*All other 4-loop corrections to B*_{i}* ^{j}* will be proportional to the
first term in (26) and are subleading in 1/k

^{2}; they can be absorbed by a rescaling ofµ

_{i}

^{j}*. The constant a*

_{1}is simply given by the 4-loop wave function renormalization.

=⇒ *a*_{1}= −_{128}^{1} .

### The two-loop Zamolodchikov metric

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

*g(c, ¯c*^{0}) =*f(k)c**mnpq**c*¯^{0}* ^{mnpq}*−

*N*

^{2}

16*c** _{ijmn}*α¯

*α*

^{mnpq}*pqrs*¯

*c*

^{0}

*+ · · · whereasα*

^{rsij}*ijkl*’s are constrained by

µ_{i}* ^{j}* =

*N*

^{2}α

*α¯*

_{imnp}

^{jmnp}− *N*^{4}
128

α* _{imnk}*α¯

*α*

^{mnpq}*pqrs*α¯

*+ α*

^{rskj}*α¯*

_{kmni}*α*

^{mnpq}*pqrs*α¯

*+ · · ·*

^{rsjk}=*r*(k)δ*i**j*

*Using dµ*_{i}* ^{j}* = ι

_{v}*i**j*ω, we get
ω =*N*^{2}*d*α*mnpq*∧*d*α¯* ^{mnpq}*−

*N*

^{4}

32α* _{rsij}*α¯

^{pqij}*dα*

*mnpq*∧

*d*α¯

*+ · · · .*

^{mnrs}*g(c, ¯c*

^{0})is not the same as the natural symplectic metric on the quotient spaceM =

*V*//U(M)defined using the symplectic formω.

25 / 28

We also provide a check on the existence of such contact
*term by explicitly computing F*(c1,*c*_{2}, ¯*c*_{3};*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,*c*_{2}, ¯*c*_{3};*x*,*y*,*z) ∼(c*1)*ijkl*(c2)*mnpq*¯*c*_{3}* ^{ijmn}*α¯

*1*

^{klpq}|x−*z|*^{2}|y −*z|*^{2}
Z

*d*^{3}*w* 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

*d*^{3}*w* 1

|x−*w|*^{2}|y−*w*|^{4} = −π

4δ^{3}(x −*y*). (28)
*This gives the contact term in F*(c1,*c*_{2}, ¯*c*_{3};*x*,*y*,*z).*

27 / 28

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, ¯c*^{0})at strong ’t Hooft coupling limit?