PERTURBATIVE STUDY OF THE LEIGH-STRASSLERDEFORMED N = 4 SYM

K. Madhu

Dept. of Physics, IIT Madras

ISM – p.1

K. Madhu and S. Govindarajan, JHEP 0705 (2007) 038[arXiv:hep-th/0703020].

K. Madhu and S. Govindarajan, To appear.

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Introduction

The AdS/CFT correspondence as a gauge theory-stringtheory duality, was conjectured for N = 4supersymmetric Yang-Mills theory

This Leigh-Strassler theory is a marginal deformation ofN = 4 SYM is interesting to extend the correspondenceto cases with lesser supersymmetry, i.e, N = 1supersymmetry.

The gravity dual of this is not yet known in generality,however special cases like the β-deformation is betterstudied. (Lunin, Maldacena)

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Introduction

Chiral primary states in superconformal gauge theoryare particularly useful in studying the AdS/CFTcorrespondence.

We construct chiral primary states of the LS theory uptodimension 6.

These states are found to exist in representations of adiscrete symmetry group of the theory, the trihedralgroup ∆(27).

We study the perturbative properties of the LS theorylike conformal invariance, holomorphicity andsymmetries to higher order.

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Leigh-Strassler deformation of N = 4 SYM

Consider the LS deformation of SU(N) N = 4 SYM

L =

∫

d2θd2θTr(

e−gV ΦegV Φ)

+[ 1

2g2

∫

d2θTr(

WαWα

)

+ ih

∫

d2θTr(

qΦ1Φ2Φ3 − qΦ1Φ3Φ2

)

+ih′

3

∫

d2θtr(

Φ31 + Φ3

2 + Φ33

)

+ c.c.]

The theory is conformal invariant provided the followingcondition is obeyed

|h|2(

1 +1

N2(q − q)2

)

+ |h′|2N2 − 4

2N2= g2 .

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The trihedral group: ∆(27)

The theory has a symmetry of the trihedral group∆(27) ∼ (Z3)R × Z3 o C3 acts on the fields (Aharony et al.)

Z3 : Φ1 −→ Φ1 ,Φ2 −→ ωΦ2 ,Φ3 −→ ω2Φ3

C3 : Φ1 −→ Φ2 −→ Φ3 −→ Φ1

with ω, a non-trivial cube-root of unity and (Z3)R is asub-group of U(1)R.

∆(27) has nine one-dimensional representations LQ,j

and two three-dimensional representations V1 and V2.

In particular, there are polynomials which exist in theabove mentioned representations, which we make useof in our work.

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Representations of ∆(27)

∆(27) has nine one-dimensional representations LQ,j

and two three-dimensional representations V1 and V2.

The generators h and τ act on the one-dimensionalrepresentation v as follows:

h · v = ωQ v , τ · v = ωj v (1)

where v ∈ LQ,j. The ‘charges’ Q = 0, 1, 2 and j = 0, 1, 2

both are clearly valued modulo three. The singletcorresponds to L0,0 in this notation.

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Representations of ∆(27)

∆(27) has nine one-dimensional representations LQ,j

and two three-dimensional representations V1 and V2.

On three-dimensional representation Va with a = 1, 2

h ·

v0

v1

v2

=

1 0 0

0 ωa 0

0 0 ω2a

v0

v1

v2

τ ·

v0

v1

v2

=

0 0 1

1 0 0

0 1 0

v0

v1

v2

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Polynomials as irreps of ∆(27)

Consider three complex variables (z1, z2, z3) thattransform in the three-dimensional representation V1 of∆(27).

For ∆0 = 0 mod 3, polynomials are in theone-dimensional representations LQ,j. For example,

(z31 + ωjz3

2 + ω2jz33) ∈ L0,j ,

with j = 0 giving rise to a singlet. Similarly, z1z2z3 is alsoa singlet.

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Polynomials as irreps of ∆(27)

Consider three complex variables (z1, z2, z3) thattransform in the three-dimensional representation V1 of∆(27).

For ∆0 6= 0 mod 3, polynomials are in the threedimensional representations Va, where a = ∆0 mod 3.For example,

z2z3

z3z1

z1z2

∈ V2 .

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Chiral Primary StatesChiral primary states have zero anomalous dimensions.

We will consider states made up of scalar fields.

The anomalous dimension for such operators can bewritten as a modulus square.

We find anomalous dimensions of operators Odiagramatically by computing 〈O(x)O(0)〉 at planar(N −→ ∞), one-loop order. This calculation is done incomponent fields.

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Interaction potential LS theory

In terms of component fields VF is the F-term potential

VF = Tr(

|h′|2Z12Z2

1 + hh′[Z2, Z3]qZ12− hh′[Z2, Z3]qZ

21

−|h|2[Z2, Z3]q[Z2, Z3]q

)

−1

N

[

|h′|2 Tr(Z12) Tr(Z2

1)

+ hh′ Tr([Z2, Z3]q) Tr(Z12) − hh′ Tr(Z2

1) Tr([Z2, Z3]q)

+|h|2 Tr([Z2, Z3]q)(

Tr[Z2, Z3]q)

]

+ cyclic permutations

VF has a double trace interaction which is suppressedby a factor 1

N. We will see that this is important even in

planar (N −→ ∞) limit.

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Anomalous dimensions at planar one-loop

For the simplest candidate operator O = Tr(Zk1 Z l

2Zm3 ),

we compute 〈O(x)O(0)〉 to one-loop at large N .

We observe that the contributions to anomalousdimensions from all the interactions other than VF

vanishes when we impose conformal invariancecondition. This is a demonstration of the holomorphicityof the theory. (d’Hoker et al.)

Dimension ∆0 = 2 operators get a contribution from the1N

suppressed term in VF . This cancels exactly with thecontributions from the rest of the terms. Hencedimension ∆0 = 2 operators are always protected.

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Chiral primaries of LS theory

There are no further chiral primaries of LS theory of theabove simple form.

We need to consider linear combinations ofpermutations of monomials constructed out ofoperators.

These operators are in the polynomials representationsof trihedral ∆(27).

For example, dimension ∆0 = 4 operator looks like

O14 = tr

(

Z42 + b Z3

1Z2 + c Z21Z2

3 + c1Z1Z3Z1Z3

+d Z1Z22Z3 + d1Z1Z2Z3Z2 + d2Z3Z

22Z1

+fZ33Z2

)

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Chiral Primaries of Leigh-Strassler theory

This operator is in the three dimensional representationV1 of ∆(27).

Calculating the anomalous dimensions and writing it asa modulus square, we find that there exists a uniquesolution for generic values of couplings to the conditionfor vanishing of anomalous dimensions. Thus we obtainthe chiral primary at dimension ∆0 = 4.

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Chiral Primaries at planar one-loop

At dimension ∆0 = 6 we find that there are operators

O(0,0)6 in the L0,0 representation of ∆0 = 4 that are

protected. There are precisely two independentsolutions, for generic values of couplings.

For operators in the Li,j, i 6= 0 and/or j 6= 0 there are nogeneric solutions. But solutions exist on specificsub-loci of the coupling space.

In general, if an operator has dimension ∆0 = a mod 3,for generic values of couplings, when a = 0 mod 3, thereare two independent chiral primaries in theone-dimensional L0,0 representation of ∆(27). Whena 6= 0 mod 3, the chiral primaries are in thethree-dimensional representation Va.

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Chiral Primaries at planar one-loop

We can organize the chiral primaries of N = 4 SYM, theβ-deformed theory as well, since the ∆(27) symmetry ofthe LS theory was obtained by breaking down thebigger invariance groups of these theories. For ∆0 > 2,we have the following conjecture :

Scaling dim. ∆0 = 3r ∆0 = a mod 3

N = 4 theory L0,0 ⊕r(r+1)

2

[

⊕i,j Li,j

] (∆0+1)(∆0+2)6 Va

β-def. theory L0,0 ⊕j L0,j Va

LS theory 2L0,0 Va

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Anomalous dimension of chiral superfield

Using SU(N) gauge invariance and the ∆(27) symmetrywe can argue that γI

J must be proportional to δIJ .

The only gauge invariant SU(N) tensor is δab . Hence

gauge invariance of the matrix γIJ implies γI

J ≡ γijδ

ab .

∆(27) invariance implies c112 = . . . = 0 andc111 = c222 = c333

For the quantum theory to preserve ∆(27) symmetry,β(c112) = 0. and β(c111) = β(c222).

Using the JJN β-function

β(YIJK) ∼ YLJK γLI + YILK γL

J + YIJL γLK

we easily see that γij ≡ γδi

j.

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Conformal invariance of LS theory

Hence in LS theory, there is a single condition in thecoupling space which ensures marginality of thecouplings. This holds true at two-loop too.

At three-loop the anomalous dimension has acontribution (Jack, Jones, North)

However, there exists redefinitions of couplingconstants that ensure the vanishing of anomalousdimension at three-loop also.

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Effective Superpotential

The contributions from the D-terms to the effectivesuperpotential at two-loop are from the followingFeynman diagrams

The contribution from this is proportional to W once weimpose the conformal invariance condition.

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Effective Superpotential

There are no corrections possible at one-loop fromchiral interaction vertices.

Holomorphy restricts the possible diagrams that cancontribute to the effective action. The only diagram thatgives a non-trivial contribution (that is not proportaionalto the classical superpotential)

This is a finite contribution

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Effective Superpotential

This contribution is related to the three-loop anomalousdimension γ(3) calculated. It is expected that thecoupling constant redefinitions that makeγ(3) vanish, remove this contribution. Thus the two-loopeffective action is expected to be identical to the treelevel action

Hence holomorphy and conformal invariance can bepreserved.

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Summary of results

The symmetry group trihedral ∆(27) of the LS theoryhelps to classify the chiral primary states withdimension ∆0 > 2 in its representations, for genericvalues of the couplings. The multi trace interactions inVF ensure that dimension 2 operators are alwaysprotected

∆(27) is essential in preserving the conformalinvariance of the theory. The effective Kähler potentialas well as the superpotential shows that ∆(27) ispreserved in quantum theory as well.

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Summary of results

We observe that there are coupling constantredefinitions which ensure vanishing of anomalousdimensions to three-loop, preserving conformalinvariance.

The same coupling constant redefinitions can eliminatetwo-loop corrections to the superpotential hence alsopreserving the holomorphicity of the theory.

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