Marginal Deformations and Non-Integrability

Konstantinos Zoubos

University of Pretoria

NITheP Associates Meeting

Stellenbosch

19/09/2014

Based on arXiv:1311.3241 with D. Giataganas and L. Pando Zayas

Motivation: Non-perturbative QFT

• We would like to understand Quantum Field Theory atstrong coupling• Relevant for QCD: Chiral Symmetry Breaking,

Confinement• QCD is a Yang-Mills theory:

S = −14

∫d4x Tr(FµνFµν)+· · · , Fµν = ∂µAν−∂νAµ−ig[Aµ,Aν ]

Aµ = Aaµ(T a)i

j is the gauge potential , T ∈ SU(3)

• Computations difficult when g is not small• One approach is Lattice QFT⇒ Simulation• We will use analytic methods coming from String Theory

N = 4 Super-Yang-Mills

• Add more (super)symmetry

• Field content:

Aµ, ψ1,2,3,4α ,X ,Y ,Z All in adjoint of SU(N)

• SuperpotentialWN=4 = gTr(X [Y ,Z ])

• Potential V = |∂W∂X |2 + |∂W∂Y |

2 + |∂W∂Z |2

• This is a UV finite theory⇒ Conformally invariant

• Conformal group:

[Pµ,Pν ] = 0 , [Pρ,Lµν ] = i(ηρµPν − ηρνPµ) ,

[Lµν ,Lρσ] = i(ηνρLµσ + ηµσLνρ − ηµρLνσ − ηνσLµρ)

[D,Pµ] = iPµ , [D,Kµ] = −iKµ , [Kµ,Kν ] = 0 ,[Kµ,Pν ] = 2i(ηµνD − Lµν) , [Kρ,Lµν ] = i(ηρµKν − ηρνKµ)

Integrability in N = 4 SYM

• In 2002, J. Minahan and K. Zarembo discoveredintegrability in N = 4 SYM in the planar limit N →∞• Observables: Gauge invariant operators, here in X ,Y

scalar sector

O = Tr(XYXXYY · · · ) L fields

• These can be mapped to a spin chain:• The Dilatation operator is mapped to an integrable

Hamiltonian (XXX Heisenberg chain)

D = xµ∂µ ⇒ H =L∑

i=1

~Si · ~Si+1

• Anomalous Dimensions↔ Energies of states

Boundaries of Integrability

• Conformal Invariance does not imply integrability• Can we find CFT’s that move from integrable to

non-integrable on varying a parameter?• Leigh-Strassler deformations of N = 4 SYM

WN=4 = gTr(X [Y ,Z ]) −→ WLS = κTr(

X [Y ,Z ]q +h3(X 3+Y 3+Z 3))

q-commutator: [Y ,Z ]q = YZ − qZY

• β-deformation: q = e2πiβ (κ = g,h = 0)• Real β integrable. Complex β does not correspond to an

integrable spin chain [Berenstein-Cherkis ’04]

• Proof?

The AdS/CFT correspondence

• N = 4 SYM is equivalent to String Theory on AdS5 × S5

• Any observable in gauge theory can be mapped to one inthe higher-dimensional space

• Large N limit⇒ classical string theory

• Strong gauge coupling⇒ classical supergravity

• Integrability of N = 4 SYM implies integrable string motion

The Lunin-Maldacena geometry

• The dual of the β deformations was constructed in 2005• Deformed 5-sphere

ds2 = R2√

H

ds2AdS5 +

3∑i=1

(dρ2i + Gρ2

i dφ2i ) + (γ2 + σ2)Gρ2

1ρ22ρ

23

(3∑

i=1

dφi

)2

G =1

1 + (γ2 + σ2)Q, Q = ρ2

1ρ22+ρ

22ρ

23+ρ

21ρ

23, H = 1+ σ2Q , β = γ− iσ

• B-field + dilaton fields as well• We will consider integrability of classical string motion on

this background• Aim: Holographically show that the complex-beta

Leigh-Strassler theory is not integrable

Analytic Non-integrability

• Consider a system of equations ~x = ~f (~x)

• Find one solution x = x(t)• Linearise around x• If the linearised system has no integrals of motion, neither

does the full system• 2d Hamiltonian systems: Integrals of motion↔ differential

Galois theory• Kovacic algorithm: Determines if there are Liouvillian

solutions• If no solution: Hamiltonian system is not integrable• If ∃ solution: Inconclusive• We will reduce string motion on LM to a 2d Hamiltonian

system and apply the Kovacic algorithm [Basu-Pando Zayas]

Galois Theory - an example

• Consider the polynomial x2 − 4x + 1 = 0• Roots x± = 2±

√3

• Write relations between the roots with rational coefficients

x+ + x− = 4 , x+x− = 1

• The Galois group is the permutation group of these rootsthat preserves these relations• If the Galois group is not solvable we cannot express the

roots in terms of radicals• Explains why no general formula for degree ≥ 5• Differential Galois theory: Differential equations instead of

polynomials

Rewrite LM

• Metric

ds2 =√

H(− cosh2 ρdt2 + dρ2

)+√

H

dα2 + sin2 αdθ2 + G∑

i,(j<k)

ρ2i

(1 +

(γ2 + σ2

)ρ2

j ρ2k

)dφ2

i

+ 2√

HG(γ2 + σ2

)ρ2

1ρ22ρ

23(dφ1dφ2 + dφ1dφ3 + dφ2dφ3)

• B-field

B = R2

γG∑i<j

ρiρj dφi ∧ dφj − σρ1ρ2

(1− ρ2

3

)(dθ ∧ (dφ1 + dφ2 + dφ3))

.

String σ-model• String action

S = −R2

2

∫dτ

dσ2π

[γαβGMN∂αX M∂βX N − εαβBMN∂αX M∂βX N

]

• String ansatz

t = t(τ) , ρ = ρ(τ) , α = α(τ, σ) , θ = θ(τ, σ) , φi = φi(τ, σ)

• Substituting:

S =− R2

4π

∫dσdτ

√H(

cosh2 ρ t2 +(ρ′2 − ρ2

))+√

H((α′2 − α2

)+ sin2 α

(θ′2 − θ2

))+∑

i

Gii

(φ′2

i − φ2i

)+ 2

∑i,j,(i<j)

Gij

(φ′

iφ′j − φi φj

)− 2

∑i,j,(i<j)

Bij

(φiφ

′j − φ′

i φj

)− 2

∑i

Bθi

(θφ′

i − θ′φi

).

Pointlike String

t = t(τ) , ρ = ρ(τ) , α = α(τ) , θ = θ(τ) ,

• Expected to be integrable

2Leff =κ2√

H+√

Hα2 +√

H sin2 αθ2,

• Fix plane α = π2 , θ

2 = κ2

H

• Variation along normal plane α = π2 + η(t)

• Normal Variational Equation

2κ2H0((

1− z2)η′′(z)− zη′(z) + η(z))

= 0.

• Integrable!

Extended String

t = t(τ) , α = α(τ) , θ = θ(τ) , φ1 = 0 , φ2 = mσ , φ3 = 0 ,

• Effective lagrangian

2Leff =κ2√

H+√

Hα2 +√

H sin2 αθ2 −√

HA2m2 + 2Bθφ2 θm

• Fix plane θ = 0 , α2 = κ2

H , take β = iσ• θ = 0 + η(τ) −→ NVE

η(z)′′+2zη(z)′+

m[m((σ2 + 2

)z2 + z4 + 1

)− 4κσz

(z2 + 1

)]κ2 (z2 + 1)

4 η(z) = 0

• Kovacic: Not Integrable! (unless σ = 0 or m = 0)

Numerical Analysis: Poincaré Sections

• Study the string hamiltonian numerically

H = − κ2

2√

H+

p2α

2√

H+

p2θ

2√

H sin2 α− Bθφ2 m√

H sin2 αpθ+

12

G22m2+B2θφ2

m2

2√

H sin2 α

• Allows to consider general complex β• Poincaré sections for γ = 1, σ = 0.001, σ = 2.0, σ = 10.0

Phase Space Trajectories

• σ = 0 , γ = 0.01 , γ = 1 , γ = 100

• γ = 0 , σ = 0.001 , σ = 1 , σ = 10

• Numerics confirms the picture we found analytically

Conclusions

• Showed, both analytically and numerically, that stringmotion of the dual background to the imaginary-βLeigh-Strassler theories is non-integrable

• Assuming AdS/CFT, shows that these theories are notintegrable at strong coupling

• Matches expectations from weak coupling

• Nice application of analytic non-integrability approach

Outlook

• Study other theories, such as the h-deformation

• Check special 1-loop integrable points

• Implications of non-integrability for QFT?

• Study other backgrounds, e.g. with dynamical flavours

• Other heavy objects, e.g. D-branes

• Keep mapping the limits of integrability in 4d QFT