1) The document discusses integrability in quantum field theories and string theories, specifically N=4 supersymmetric Yang-Mills theory (SYM). 2) It was previously shown that N=4 SYM is integrable in the planar limit through a mapping to spin chains. Deformations of N=4 SYM called Leigh-Strassler deformations were also studied. 3) The document analyzes string motion on the gravity dual background known as the Lunin-Maldacena geometry, which corresponds to Leigh-Strassler deformations. Through analytic and numerical methods, it is shown that string motion on this background is non-integrable for complex deformations, indicating

1. 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

2. Motivation: Non-perturbative QFT
We would like to understand Quantum Field Theory at
strong coupling
Relevant for QCD: Chiral Symmetry Breaking,
Confinement
QCD is a Yang-Mills theory:
S =
1
4
Z
d4x Tr(FF)+ ; F = @A@Aig[A;A]
A = Aa
(Ta)i
j is the gauge potential ; T 2 SU(3)
Computations difficult when g is not small
One approach is Lattice QFT ) Simulation
We will use analytic methods coming from String Theory

3. N = 4 Super-Yang-Mills
Add more (super)symmetry
Field content:
A; 1;2;3;4
;X;Y; Z All in adjoint of SU(N)
Superpotential WN=4 = gTr(X[Y; Z])
Potential V = j @W
@X j2 + j @W
@Y j2 + j @W
@Z j2
This is a UV finite theory ) Conformally invariant
Conformal group:
[P;P] = 0 ; [P; L] = i(P P) ;
[L; L] = i(L + L L L)
[D;P] = iP ; [D;K] = iK ; [K;K] = 0 ;
[K;P] = 2i(D L) ; [K; L] = i(K K)

4. Integrability in N = 4 SYM
In 2002, J. Minahan and K. Zarembo discovered
integrability in N = 4 SYM in the planar limit N ! 1
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 = x@ ) H =
XL
i=1
~Si ~Si+1
Anomalous Dimensions $ Energies of states

5. 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 = Tr
X[Y; Z]q +
h
3
X3+Y3+Z3
q-commutator: [Y; Z]q = YZ qZY

6. -deformation: q = e2i

7. ( = g; h = 0)
Real

8. integrable. Complex

9. does not correspond to an
integrable spin chain [Berenstein-Cherkis ’04]
Proof?

10. 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 in
the higher-dimensional space
Large N limit ) classical string theory
Strong gauge coupling ) classical supergravity
Integrability of N = 4 SYM implies integrable string motion

11. The Lunin-Maldacena geometry
The dual of the

12. deformations was constructed in 2005
Deformed 5-sphere
ds2 = R2
p
H
2
4ds2
AdS5 +
X3
i=1
(d2i
+ G2i
d2i
) + (~
2 + ~2)G21
22
23
X3
i=1
di
!23
5
G =
1
1 + (~
2 + ~2)Q ; Q = 21
22
+22
23
+21
23
; H = 1+~2Q ;

13. = ~
i ~
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

14. 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 9 solution: Inconclusive
We will reduce string motion on LM to a 2d Hamiltonian
system and apply the Kovacic algorithm [Basu-Pando Zayas]

15. Galois Theory - an example
Consider the polynomial x2 4x + 1 = 0
p
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 roots
that 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

16. Rewrite LM
Metric
ds2 =
p
H
cosh2 dt2 + d2
+
p
H
0
@d2 + sin2 d2 + G
X
i;(jk)
2i
1 +
~
2 + ~2
2j
2k
d2i
1
A
p
HG
+ 2
~
2 + ~2
21
22
23
(d1d2 + d1d3 + d2d3)
B-field
B = R2
0
@~
G
X
ij
ij di ^ dj ~12
1 23
1
A:
(d ^ (d1 + d2 + d3))

17. String -model
String action
S =
R2
2
Z
d
d
2
h

18. GMN@XM@

19. XN

20. BMN@XM@

21. XN
i
String ansatz
t = t( ) ; = ( ) ; = (; ) ; = (; ) ; i = i (; )
Substituting:
S =
R2
4
Z
dd
p
H
cosh2 _t2 +
02 _2
+
p
H
02 _ 2
+ sin2
02 _2
+
X
i
Gii
02
i _
2i
+ 2
X
i;j;(ij)
Gij
0i
0j
_
i _
j
2
X
i;j;(ij)
Bij
_
i0j
0i
_
j
2
X
i
Bi
_0i
0 _
i
:

22. Pointlike String
t = t( ) ; = ( ) ; = ( ) ; = ( ) ;
Expected to be integrable
2Leff =
2
p
H
+
p
H_ 2 +
p
H sin2 _2;
Fix plane = 2
; _2 = 2
H
Variation along normal plane = 2
+ (t)
Normal Variational Equation
22H0
1 z2
00(z) z0(z) + (z)
= 0:
Integrable!

23. Extended String
t = t( ) ; = ( ) ; = ( ) ; 1 = 0 ; 2 = m ; 3 = 0 ;
Effective lagrangian
2Leff =
2
p
H
+
p
H_ 2 +
p
H sin2 _2
p
HA2m2 + 2B2
_m
Fix plane = 0 ; _ 2 = 2
H , take

24. = i ~
= 0 + ( ) ! NVE
(z)00+
2
z (z)0+
m
m
~2 + 2
z2 + z4 + 1
4~z
z2 + 1
2 (z2 + 1)
4 (z) = 0
Kovacic: Not Integrable! (unless ~ = 0 or m = 0)

25. Numerical Analysis: Poincaré Sections
Study the string hamiltonian numerically
H =
2
2
p
H
+
p2
2
p
H
+
p2
p
H sin2
2
B2m
p
H sin2
p+
1
2
G22m2+
B2
2m2
p
H sin2
2
Allows to consider general complex

26. Poincaré sections for ~
= 1, ~ = 0:001; ~ = 2:0; ~ = 10:0

27. Phase Space Trajectories
~ = 0 , ~
= 0:01 ; ~
= 1 ; ~
= 100
~
= 0 , ~ = 0:001 ; ~ = 1 ; ~ = 10
Numerics confirms the picture we found analytically

28. Conclusions
Showed, both analytically and numerically, that string
motion of the dual background to the imaginary-

29. Leigh-Strassler theories is non-integrable
Assuming AdS/CFT, shows that these theories are not
integrable at strong coupling
Matches expectations from weak coupling
Nice application of analytic non-integrability approach

30. 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