1. Integrable Conformal Field Theories in Higher Dimensions
Workshop: ‘’Chaos and Integrability inMulti-Component Systems”
Far East Federal University, Vladivostok, 2-8 October 2017
Integrable Conformal Field Theories in Higher Dimensions
Vladimir Kazakov (ENS, Paris)
Collaborations with
Ӧ. Gürdogan arXiv:
J. Caetano and Ӧ. Gürdogan arXiv:
D. Grabner, N. Gromov, G. Korchemsky arXiv:1709……..
N. Gromov, G. Korchemsky, S. Negro, G. Sizov arXiv:
D. Chicherin, F. Loebbert, D. Mueller, D. Zhang arXiv:
arXiv:

2. Outline Planar N=4 SYM is a 4D CFT. Integrable for all couplings.
Anomalous dimensions, computable via Quantum Spectral Curve (QSC)
Big progress for computations of structure constants, correlators, ǪǬ-potential, amplitudes,1/N2, etc.
β-deformation leads to N=1 SYM – also integrable CFT.
Integrable deformations, e.g. ɤ-deformation. Still CFT at a complex coupling.
Gurdogan, V.K. 2015
Double scaling limit of N=4 SYM: strong ɤ-deformation & weak coupling leads to non-unitary, “chiral” CFT dominated by very few specific integrable (computable!) multi-loop 4D Feynman graphs:
Gurdogan, V.K. 2015
“fishnet’’
(massless, 4-scalar int.)
A. Zamolodchikov 1980
“brick wall’’
(Yukawa interaction)
Caetano, Gurdogan, V.K. 2016
mixture of both
Basis of integrability: conformal SU(2,2) spin chain: a window to full AdS/CFT integrability
Anomalous dimensions of BMN-vacuum operators dominated by ``wheel” graphs, via QSC 2

3. ɣ-twisted N=4 SYM and double scaling limit
ɣ-twisted N=4 SYM Lagrangian: product of matrix fields → star-product
Leigh, Strassler
Frolov, Tseytlin
Beisert, Roiban
Lunin, Maldacena
where
Cartan charges
of R-symmetry
- twists
Breaks R-symmetry and all supersymmetry: PSU(2,2|4)→SU(2,2)×U(1)3
Generates double-trace terms breaking conformal invariance
β-functions have interesting complex zero, e.g. at one loop:
Sieg, Wilhelm, Fokken 2013
Sieg, Wilhelm 2016
It is all-loop non-unitary CFT! At large N, ‘t Hooft coupling g2 doesn’t run
Integrable! All anomalous dimensions computable from QSC formalism

4. Double scaling limit of ɣ-twisted N=4 SYM
Gurdogan, V.K. 2015
Double scaling limit: strong twist, weak coupling

5. Special case: bi-scalar chiral “CFT”
Gurdogan, V.K. 2015
Special case: bi-scalar chiral “CFT”
Special case:
Zero dimensional analogue:
Kostov, Staudacher 1995
Propagators
Vertex:
Missing
“anti-chiral” vertex
Very limited number of planar graphs
Sieg, Wilhelm, Fokken 2013
No mass or vertex renormalization in planar limit!
Sieg, Wilhelm 2016
ξ dosn’t run in planar limit, but tr2 couplings do run. Still CFT at critical point!

6. Operators, correlators, graphs…
Caetano, Gurdogan, V.K , 2016
Operators, correlators, graphs…
Operators with M magnons in the “vacuum” of
Correlators
anomalous dimension
“wheel” graphs
UV- reduction
BMN vacuum
(non-protected)
Typical “fishnet” structure in the bulk. Integrable!
“spiderweb” graphs
Multi-magnon
spiral graphs
Particular case, 4-point function, computed
explicitly by conformal bootstrap
Basso, Dixon 2017

7. Wheel graphs at any L and dimension of
from integrability
Broadherst 1980
Gurdogan, V.K. 2015
Ahn, Bajnok, Bombardelli, Nepomechie 2013

8. Fishnet amplitudes in bi-scalar model
Chicherin, V.K., Loebbert, Mueller, Zhong 2017
Fishnet amplitudes in bi-scalar model
Single-trace correlator defined by a single multi-loop graph
Typical structure –
regular square lattice – “fishnet”:
A disc cut out of square lattice
Dual graph with light-like momenta
defines bi-scalar amplitude
Manifestly finite and dually conformal invariant. Integrable?

9. Yangian invariance of bi-scalar amplitudes
Chicherin, V.K., Loebbert, Mueller, Zhong (2017)
Conformal Lax operator
Chicherin, Derkachev, Isaev (2012)
SU(4) generators in fundamental
generators in irrep (0,Δ,0)
- diagonal action on unity
Chicherin, Derkachev, Kirschner (2009)
- intertwining
Monodromy around graph (“lasso”)
can be undone
by these operations
Feynman graph |G > invariant under action of monodromy around boundary

10. Examples: Yangian PDE on cross and double-cross integrals
Expanding in u we get Yangian generators of level 1, etc.
Differential relation of Yangian symmetry for “cross” integral
- generators of conformal algebra
For “double-cross”:

11. Fishnet graphs and integrable conformal SU(2,2) spin chain
Gurdogan, V.K. 2015
Gromov, V.K , Korchemsky, Negro, Sizov 2017
Fishnet graphs and integrable conformal SU(2,2) spin chain
Transfer-matrix generating a wheel (fishnet) graph
Integrability: graph-generating transfer-matrix emerges at special value of
spectral parameter of T-matrix in physical irrep on 4D conformal group
Hence integrability of bi-scalar model is demonstrated explicitly in all orders!
To compute dimensions we use the Quantum Spectral Curve of N=4 SYM

12. Quantum Spectral Curve of N=4 SYM
Gromov, V.K., Leurent, Volin 2014
V.K., Lerent, Volin 2015
Quantum Spectral Curve of N=4 SYM
QSC eqs. close on a finite number of Baxter functions of spectral parameter
The multi-index I labeled by Hasse diagram - N-hypercube (N=8 - rank of PSU(2,2|4) )
gl(2)
gl(3)
gl(4)
gl(8)
Parametrize all Q-functions through one-index Q-functions
through determinant formula
Plücker QQ-relations on each face:
Grassmanian structure!
Krichever, Lupan, Wiegmann, Zabrodin, 1994

13. (K|M)-graded Q-system and Heizenberg spin chain
Tsuboi
V.K., Sorin, Zabrodin
V.K., Leurent, Volin.
gl(3)
gl(2|1)
Nicest formulation of solution for spectrum of GL(N) Heisenberg spin chain:
impose polynomiality of Q-functions
“Tilting” the hypercube, i.e. fixing two diametrically opposite Q-functions on Hasse diagram we get Q-system for Heisenberg gl(K|M) super-spin chain!
and all other Q-functions are (twisted) polynomials. Then the energy is

14. Qantum Spectral Curve of AdS/CFT
Gromov, V.K., Leurent, Volin 2013
V.K., Leurent, Volin 2014
Q-functions on (4|4) Hasse diagram with nice analyticity on defining sheets
Hodge dual single index
Q-functions
single index
Q-functions
Q-functions live on infinitely branching Riemann surface with fixed equidistant cuts

15. Qantum Spectral Curve: large u, ɤ-twist and RH conditions
Gromov, V.K., Leurent, Volin 2013
V.K., Leurent, Volin 2014
Large u asymptotics fixed by PSU(2,2|4) Cartan charges
ɤ-deformed QSC
Monodromy around branchpoints (inspired by classical finite gap solution)
given by “gluing” relations at the main cut:
V.K.,Marshakov,Minahan,Zarembo
Beisert,V.K.,Sakai,Zarembo
Gromov, V.K. (2010) – in classical limit
Gromov, Sizov, Levkovich-Maslyk (2015)
These Riemann-Hilbert conditions fix all physical solutions for Q-system and also conformal dimensions Δ(g2) (finite set of operators with given charges)

16. Double Scaling in QSC Consider BMN operators
Gromov, V.K., Korchemsky, Negro, Sizov 2017
Double Scaling in QSC …
Consider BMN operators
in bi-scalar model. Construct QSC in DS limit:
Extra “Left-Right” symmetry for such operators
Analyticity: asymptotics and Zhukovsky cuts:
one short cut
short cuts in lower half-plane
Double scaling limit of QSC Baxter functions:
At week coupling, cuts generate poles – the only source of singularities at finite u:
QSC gluing relations become for this state

17. Solution for L=3 BMN vacuum
Gromov, V.K., Korchemsky, Negro, Sizov 2017
In double scaling limit, QSC gives two 2nd order Baxter equation for
resepctively, for and
which is equivalent to the following quantization condition at u=0
These q-functions should obey “pure” large u asymptotics
Solved numerically for anomalous dimensions with virtually arbitrary precision at all relevant couplings
Perturbative solution in ξ gives the exact periods of “wheel” graphs

18. L=3 BMN vacuum and all-loop wheel graphs
Gromov, V.K , Korchemsky, Negro, Sizov (2017)
Ahn, Bajnok, Bombardelli, Nepomechie 2013
Broadherst 1980
E.Panzer, 2015
Riemann (multi)-zeta numbers
Generalization to any number of spokes L is in work
(Baxter equation is available)

19. Baxter equation for O(x)=
Baxter eq. can be obtained from Yang-Baxter algebra of SU(2,2) spin chain
We need to know only the symmetries of coefficients and asymptotics of q(u)
This fixes many coefficients. The rest fixed by auxiliary quantization conditions
Quantization condition should supply Baxter eq. Can be obtained from QSC

20. Conclusions and prospects
Double scaling limit of twisted N=4 SYM theory produces new 4D chiral CFTs
Planar limit dominated by “fishnet” graphs, explicitly integrable at each loop order
QSC computes exactly wheel graphs (done for J=3 spokes but also J>3 possible)
How to compute them from conformal SU(2,2) spin chain?
Structure constants? 1/N corrections? β-functions (study of scale dependence)? quark-antiquark potential ……. in DS limit (talk of Kolya Gromov)
Bi-scalar amplitudes are finite and exhibit explicit Yangian invariance
Similar observation for DS limit of ɤ-deformed 3D ABJM – 6-scalar chiral interaction
Gromov, Levkovich-Maslyuk 2016
Chicherin, V.K., Loebbert, Mueller, Zhong (2017)
Caetano, Gurdogan, V.K 2016
6D tri-scalar theory: regular hexagonal graphs.
Twisted 6D SYM?
Torrens, Mamroud 2017
Our observations shed some light on the origins of integrability of planar SYM

21. END
biobuild.mlsoc.vt.edu

22. Dimensions from Asymptotic Bethe Ansatz
Eden, Beisert, Staudacher, 2005
Caetano, Gurdogan, V.K 2016
In bi-scalar model, rapidities live in “mirror” plane and the double-scaled
ABA in SU(2) subsector (made of )
Where σ is mirror-mirror ABA dressing phase in g=0 limit
The anomalous dimension are given by
Helps to compute unwrapped magnon graphs entering the mixing matrix:
“Minimally connected” Feynman graph is computed directly
Georgoudis, Goncalves, Pereira

23. Special L=2 case Naively, it is diverging already at one loop.
Grabner, Gromov, V.K., Korchemsky (to appear)
Naively, it is diverging already at one loop.
Finite after renormalization by double-trace vertices
Anomalous dimension, computed by QSC and confirmed by renormalized perturbation theory, is given by simple formula:
coincide at ξ=0,
the reason of singularity!
Perturbatively
Bi-scalar theory is conformal at certain values of double square couplings
Operators of Length=2 have finite dimensions
This picture should persist at the (complex) critical point of ɤ-deformed N=4 SYM

24. Numerics for L=3 “BMN vacuum”
Around ξ3=0.2 dimension becomes imaginary: phase transition, finite convergence radius.
What happens to the string dual?

25. Operators of length=3+2n and log-multiplets
Gromov, V.K , Negro, Korchemsky, Sizov (to appear this week)
Our Baxter eq. fixes only the charges of operator (0,Δ,0|J=3,0), not its length.
It also describes operators of any length L=3+2n1+2n2 of the type
Protected operators:
Mixing in multiplet of operators of length L=5:
protected
Non-unitary mixing matrix. “Diagonalisation” means bringing to Jordan form
Lower block contains usual operators with dimensions
Upper block is Jordan cell,
it gives log-conformal correlators:
Gurarie 1993
Caetano 2016

26. Perturbative expansion and numerics for operators of length L>3
At L=3+4n+2 Baxter equation renders two complex conjugate dimensions
At L=3+4n dimensions are real (until they hit the unitary bound Δ(ξc )=2)
Numerics:
Large ξ limit also studied.
Described by a classical integrable system
of 3 non-compact spins