Different faces of integrability in the gauge theories or in hunting for the symmetries
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Transcript of Different faces of integrability in the gauge theories or in hunting for the symmetries
Different faces of integrability Different faces of integrability in the gauge theories or in in the gauge theories or in
hunting for the symmetrieshunting for the symmetries
Isaac Newton Institute, October 8Isaac Newton Institute, October 8
Some history of the hidden integrabilitySome history of the hidden integrability
Matrix models for the quantum gravity –Douglas, Gross-Migdal, Brezin-Matrix models for the quantum gravity –Douglas, Gross-Migdal, Brezin-Kazakov (89-91)Kazakov (89-91)
Regge limit of scattering amplitudes in QCD- Lipatov,Korchemsky-Regge limit of scattering amplitudes in QCD- Lipatov,Korchemsky-Faddeev(93-94)Faddeev(93-94)
Topological gauge theories in D=2(YM) and D=3 (Chern-Simons)Topological gauge theories in D=2(YM) and D=3 (Chern-Simons) Nekrasov-A.G.(94-95)Nekrasov-A.G.(94-95) N=2 SUSY Yang-Mills theories- Krichever-Marshakov-Mironov-Morozov-N=2 SUSY Yang-Mills theories- Krichever-Marshakov-Mironov-Morozov-
A.G., Witten-Donagi(95) A.G., Witten-Donagi(95) Anomalous dimensions from integrability – Braun- Derkachev-Manashev-Anomalous dimensions from integrability – Braun- Derkachev-Manashev-
Belitsky-Korchemsky (in simplest one-loop cases in QCD-98-99)Belitsky-Korchemsky (in simplest one-loop cases in QCD-98-99) Anomalous dimensions in N=4 SYM Minahan- Zarembo, Beisert-Anomalous dimensions in N=4 SYM Minahan- Zarembo, Beisert-
Staudacher (02-03)Staudacher (02-03) Integrability of the dual sigma model for N=4 SYM- Bena-Polchinski-Roiban Integrability of the dual sigma model for N=4 SYM- Bena-Polchinski-Roiban
(04)(04) Matching of YM and stringy answers; Tseytlin-Frolov ;Minahan-Zarembo-Matching of YM and stringy answers; Tseytlin-Frolov ;Minahan-Zarembo-
Kazakov-Marshakov and many others (04-….)Kazakov-Marshakov and many others (04-….) Proposal for the all-loop result ; Beisert-Eden-Staudacher(06)Proposal for the all-loop result ; Beisert-Eden-Staudacher(06)
Integrability: what does it mean?Integrability: what does it mean?
Dynamical system with N degrees of Dynamical system with N degrees of freedom should have N conserved freedom should have N conserved integrals of motion {H,Iintegrals of motion {H,Inn}=0. They }=0. They commute that is one can consider the commute that is one can consider the different “time” directionsdifferent “time” directions
If number of the conserved integrals is If number of the conserved integrals is infinite - integrable field theories. Many infinite - integrable field theories. Many examples but mainly in (1+1) dimensionsexamples but mainly in (1+1) dimensions
Universality of the integrabilityUniversality of the integrability
Plasma, Hydrodynamics - KdV, KP Plasma, Hydrodynamics - KdV, KP equationsequations
2D Quantum gravity-matrix models – 2D Quantum gravity-matrix models – KdV,KP hierarchies KdV,KP hierarchies
Gauge theories in D=2,3,4 ; Quantum Hall Gauge theories in D=2,3,4 ; Quantum Hall effect in different geometry; Black holes -effect in different geometry; Black holes -Toda,Calogero and Ruijsenaars systemsToda,Calogero and Ruijsenaars systems
Evolution equations in D=4-spin chains Evolution equations in D=4-spin chains with the different groupswith the different groups
Integrability versus group theoryIntegrability versus group theory
Phase spaces of the integrable systems are closely Phase spaces of the integrable systems are closely related to the group -like manifolds which admit the related to the group -like manifolds which admit the Poisson structurePoisson structure
Examples of the finite dimensional “group” phase Examples of the finite dimensional “group” phase manifolds parameters: Coadjoint orbit ->T*gmanifolds parameters: Coadjoint orbit ->T*gT*G-T*G->Heisenberg Double>Heisenberg Double
More general integrable systems involves the phase More general integrable systems involves the phase spaces with additional parameters.spaces with additional parameters.
Finite dimensional examples :quantum groups(1 Finite dimensional examples :quantum groups(1 parameter), Sklyanin algebra(2 parameters),Mukai-parameter), Sklyanin algebra(2 parameters),Mukai-Odesskii algebra (many free parameters)Odesskii algebra (many free parameters)
Integrability versus group theoryIntegrability versus group theory
Poisson structure is closely related to the geometric objects. Example – intersection of N quadrics Qk in CP(N+2) with homogenious coordinates xk.
Complicated polynomial algebras induced by geometry. The quadrics are Casimir operators of this algebra. A lot of Casimirs and free parameters.
Integrability versus Integrability versus groupgroup theory theory
Infinite dimensional examples; Kac-Moody algebra,Infinite dimensional examples; Kac-Moody algebra,
Virasoro algebra. Parameters: central charges and Virasoro algebra. Parameters: central charges and parameters of representationparameters of representation
Parameters of the “group” phase spaces are mapped into Parameters of the “group” phase spaces are mapped into the parameters of the integrable systemsthe parameters of the integrable systems
Generic situation: Integrable system follows Generic situation: Integrable system follows from the free motion on the group-like from the free motion on the group-like manifolds with possible constraintsmanifolds with possible constraints
Integrability versus group theoryIntegrability versus group theory
Examples; KdV- free rotator on the coadjoint Examples; KdV- free rotator on the coadjoint Virasoro orbit Virasoro orbit uutt=uu=uuxx+u+uxxxxxx
Calogero and Toda systems - free motionCalogero and Toda systems - free motion on the T*(SU(N)) with the simple constrainton the T*(SU(N)) with the simple constraint
Relativistic Calogero system(Ruijsenaars)-Relativistic Calogero system(Ruijsenaars)-free motion on the Heisenberg Double with free motion on the Heisenberg Double with
constraintconstraint
Examples Examples
Potential of the integrable Calogero many-body system
Ruijsenaars many-body system
Integrability versus moduli spacesIntegrability versus moduli spaces
General comment:General comment:
Consider the solution to the equation of motion in some Consider the solution to the equation of motion in some gauge theorygauge theory
F=0, 3d Chern-Simons gauge theoryF=0, 3d Chern-Simons gauge theory
F=*F self-duality equation in 4d Yang-MillsF=*F self-duality equation in 4d Yang-Mills
F=*dZ BPS condition for the stable objects in SUSY YMF=*dZ BPS condition for the stable objects in SUSY YM
Solutions to these equations have nontrivial moduli spaces Solutions to these equations have nontrivial moduli spaces which enjoy the rich symmetry groups and provide the which enjoy the rich symmetry groups and provide the phase space for the integrable systemsphase space for the integrable systems
Integrability versus Riemann SurfacesIntegrability versus Riemann Surfaces
General comment: Solutions to the integrable systems General comment: Solutions to the integrable systems are parameterized by the Riemann surfaces (in general are parameterized by the Riemann surfaces (in general of infinite genus) which are related to the complex of infinite genus) which are related to the complex Liouville tori. In many interesting situations these Liouville tori. In many interesting situations these surfaces have finite genus. surfaces have finite genus.
Moduli of the complex structures of these Riemann surfaces are related to the integrals of motion. Summation over solutions=integration over the moduli
2D Yang-Mills on the cylinder2D Yang-Mills on the cylinder
CConsider SU(N) gauge theoryonsider SU(N) gauge theory
Heavy fermion at rest
Theory has no dynamical field degrees of freedom. However there are N quantum mechanical degrees of freedom from the holonomy of the connection.
A=diag(x1,…….,xn),
E=diag(p1,……,pn) + nondiag
Standard YM Hamiltonian H=Tr E^2 yields the Calogero integrable system with trigonometric long-range interaction
2
2D Yang-Mills theory and Calogero system2D Yang-Mills theory and Calogero system
What is the meaning of the time variables? What is the meaning of the time variables? The “first” time is the inverse coupling constantThe “first” time is the inverse coupling constant Higher “times” tHigher “times” tk k - - chemical potentials for the powers of chemical potentials for the powers of
the electric fieldthe electric field This is the generic situation – evolution parameters in This is the generic situation – evolution parameters in
the integrable systems relevant for the gauge theories the integrable systems relevant for the gauge theories are the couplings for the operators are the couplings for the operators
S=SS=S00 + t + tk k OOk k with some operators O with some operators Okk
In theories with running coupling tIn theories with running coupling t0 0 =log(scale) that is =log(scale) that is integrability is some property of RG evolutionintegrability is some property of RG evolution
Chern-Simons theory and Ruijsenaars Chern-Simons theory and Ruijsenaars systemsystem
Consider SU(N) Chern-Simons theory on the torus with Consider SU(N) Chern-Simons theory on the torus with marked point (Wilson line along the time direction)marked point (Wilson line along the time direction)
The phase space is related to the moduli space of flat connections on the torus. Coordinates follows from the holonomy along A-cycle and momenta from holonomy along B-cycle. The emerging dynamical system on the moduli space – relativistic generalization of the Calogero system with N degrees of freedom. When one of the radii degenerates Ruijsenaars system degenerates to the Calogero model. These are examples of integrability in the perturbed topological theory.
Integrability in N=2 Supersymmetric gauge Integrability in N=2 Supersymmetric gauge theoriestheories
In N=2 theory there are physical variables protected by In N=2 theory there are physical variables protected by holomorphy; low-energy effective actions and spectrum holomorphy; low-energy effective actions and spectrum of stable particlesof stable particles
All these holomorphic data are fixed by finite-All these holomorphic data are fixed by finite-dimensional integrable system which captures the one-dimensional integrable system which captures the one-loop perturbative correction and contribution from the loop perturbative correction and contribution from the arbitrary number of instantons to the tree Lagrangian arbitrary number of instantons to the tree Lagrangian
Theory involves naturally two moduli spaces. Moduli Theory involves naturally two moduli spaces. Moduli space of vacua is parameterized by the vacuum space of vacua is parameterized by the vacuum condensates. Also moduli space of instantons. condensates. Also moduli space of instantons.
Integrability in N=2 SUSY theoriesIntegrability in N=2 SUSY theories
Seiberg and Witten found solution for the holomorphic data in terms Seiberg and Witten found solution for the holomorphic data in terms of the family of the Riemann surfaces of the genus (N-1) with some of the family of the Riemann surfaces of the genus (N-1) with some additional data (meromorphic differential) bundled over the moduli additional data (meromorphic differential) bundled over the moduli space of the vacua space of the vacua
Vacuum expectation values of the complex scalars parameterize the moduli space of the Riemann surfaces.
Mapping into the integrable systemMapping into the integrable system
Time variable in the integrable system t= log (IR scale)Time variable in the integrable system t= log (IR scale) Riemann surface = solution to the classical equations of motionRiemann surface = solution to the classical equations of motion Moduli space of vacua = half of the phase space of the integrable Moduli space of vacua = half of the phase space of the integrable
systemsystem Masses of the stable particles= “action” variablesMasses of the stable particles= “action” variables All N=2 gauge theories with the different matter content have the All N=2 gauge theories with the different matter content have the
corresponding integrable system under the carpetcorresponding integrable system under the carpet
Gauge theories with N=2 SUSY versus integrable systems
Integrability and N=2 gauge theoriesIntegrability and N=2 gauge theories
The very surface has even more “physical” interpretation The very surface has even more “physical” interpretation – this is the surface we would live on if we would enjoy – this is the surface we would live on if we would enjoy N=2 SUSY. Any “N=2 citizen” lives on the 5+1 N=2 SUSY. Any “N=2 citizen” lives on the 5+1 worldvolume of the soliton(M5 brane) in higher worldvolume of the soliton(M5 brane) in higher dimensions which looks as R(3,1)+(Riemann surface).dimensions which looks as R(3,1)+(Riemann surface).
Is it possible to derive integrable system Is it possible to derive integrable system “microscopically”? Yes, it follows from the consideration “microscopically”? Yes, it follows from the consideration of the instanton moduli space (Nekrasov 04).of the instanton moduli space (Nekrasov 04).
Hence we have situation when integrability related with Hence we have situation when integrability related with RG flows involves the summation over nonperturbative RG flows involves the summation over nonperturbative solutions. Symmetries behind moduli spaces. solutions. Symmetries behind moduli spaces.
Anomalous dimensions in the gauge Anomalous dimensions in the gauge theories and Integrabilitytheories and Integrability
Time variable T= log(RG scale), that is once again Time variable T= log(RG scale), that is once again integrability behind the RG evolutionintegrability behind the RG evolution
One loop renormalization of the composite operators in YM theory is governed by the integrable Heisenberg spin chains
Example of the operator TrXXXZXZZZXXX, the number of sites in the chain coincides with the number of fields involved in the composite operator
Anomalous dimensions and integrabilityAnomalous dimensions and integrability
Acting by the spin chain Hamiltonian on the set of Acting by the spin chain Hamiltonian on the set of operators one gets the spectrum of anomalous operators one gets the spectrum of anomalous dimensions upon the diagonalization of the mixing dimensions upon the diagonalization of the mixing matrix. The RG equation because of integrability has matrix. The RG equation because of integrability has hidden conserved quantum numbers – eigenvalues of hidden conserved quantum numbers – eigenvalues of the higher Hamiltonians commuting with dilatationthe higher Hamiltonians commuting with dilatation
In N=4 SuperYM spin chain responsible for one-loop In N=4 SuperYM spin chain responsible for one-loop evolution has the symmetry group SO(6)*SO(2,4) which evolution has the symmetry group SO(6)*SO(2,4) which is the global symmetry group of the N=4 SYMis the global symmetry group of the N=4 SYM
Higher loops integrable system involves the interaction Higher loops integrable system involves the interaction between nearest L neighbors at L loop order between nearest L neighbors at L loop order
Anomalous dimensions and integrabilityAnomalous dimensions and integrability
Gauge-string duality ; N=4 SYM is dual to the Gauge-string duality ; N=4 SYM is dual to the superstring theory in superstring theory in
String tension is proportional to the square root of t’Hooft coupling
That is weak coupling in the gauge theory correspond to the deep quantum regime in the string sigma model while strong coupling corresponds to the quasiclassical string(Maldacena 97). Could gauge/string duality explain the origin of integrability? The answer is partially positive. Stringy sigma model on this background is CLASSICALLY integrable.
Anomalous dimensions and Anomalous dimensions and integrabilityintegrability
Hamiltonian of the string = Dilatation operator in the Hamiltonian of the string = Dilatation operator in the gauge theorygauge theory
That is derivation of the spectrum of anomalous That is derivation of the spectrum of anomalous dimensions is equivalent to the derivation of the dimensions is equivalent to the derivation of the spectrum of the quantum string in the fixed backgroundspectrum of the quantum string in the fixed background
The main problem – there is no solution to the The main problem – there is no solution to the QUANTUM sigma model in this background yet. That is QUANTUM sigma model in this background yet. That is no exact quantum spectrum we look for.no exact quantum spectrum we look for.
The hint – consider the operators with large quantum The hint – consider the operators with large quantum numbers (R charge,Lorentz spin S e.t.c.). The numbers (R charge,Lorentz spin S e.t.c.). The corresponding string motion is quasiclassical!corresponding string motion is quasiclassical!
Anomalous dimensions and integrabilityAnomalous dimensions and integrability
In this “forced” quasiclassical regime the comparison can In this “forced” quasiclassical regime the comparison can be made between perturbative YM calculations and be made between perturbative YM calculations and stringy answers. Complete agreement where possible.stringy answers. Complete agreement where possible.
First predictions from integrability for the all-loop First predictions from integrability for the all-loop answers for the simplest object – anomalous dimension answers for the simplest object – anomalous dimension of the operators with the large Lorentz spin S of the operators with the large Lorentz spin S
F(g) Log S (Beisert-Eden-Staudacher) F(g) Log S (Beisert-Eden-Staudacher) There are a lot of higher conserved charges commuting There are a lot of higher conserved charges commuting
with dilatation. Their role is not completely clear yet.with dilatation. Their role is not completely clear yet. They imply the hidden symmetries behind the They imply the hidden symmetries behind the
perturbative YM ( Yangian symmetry,Dolan-Nappi-Witten perturbative YM ( Yangian symmetry,Dolan-Nappi-Witten e.t.c.)e.t.c.)
Integrability and the scattering amplitudesIntegrability and the scattering amplitudes
At the weak coupling the scattering amplitudes in the At the weak coupling the scattering amplitudes in the Regge limit are governed by the complex integrable Regge limit are governed by the complex integrable system SL(2,C) Heisenberg spin chain. Number of system SL(2,C) Heisenberg spin chain. Number of reggeons = number of sites in the spin chain. Pomeron-reggeons = number of sites in the spin chain. Pomeron-spin chain with 2 sites, Odderon- spin chain with 3 sitesspin chain with 2 sites, Odderon- spin chain with 3 sites
Time variable in the integrable evolution Time variable in the integrable evolution T= log (scale)=log s, where s-kinematical invariant of the T= log (scale)=log s, where s-kinematical invariant of the scattering problemscattering problem
There is holomorphic factorization of the HamiltonianThere is holomorphic factorization of the Hamiltonian
(Lipatov)(Lipatov)
Integrability and the scattering amplitudesIntegrability and the scattering amplitudes
Scattering with the mutireggeon exchanges
Integrability and the scattering amplitudesIntegrability and the scattering amplitudes
The integrability is the property of the evolution The integrability is the property of the evolution equations (BFKL) once againequations (BFKL) once again
Spectrum of the integrable system defines the Spectrum of the integrable system defines the asymptotic behavior of the scattering amplitudesasymptotic behavior of the scattering amplitudes
Hk is the Hamiltonian of the spin chain with k sites
Integrability and scattering amplitudesIntegrability and scattering amplitudes
Many questions; What happens with integrability (upon Many questions; What happens with integrability (upon the resummation of the gluons to reggeons) at higher the resummation of the gluons to reggeons) at higher loops. What is the meaning of higher conserved loops. What is the meaning of higher conserved charges? E.t.c.charges? E.t.c.
From the stringy side some progress as well. Attempts to From the stringy side some progress as well. Attempts to identify the stringy configurations responsible for the identify the stringy configurations responsible for the scattering amplitudes ( Alday-Maldacena). However no scattering amplitudes ( Alday-Maldacena). However no clear identification yet similar to the clear identification yet similar to the string energy=anomalous dimensionsstring energy=anomalous dimensions
ConclusionConclusion
Integrability is very general phenomenon behind the Integrability is very general phenomenon behind the evolution equations (T= log (scale)) and moduli spaces evolution equations (T= log (scale)) and moduli spaces in many different topological and nontopological gauge in many different topological and nontopological gauge theoriestheories
Perfect matching with gauge/string duality when possiblePerfect matching with gauge/string duality when possible First predictions for the all-loop answers in N=4 SYM First predictions for the all-loop answers in N=4 SYM
theorytheory Prediction for the hidden symmetries in YM gauge theory Prediction for the hidden symmetries in YM gauge theory
(Yangian e.t.c.) Meaning of higher charges in the RG (Yangian e.t.c.) Meaning of higher charges in the RG evolution not clear enoughevolution not clear enough
Just the very beginning of the story. A lot to be done…..Just the very beginning of the story. A lot to be done…..