Yangian%20Symmetry%20in%20Yang-Mills%20Theories

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Yangian Symmetry in Yang-Mills Theories

• S. G. Rajeev
• Seminar at Cornell University
• Dec 8th 2004

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Work in Collaboration with Abhishek Agarwal
hep-th/0405116,hep-th/0409180

• Also earlier work with Herbert Lee,
• Teoman Turgut and Govind Krishnaswamy

Yangian symmetry was originally proposed for
Yang-Mills theories in L. Dolan, C.R.Nappi and
E. Witten hep-th/0308089,0401243 We extend the
idea to one and two loops of the quantum theory.
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What is a Yangian?
D. Bernard and A. Leclair hep-th/9205064
It is a deformation of the associative algebra
defined by the commutation relations
The most familiar deformation is the KacMoody
algebra
But there is another one if we restrict the range
of the indices to non-negative values. To
understand this, first note that the above
algebra is generated by
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Finite Presentation
These satisfy the obvious relations
The higher generators are given by repeated
commutators of these. However since there are
several ways of doing that certain consistency
relations need to be imposed
These Serre relations give a presentation of
the algebra in terms of a finite number of
generators and relations.
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The Co-product
Equally important to representation theory is the
co-product
The co-product allows us to form new
representations by taking tensor products of old
ones. Physically it is analogous to the rules for
addition of angular momentum. Note that the
order of composition doesnt matter as the
co-product is co-commutative.
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Lack of Co-Commutativity
It is possible now to modify this structure so
that the co-product is no longer co-commutative
This is the rule for addition of certain
non-local conserved charges of quantum spin
chains and matrix models. There is a preferred
order for combining spins (and matrices) so that
the rule for addition of charges need not be
co-commutative.
Can the multiplication law be changed so that
this new co-product is still a homomorphism? No
change is needed in the relations
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The Terrific Relations
There is a modification of the Serre relations
which are preserved by this new co-product
The Hopf algebra defined by these relations is
the Yangian.
It is a true quantum group neither
commutative nor co-commutative. A new kind of
symmetry that explains the integrability of many
quantum systems spin chains and matrix models.
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Yangian Symmetry of Quantum Spin Chains
A spin chain is a sequence of L spins arranged
on a line with the last one connected to the
first. Each spin can take N possible values. A
typical hamiltonian (XXX Heisenberg chain) would
be

There is a Yangian symmetry in this system that
explains the exact solvability of these spin
chains by the celebrated Bethe ansatz.
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Cut and Paste
Operators
They check if the lower sequence appears in the
beginning of the list of spin states if it
does, it is cut out and replaced by the upper
sequence. (Recall that by cyclic symmetry we can
bring any spin to the beginning of the list.)
Otherwise we get zero. Rather like the cut and
paste function of a text editor.
The Heisenberg Hamiltonian is
More complicated hamiltonians can be written as
linear combinations of these operators.
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The Commutation Relations of the cut and paste
Operators
These operators satisfy an interesting Lie
algebra C.W.H.Lee and S. G. Rajeev Phys. Rev.
Lett. 80,2285-2288(1998)
where the structure constants have a graphical
interpretation.
There is a sophisticated theory explaining the
integrability of the Heisenberg spin chain, in
terms of transfer matrices and Yang-Baxter
relations. It was found that there is an
underlying Yangian symmetry.
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The Generators of the Yangian for the
Heisenberg Spin Chain
There is an obvious unitary symmetry in the
Heisenberg spin chain with the conserved quantity
It is less obvious that there is another symmetry
the sum being over all possible sequences. This
follows by expanding the transfer matrix around
the point at infinity in the spectral parameter.
The Serre relations follow from the fact the
transfer matrix of the spin chain satisfies the
co-product rules but they can also be verified
directly using the commutation relations of the
cut and paste operators.
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Matrix Models and Spin Chains
S. G. Rajeev and C.W.H. Lee Nuclear Physics B,
529, 656-688(1998).
A matrix model is a quantum system whose degrees
of freedom are matrices. The basic operators
satisfy the canonical commutation relations
The hamiltonian is a unitary invariant operator
such as
The states that survive the large N limit are
Note that these states are in one-one
correspondence with the states of a quantum spin
chain.
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Cut and Paste Operators in Large N Matrix Models
The effect of these operators on the states above
is exactly those of the cut and paste operators
on spin chains. Thus there is an equivalence
between the large N limit of matrix models and
quantum spin chains. Certain matrix models go
over to integrable spin chains. For example, the
Heisenberg spin chain corresponds to
Thus these matrix models can be solved at least
in the large N limit by the Bethe ansatz. Several
such examples were given in C.W.H.Lee and S. G.
Rajeev Phys. Rev. Lett. 80,2285(1998)
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Yangian Symmetry in Matrix Models
Using the equivalence of matrix models to spin
chains, it should be possible to translate the
Yangian symmetries into the language of matrix
models. Matrix models are more general objects
than spin chains, since the equivalence is only
true at large N. Also they are prototypes of
Yang-Mills theories.
Is there a further deformation of the Yangian
which is also a symmetry of the finite N matrix
model?
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Matrix Approach to String Theory
T. Banks, W. Fishler, S.H. Shenker and L.
Susskind, Phys. Rev. D55, 5112(1995) R.
Dijkgraaf, E. Verlinde, H. Verlinde
hep-th/9703030 N. Kim and J.Plefka
hep-th/0207034
One of the approaches to string theory is through
the large N limit of matrix models. That matrix
models can be integrable and have hidden
symmetries suggest that string theory might be
more tractable than it looks at first sight. For
example, string theory in flat space is expected
to be equivalent to the matrix model with
lagrangian
The dots representing SUSY completion.
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Mass deformation of Matrix Models
However this theory is hard to study since it
doesnt have a minimum for its potential there
is a degeneracy which must be lifted by some
quantum effect. If we add the mass term we get
a theory that can be studied perturbatively
This should represent string theory in a plane
wave background
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Maximal Super-Yang-Mills Theory
Understanding Yang-Mills theories is the great
challenge for theoretical physics. The Yang-Mills
theory with the best chance of being integrable
is the maximally supersymmetric one, with a set
of four fermions and six scalars for each gauge
boson. We dont yet know what it means for such a
theory to be integrable. But certain limiting
cases are integrable. And these have Yangian
symmetries.
Maldacena has conjecutured that this theory is
equivalent to a string theory in the AdS
background. We dont yet know how to formulate
such a string theory. However both major
approaches (sigma models and matrix models) lead
to theories with Yangian symmetries.
Yangian (more generally Hopf,) symmetries could
be key to proving such an equivalence. Much like
the use of current algebra in proving the
Bose-Fermi correspondence in two dimensional
field theory.(Polchinski, Roiban..sigma model
approach to AdS string)
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The Dilatation Operator of N4 SYM
Although N4 SUSY YM has zero beta function (no
coupling constant renormalization) its gauge
invariant obsevables have anomalous dimension.
In fact the anomalous dimensions form an infinite
dimensional matrix which can be computed in
perturbation theory.
N. Beisert, C. Kristjansen, M. Staudacher
hep-th/0303060N. B., M.S. hep-th/0307042J.A.
Minahan, K. Zarembo hep-th/0212208 V.A. Kazakov,
A. Marshakov, J.M.,K. Z. hep-th/0402207
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Integrability of N4 SUSY YM
The analogue of the mass spectrum in a conformal
field theory is the set of eigenvalues of its
dilatation operator i.e., the anamolous
dimension matrix. At least at one loop the
dilatation operator can be diagonalized by the
Bethe ansatz. There are indications that it
persists to higher loops.
The integrability is explained by the Yangian
symmetry.
In our papers S.G.R.and Abhishek Agarwal
hep-th/0405116,0409180 we construct the Yangian
generators directly in terms of the scalar field
variables (matrix variables) and show that the
Serre relations are satisied at large N using the
cut and pasteoperators. Also we construct
deformations to the Yangian charges that extend
the symmetry to the two-loop dilatation operator.
What happens beyond that is not yet known.
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Will Integrability survive to Realistic Theories?
Most realistic systems in nature are not
integrable,but we find that studying a limiting
case that is integrable is usually a good
starting point. N4 SUSY-YM could be like the
Kepler problem while QCD is like celestial
mechanics.
The Birkoff procedure in mechanics allows us to
extend conserved quantities to any perturbation
of a classical system order by order in
perturbation theory. It is only for integrable
systems that this procedure converges. That there
are such perturbations in some regions of the
phase space was eventually established by the KAM
theorem. Realistic theories could have some
sectors are integrable and others that are not.
(QCD dilatation dynamics appears to be this
way.)
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Dilatation Dynamics of QCD
The full anomalous dimension matrix at one loop
of QCD has just been calculatedN. Beisert, G.
Ferretti, R. Heise, K. Zarembo hep-th/0412029.
It is an SU(2,2) spin chain but not integrable.
(For N4 SYM we would get an SU(2,24) integrable
spin chain.)
Nevertheless they are able to determine the
ground state (the operator with the smallest
anomalous dimension) as well as the low lying
excitations using a Bether ansatz
quasi-integrable system.
There were indications of integrability much
earlier in the related study of structure
functions e.g., MULTI-COLOR QCD AT HIGH ENERGIES
AND ONE-DIMENSIONAL HEISENBERG MAGNET L.D.
Faddeev , G.P. Korchemsky (1994). There is much
more work, see for references A.V. Belitsky, G.P.
Korchemsky, D. Muller hep-th/0412054
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Beyond Perturbation Theory
When an infinite number of operators mix even
one loop anomalous dimensions can lead to
sophisticated dynamical problems. If some kind
of gauge-string duality holds we can translate
the problem of solving the full Callan-Symanzik
equation (dilatation dynamics) to a matrix
model or a sigma model at least in the large N
limit ( classical limit of string theory). While
realistic systems like QCD are unlikely to be
integrable, there might be supersymmetric
variants which are. Then we can study the
formation of hadronic bound states as a problem
in this dynamics at short distances we have the
boundary conditions of a free theory and at long
distances we get out the hadronic states.
Although only a dream in QCD, I have done exactly
this in a quantum mechanical toy model with
asymptotic freedom. It is possible to determine
explicitly the operator that represents the
violation of scale invariance due to
renormalization.
Quantum Field Theory is back!