56 research outputs found
Planar N=4 gauge theory and the Inozemtsev long range spin chain
We investigate whether the (planar, two complex scalar) dilatation operator
of N=4 gauge theory can be, perturbatively and, perhaps, non-perturbatively,
described by an integrable long range spin chain with elliptic exchange
interaction. Such a chain was introduced some time ago by Inozemtsev. In the
limit of sufficiently ``long'' operators a Bethe ansatz exists, which we apply
at the perturbative two- and three-loop level. Spectacular agreement is found
with spinning string predictions of Frolov and Tseytlin for the two-loop
energies of certain large charge operators. However, we then go on to show that
the agreement between perturbative gauge theory and semi-classical string
theory begins to break down, in a subtle fashion, at the three-loop level. This
corroborates a recently found disagreement between three-loop gauge theory and
near plane-wave string theory results, and quantitatively explains a previously
obtained puzzling deviation between the string proposal and a numerical
extrapolation of finite size three-loop anomalous dimensions. At four loops and
beyond, we find that the Inozemtsev chain exhibits a generic breakdown of
perturbative BMN scaling. However, our proposal is not necessarily limited to
perturbation theory, and one would hope that the string theory results can be
recovered from the Inozemtsev chain at strong 't Hooft coupling.Comment: 31 pages, no figure; v1: one reference added, minor changes; v2:
slightly extended discussion of rapidity, references adde
Strong coupling from the Hubbard model
It was recently observed that the one dimensional half-filled Hubbard model
reproduces the known part of the perturbative spectrum of planar N=4 super
Yang-Mills in the SU(2) sector. Assuming that this identification is valid
beyond perturbation theory, we investigate the behavior of this spectrum as the
't Hooft parameter \lambda becomes large. We show that the full dimension
\Delta of the Konishi superpartner is the solution of a sixth order polynomial
while \Delta for a bare dimension 5 operator is the solution of a cubic. In
both cases the equations can be solved easily as a series expansion for both
small and large \lambda and the equations can be inverted to express \lambda as
an explicit function of \Delta. We then consider more general operators and
show how \Delta depends on \lambda in the strong coupling limit. We are also
able to distinguish those states in the Hubbard model which correspond to the
gauge invariant operators for all values of \lambda. Finally, we compare our
results with known results for strings on AdS_5\times S^5, where we find
agreement for a range of R-charges.Comment: 14 pages; v2: 17 pages, 2 figures, appendix and references added;
typos fixed, minor changes; v3 fixed figures; v4 more references added, minor
correctio
Aspects of Integrability in N =4 SYM
Various recently developed connections between supersymmetric Yang-Mills
theories in four dimensions and two dimensional integrable systems serve as
crucial ingredients in improving our understanding of the AdS/CFT
correspondence. In this review, we highlight some connections between
superconformal four dimensional Yang-Mills theory and various integrable
systems. In particular, we focus on the role of Yangian symmetries in studying
the gauge theory dual of closed string excitations. We also briefly review how
the gauge theory connects to Calogero models and open quantum spin chains
through the study of the gauge theory duals of D3 branes and open strings
ending on them. This invited review, written for Modern Physics Letters-A, is
based on a seminar given at the Institute of Advanced Study, Princeton.Comment: Invited brief review for Mod. Phys. Lett. A based on a talk at I.A.S,
Princeto
Higher Charges in Dynamical Spin Chains for SYM Theory
We construct, to the first two non-trivial orders, the next conserved charge
in the su(2|3) sector of N=4 Super Yang-Mills theory. This represents a test of
integrability in a sector where the interactions change the number of sites of
the chain. The expression for the charge is completely determined by the
algebra and can be written in a diagrammatic form in terms of the interactions
already present in the Hamiltonian. It appears likely that this diagrammatic
expression remains valid in the full theory and can be generalized to higher
loops and higher charges thus helping in establishing complete integrability
for these dynamical chains.Comment: 14 pages; V2: Appendix added with diagrammatic expression for H_{3,2
On the breakdown of perturbative integrability in large N matrix models
We study the perturbative integrability of the planar sector of a massive
SU(N) matrix quantum mechanical theory with global SO(6) invariance and
Yang-Mills-like interaction. This model arises as a consistent truncation of
maximally supersymmetric Yang-Mills theory on a three-sphere to the lowest
modes of the scalar fields. In fact, our studies mimic the current
investigations concerning the integrability properties of this gauge theory.
Like in the field theory we can prove the planar integrability of the SO(6)
model at first perturbative order. At higher orders we restrict ourselves to
the widely studied SU(2) subsector spanned by two complexified scalar fields of
the theory. We show that our toy model satisfies all commonly studied
integrability requirements such as degeneracies in the spectrum, existence of
conserved charges and factorized scattering up to third perturbative order.
These are the same qualitative features as the ones found in super Yang-Mills
theory, which were enough to conjecture the all-loop integrability of that
theory. For the SO(6) model, however, we show that these properties are not
sufficient to predict higher loop integrability. In fact, we explicitly
demonstrate the breakdown of perturbative integrability at fourth order.Comment: 27 page
On the Scattering Phase for AdS_5 x S^5 Strings
We propose a phase factor of the worldsheet S-matrix for strings on AdS_5 x
S^5 apparently solving Janik's crossing relation.Comment: 9 pages, v2: revised conclusions about agreement with perturbative
string theory; minor changes, v3: resolution to above problems indicated, to
appear in Mod. Phys. Lett.
Open Spin Chains in Super Yang-Mills at Higher Loops: Some Potential Problems with Integrability
The super Yang-Mills duals of open strings attached to maximal giant
gravitons are studied in perturbation theory. It is shown that non-BPS baryonic
excitations of the gauge theory can be studied within the paradigm of open
quantum spin chains even beyond the leading order in perturbation theory. The
open spin chain describing the two loop mixing of non-BPS giant gravitons
charged under an su(2) of the so(6) R symmetry group is explicitly constructed.
It is also shown that although the corresponding open spin chain is integrable
at the one loop order, there is a potential breakdown of integrability at two
and higher loops. The study of integrability is performed using coordinate
Bethe ansatz techniques.Comment: 28 pages. References added in revised versio
New Integrable System of 2dim Fermions from Strings on AdS_5 x S^5
We consider classical superstrings propagating on AdS_5 x S^5 space-time. We
consistently truncate the superstring equations of motion to the so-called
su(1|1) sector. By fixing the uniform gauge we show that physical excitations
in this sector are described by two complex fermionic degrees of freedom and we
obtain the corresponding Lagrangian. Remarkably, this Lagrangian can be cast in
a two-dimensional Lorentz-invariant form. The kinetic part of the Lagrangian
induces a non-trivial Poisson structure while the Hamiltonian is just the one
of the massive Dirac fermion. We find a change of variables which brings the
Poisson structure to the canonical form but makes the Hamiltonian nontrivial.
The Hamiltonian is derived as an exact function of two parameters: the total
S^5 angular momentum J and string tension \lambda; it is a polynomial in 1/J
and in \sqrt{\lambda'} where \lambda'=\frac{\lambda}{J^2} is the effective BMN
coupling. We identify the string states dual to the gauge theory operators from
the closed su(1|1) sector of N=4 SYM and show that the corresponding near-plane
wave energy shift computed from our Hamiltonian perfectly agrees with that
recently found in the literature. Finally we show that the Hamiltonian is
integrable by explicitly constructing the corresponding Lax representation.Comment: 35 pages;v2:typos corrected, references adde
Planar N=4 Gauge Theory and the Hubbard Model
Recently it was established that a certain integrable long-range spin chain
describes the dilatation operator of N=4 gauge theory in the su(2) sector to at
least three-loop order, while exhibiting BMN scaling to all orders in
perturbation theory. Here we identify this spin chain as an approximation to an
integrable short-ranged model of strongly correlated electrons: The Hubbard
model.Comment: 35 pages, 2 figures; v2: typos and references fixed, published
versio
On highest-energy state in the su(1|1) sector of N=4 super Yang-Mills theory
We consider the highest-energy state in the su(1|1) sector of N=4 super
Yang-Mills theory containing operators of the form tr(Z^{L-M} \psi^M) where Z
is a complex scalar and \psi is a component of gaugino. We show that this state
corresponds to the operator tr(\psi^L) and can be viewed as an analogue of the
antiferromagnetic state in the su(2) sector. We find perturbative expansions of
the energy of this state in both weak and strong 't Hooft coupling regimes
using asymptotic gauge theory Bethe ansatz equations. We also discuss a
possible analog of this state in the conjectured string Bethe ansatz equations.Comment: 23 pages, Late
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