6,701 research outputs found
Integrability of two-loop dilatation operator in gauge theories
We study the two-loop dilatation operator in the noncompact SL(2) sector of
QCD and supersymmetric Yang-Mills theories with N=1,2,4 supercharges. The
analysis is performed for Wilson operators built from three quark/gaugino
fields of the same helicity belonging to the fundamental/adjoint representation
of the SU(3)/SU(N_c) gauge group and involving an arbitrary number of covariant
derivatives projected onto the light-cone. To one-loop order, the dilatation
operator inherits the conformal symmetry of the classical theory and is given
in the multi-color limit by a local Hamiltonian of the Heisenberg magnet with
the spin operators being generators of the collinear subgroup of full
(super)conformal group. Starting from two loops, the dilatation operator
depends on the representation of the gauge group and, in addition, receives
corrections stemming from the violation of the conformal symmetry. We compute
its eigenspectrum and demonstrate that to two-loop order integrability survives
the conformal symmetry breaking in the aforementioned gauge theories, but it is
violated in QCD by the contribution of nonplanar diagrams. In SYM theories with
extended supersymmetry, the N-dependence of the two-loop dilatation operator
can be factorized (modulo an additive normalization constant) into a
multiplicative c-number. This property makes the eigenspectrum of the two-loop
dilatation operator alike in all gauge theories including the maximally
supersymmetric theory. Our analysis suggests that integrability is only tied to
the planar limit and it is sensitive neither to conformal symmetry nor
supersymmetry.Comment: 70 pages, 10 figure
The Lorentz Integral Transform (LIT) method and its applications to perturbation induced reactions
The LIT method has allowed ab initio calculations of electroweak cross
sections in light nuclear systems. This review presents a description of the
method from both a general and a more technical point of view, as well as a
summary of the results obtained by its application. The remarkable features of
the LIT approach, which make it particularly efficient in dealing with a
general reaction involving continuum states, are underlined. Emphasis is given
on the results obtained for electroweak cross sections of few--nucleon systems.
Their implications for the present understanding of microscopic nuclear
dynamics are discussed.Comment: 83 pages, 31 figures. Topical review. Corrected typo
A Note On Boundary Conditions In Euclidean Gravity
We review what is known about boundary conditions in General Relativity on a
spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in
which one specifies the boundary geometry, is actually not elliptic and in
general does not lead to a well-defined perturbation theory. It is
better-behaved if the extrinsic curvature of the boundary is suitably
constrained, for instance if it is positive- or negative-definite. A different
boundary condition, in which one specifies the conformal geometry of the
boundary and the trace of the extrinsic curvature, is elliptic and always leads
formally to a satisfactory perturbation theory. These facts might have
interesting implications for semiclassical approaches to quantum gravity.
(Submitted to a volume in honor of Roman Jackiw.)Comment: 26 pp. Dedication added to Roman Jackiw. Minor corrections in this
versio
Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
In this paper we consider the spectral and nonlinear stability of periodic
traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In
particular, we resolve the long-standing question of nonlinear modulational
stability by demonstrating that spectrally stable waves are nonlinearly stable
when subject to small localized (integrable) perturbations. Our analysis is
based upon detailed estimates of the linearized solution operator, which are
complicated by the fact that the (necessarily essential) spectrum of the
associated linearization intersects the imaginary axis at the origin. We carry
out a numerical Evans function study of the spectral problem and find bands of
spectrally stable periodic traveling waves, in close agreement with previous
numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy,
Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also
compare predictions of the associated Whitham modulation equations, which
formally describe the dynamics of weak large scale perturbations of a periodic
wave train, with numerical time evolution studies, demonstrating their
effectiveness at a practical level. For the reader's convenience, we include in
an appendix the corresponding treatment of the Swift-Hohenberg equation, a
nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation
for which the nonlinear stability analysis is considerably simpler, together
with numerical Evans function analyses extending spectral stability analyses of
Mielke and Schneider.Comment: 78 pages, 11 figure
Parity flow as -valued spectral flow
This note is about the topology of the path space of linear Fredholm
operators on a real Hilbert space. Fitzpatrick and Pejsachowicz introduced the
parity of such a path, based on the Leray-Schauder degree of a path of
parametrices. Here an alternative analytic approach is presented which reduces
the parity to the -valued spectral flow of an associated path of
chiral skew-adjoints. Furthermore the related notion of -index
of a Fredholm pair of chiral complex structures is introduced and connected to
the parity of a suitable path. Several non-trivial examples are provided. One
of them concerns topological insulators, another an application to the
bifurcation of a non-linear partial differential equation.Comment: numerous improvements, title change
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