1,017,964 research outputs found
Fixed points in the family of convex representations of a maximal monotone operator
Any maximal monotone operator can be characterized by a convex function. The
family of such convex functions is invariant under a transformation connected
with the Fenchel-Legendre conjugation. We prove that there exist a convex
representation of the operator which is a fixed point of this conjugation.Comment: 13 pages, updated references. Submited in July 2002 to Proc. AM
Essential Killing fields of parabolic geometries
We study vector fields generating a local flow by automorphisms of a
parabolic geometry with higher order fixed points. We develop general tools
extending the techniques of [1], [2], and [3]. We apply these tools to almost
Grassmannian, almost quaternionic, and contact parabolic geometries, including
CR structures, to obtain descriptions of the possible dynamics of such flows
near the fixed point and strong restrictions on the curvature. In some cases,
we can show vanishing of the curvature on a nonempty open set. Deriving
consequences for a specific geometry entails evaluating purely algebraic and
representation-theoretic criteria in the model homogeneous space. Published in
Indiana University Mathematics Journal.Comment: 50 pages. Minor corrections, references update
Combinatorial realizations of crystals via torus actions on quiver varieties
Consider Kashiwara's crystal associated to a highest weight representation of
a symmetric Kac-Moody algebra. There is a geometric realization of this object
using Nakajima's quiver varieties, but in many particular cases it can also be
realized by elementary combinatorial methods. Here we propose a framework for
extracting combinatorial realizations from the geometric picture: We construct
certain torus actions on the quiver varieties and use Morse theory to index the
irreducible components by connected components of the subvariety of torus fixed
points. We then discuss the case of affine sl(n). There the fixed point
components are just points, and are naturally indexed by multi-partitions.
There is some choice in our construction, leading to a family of combinatorial
models for each highest weight crystal. Applying this construction to the
crystal of the fundamental representation recovers a family of combinatorial
realizations recently constructed by Fayers. This gives a more conceptual proof
of Fayers' result as well as a generalization to higher level. We also discuss
a relationship with Nakajima's monomial crystal.Comment: 23 pages, v2: added Section 8 on monomial crystals and some
references; v3: many small correction
-adic Mellin Amplitudes
In this paper, we propose a -adic analog of Mellin amplitudes for scalar
operators, and present the computation of the general contact amplitude as well
as arbitrary-point tree-level amplitudes for bulk diagrams involving up to
three internal lines, and along the way obtain the -adic version of the
split representation formula. These amplitudes share noteworthy similarities
with the usual (real) Mellin amplitudes for scalars, but are also significantly
simpler, admitting closed-form expressions where none are available over the
reals. The dramatic simplicity can be attributed to the absence of descendant
fields in the -adic formulation.Comment: 60 pages, several figures. v2: Minor typos fixed, references adde
Running coupling and mass anomalous dimension of SU(3) gauge theory with two flavors of symmetric-representation fermions
We have measured the running coupling constant of SU(3) gauge theory coupled
to Nf=2 flavors of symmetric representation fermions, using the Schrodinger
functional scheme. Our lattice action is defined with hypercubic smeared links
which, along with the larger lattice sizes, bring us closer to the continuum
limit than in our previous study. We observe that the coupling runs more slowly
than predicted by asymptotic freedom, but we are unable to observe fixed point
behavior before encountering a first order transition to a strong coupling
phase. This indicates that the infrared fixed point found with the thin-link
action is a lattice artifact. The slow running of the gauge coupling permits an
accurate determination of the mass anomalous dimension for this theory, which
we observe to be small, gamma_m < 0.6, over the range of couplings we can
reach. We also study the bulk and finite-temperature phase transitions in the
strong coupling region.Comment: 17 pages, 16 figures. Substantial modifications to explain why the
fat-link result for the beta function supersedes our thin-link result; also
updated the phase diagram to reflect additional numerical work. Added
references. Final versio
Dual theory of the superfluid-Bose glass transition in disordered Bose-Hubbard model in one and two dimensions
I study the zero temperature phase transition between superfluid and
insulating ground states of the Bose-Hubbard model in a random chemical
potential and at large integer average number of particles per site. Duality
transformation maps the pure Bose-Hubbard model onto the sine-Gordon theory in
one dimension (1D), and onto the three dimensional Higgs electrodynamics in two
dimensions (2D). In 1D the random chemical potential in dual theory couples to
the space derivative of the dual field, and appears as a random magnetic field
along the imaginary time direction in 2D. I show that the transition from the
superfluid state in both 1D and 2D is always controlled by the random critical
point. This arises due to a coupling constant in the dual theory with replicas
which becomes generated at large distances by the random chemical potential,
and represents a relevant perturbation at the pure superfluid-Mott insulator
fixed point. At large distances the dual theory in 1D becomes equivalent to the
Haldane's macroscopic representation of disordered quantum fluid, where the
generated term is identified with random backscattering. In 2D the generated
coupling corresponds to the random mass of the complex field which represents
vortex loops. I calculate the critical exponents at the superfluid-Bose glass
fixed point in 2D to be \nu=1.38 and z=1.93, and the universal conductivity at
the transition \sigma_c = 0.26 e_{*}^2 /h, using the one-loop field-theoretic
renormalization group in fixed dimension.Comment: 25 pages, 6 Postscript figures, LaTex, references updated, typos
corrected, final version to appear in Phys. Rev. B, June 1, 199
Conformal Bootstrap in Mellin Space
We propose a new approach towards analytically solving for the dynamical
content of Conformal Field Theories (CFTs) using the bootstrap philosophy. This
combines the original bootstrap idea of Polyakov with the modern technology of
the Mellin representation of CFT amplitudes. We employ exchange Witten diagrams
with built in crossing symmetry as our basic building blocks rather than the
conventional conformal blocks in a particular channel. Demanding consistency
with the operator product expansion (OPE) implies an infinite set of
constraints on operator dimensions and OPE coefficients. We illustrate the
power of this method in the epsilon expansion of the Wilson-Fisher fixed point
by reproducing anomalous dimensions and, strikingly, obtaining OPE coefficients
to higher orders in epsilon than currently available using other analytic
techniques (including Feynman diagram calculations). Our results enable us to
get a somewhat better agreement of certain observables in the 3d Ising model,
with the precise numerical values that have been recently obtained.Comment: 5 pages+appendices. v2:Added--matching of HS current anomalous
dimension to O(\epsilon^3); references; minor changes, v3: comparison with
numerics updated, typos fixed. v4: Published versio
-dimensional SYK, AdS Loops, and Symbols
We study the symbol for the conformal group, and its appearance in three
seemingly unrelated contexts: the SYK model, conformal representation theory,
and perturbative amplitudes in AdS. The contribution of the planar Feynman
diagrams to the three-point function of the bilinear singlets in SYK is shown
to be a symbol. We generalize the computation of these and other Feynman
diagrams to dimensions. The symbol can be viewed as the crossing
kernel for conformal partial waves, which may be computed using the Lorentzian
inversion formula. We provide closed-form expressions for symbols in
. In AdS, we show that the symbol is the Lorentzian inversion of
a crossing-symmetric tree-level exchange amplitude, thus efficiently packaging
the double-trace OPE data. Finally, we consider one-loop diagrams in AdS with
internal scalars and external spinning operators, and show that the triangle
diagram is a symbol, while one-loop -gon diagrams are built out of
symbols.Comment: 62 pages; v2 fixed typos and references, added comments about
anomalous dimensions; v3, fixed typos, published versio
- …