3,213 research outputs found
Certain subclasses of multivalent functions defined by new multiplier transformations
In the present paper the new multiplier transformations
\mathrm{{\mathcal{J}% }}_{p}^{\delta }(\lambda ,\mu ,l) (\delta ,l\geq
0,\;\lambda \geq \mu \geq 0;\;p\in \mathrm{% }%\mathbb{N} )} of multivalent
functions is defined. Making use of the operator two new subclasses and \textbf{\ }of multivalent analytic
functions are introduced and investigated in the open unit disk. Some
interesting relations and characteristics such as inclusion relationships,
neighborhoods, partial sums, some applications of fractional calculus and
quasi-convolution properties of functions belonging to each of these subclasses
and
are
investigated. Relevant connections of the definitions and results presented in
this paper with those obtained in several earlier works on the subject are also
pointed out
On 2-form gauge models of topological phases
We explore various aspects of 2-form topological gauge theories in (3+1)d.
These theories can be constructed as sigma models with target space the second
classifying space of the symmetry group , and they are classified by
cohomology classes of . Discrete topological gauge theories can typically
be embedded into continuous quantum field theories. In the 2-form case, the
continuous theory is shown to be a strict 2-group gauge theory. This embedding
is studied by carefully constructing the space of -form connections using
the technology of Deligne-Beilinson cohomology. The same techniques can then be
used to study more general models built from Postnikov towers. For finite
symmetry groups, 2-form topological theories have a natural lattice
interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d
that is exactly solvable. This construction relies on the introduction of a
cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified
with the simplicial cocycles of as provided by the so-called
-construction of Eilenberg-MacLane spaces. We show algebraically and
geometrically how a 2-form 4-cocycle reduces to the associator and the braiding
isomorphisms of a premodular category of -graded vector spaces. This is used
to show the correspondence between our 2-form gauge model and the Walker-Wang
model.Comment: 78 page
2D Conformal Field Theories and Holography
It is known that the chiral part of any 2d conformal field theory defines a
3d topological quantum field theory: quantum states of this TQFT are the CFT
conformal blocks. The main aim of this paper is to show that a similar CFT/TQFT
relation exists also for the full CFT. The 3d topological theory that arises is
a certain ``square'' of the chiral TQFT. Such topological theories were studied
by Turaev and Viro; they are related to 3d gravity. We establish an
operator/state correspondence in which operators in the chiral TQFT correspond
to states in the Turaev-Viro theory. We use this correspondence to interpret
CFT correlation functions as particular quantum states of the Turaev-Viro
theory. We compute the components of these states in the basis in the
Turaev-Viro Hilbert space given by colored 3-valent graphs. The formula we
obtain is a generalization of the Verlinde formula. The later is obtained from
our expression for a zero colored graph. Our results give an interesting
``holographic'' perspective on conformal field theories in 2 dimensions.Comment: 29+1 pages, many figure
Polyhedra in loop quantum gravity
Interwiners are the building blocks of spin-network states. The space of
intertwiners is the quantization of a classical symplectic manifold introduced
by Kapovich and Millson. Here we show that a theorem by Minkowski allows us to
interpret generic configurations in this space as bounded convex polyhedra in
Euclidean space: a polyhedron is uniquely described by the areas and normals to
its faces. We provide a reconstruction of the geometry of the polyhedron: we
give formulas for the edge lengths, the volume and the adjacency of its faces.
At the quantum level, this correspondence allows us to identify an intertwiner
with the state of a quantum polyhedron, thus generalizing the notion of quantum
tetrahedron familiar in the loop quantum gravity literature. Moreover, coherent
intertwiners result to be peaked on the classical geometry of polyhedra. We
discuss the relevance of this result for loop quantum gravity. In particular,
coherent spin-network states with nodes of arbitrary valence represent a
collection of semiclassical polyhedra. Furthermore, we introduce an operator
that measures the volume of a quantum polyhedron and examine its relation with
the standard volume operator of loop quantum gravity. We also comment on the
semiclassical limit of spinfoams with non-simplicial graphs.Comment: 32 pages, many figures. v2 minor correction
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