3,273 research outputs found
Pivotal decompositions of functions
We extend the well-known Shannon decomposition of Boolean functions to more
general classes of functions. Such decompositions, which we call pivotal
decompositions, express the fact that every unary section of a function only
depends upon its values at two given elements. Pivotal decompositions appear to
hold for various function classes, such as the class of lattice polynomial
functions or the class of multilinear polynomial functions. We also define
function classes characterized by pivotal decompositions and function classes
characterized by their unary members and investigate links between these two
concepts
Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants
We construct a generalization of pure lattice gauge theory (LGT) where the
role of the gauge group is played by a tensor category. The type of tensor
category admissible (spherical, ribbon, symmetric) depends on the dimension of
the underlying manifold (<=3, <=4, any). Ordinary LGT is recovered if the
category is the (symmetric) category of representations of a compact Lie group.
In the weak coupling limit we recover discretized BF-theory in terms of a
coordinate free version of the spin foam formulation. We work on general
cellular decompositions of the underlying manifold.
In particular, we are able to formulate LGT as well as spin foam models of
BF-type with quantum gauge group (in dimension <=4) and with supersymmetric
gauge group (in any dimension).
Technically, we express the partition function as a sum over diagrams
denoting morphisms in the underlying category. On the LGT side this enables us
to introduce a generalized notion of gauge fixing corresponding to a
topological move between cellular decompositions of the underlying manifold. On
the BF-theory side this allows a rather geometric understanding of the state
sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we
recover.
The construction is extended to include Wilson loop and spin network type
observables as well as manifolds with boundaries. In the topological (weak
coupling) case this leads to TQFTs with or without embedded spin networks.Comment: 58 pages, LaTeX with AMS and XY-Pic macros; typos corrected and
references update
A Two Weight Inequality for the Hilbert transform Assuming an Energy Hypothesis
Subject to a range of side conditions, the two weight inequality for the
Hilbert transform is characterized in terms of (1) a Poisson A_2 condition on
the weights (2) A forward testing condition, in which the two weight inequality
is tested on intervals (3) and a backwards testing condition, dual to (2). A
critical new concept in the proof is an Energy Condition, which incorporates
information about the distribution of the weights in question inside intervals.
This condition is a consequence of the three conditions above. The Side
Conditions are termed 'Energy Hypotheses'. At one endpoint they are necessary
for the two weight inequality, and at the other, they are the Pivotal
Conditions of Nazarov-Treil-Volberg. This new concept is combined with a known
proof strategy devised by Nazarov-Treil-Volberg. A counterexample shows that
the Pivotal Condition are not necessary for the two weight inequality.Comment: 60 pages, 1 figure. v3. An important revision: The Energy Condition
is reformulated, a key concept of the proof, is reformulated. The main result
is unchanged. v4. important display corrected. v6: The earlier versions
incorrectly claimed a characterization, as was pointed out to us by S. Treil
v7. Corrections in Section
Fermion condensation and super pivotal categories
We study fermionic topological phases using the technique of fermion
condensation. We give a prescription for performing fermion condensation in
bosonic topological phases which contain a fermion. Our approach to fermion
condensation can roughly be understood as coupling the parent bosonic
topological phase to a phase of physical fermions, and condensing pairs of
physical and emergent fermions. There are two distinct types of objects in
fermionic theories, which we call "m-type" and "q-type" particles. The
endomorphism algebras of q-type particles are complex Clifford algebras, and
they have no analogues in bosonic theories. We construct a fermionic
generalization of the tube category, which allows us to compute the
quasiparticle excitations in fermionic topological phases. We then prove a
series of results relating data in condensed theories to data in their parent
theories; for example, if is a modular tensor category containing
a fermion, then the tube category of the condensed theory satisfies
.
We also study how modular transformations, fusion rules, and coherence
relations are modified in the fermionic setting, prove a fermionic version of
the Verlinde dimension formula, construct a commuting projector lattice
Hamiltonian for fermionic theories, and write down a fermionic version of the
Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted
to three detailed examples of performing fermion condensation to produce
fermionic topological phases: we condense fermions in the Ising theory, the
theory, and the theory, and compute the
quasiparticle excitation spectrum in each of these examples.Comment: 161 pages; v2: corrected typos (including 18 instances of "the the")
and added some reference
The blob complex
Given an n-manifold M and an n-category C, we define a chain complex (the
"blob complex") B_*(M;C). The blob complex can be thought of as a derived
category analogue of the Hilbert space of a TQFT, and as a generalization of
Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice
formal properties, including a higher dimensional generalization of Deligne's
conjecture about the action of the little disks operad on Hochschild cochains.
Along the way, we give a definition of a weak n-category with strong duality
which is particularly well suited for work with TQFTs.Comment: 106 pages. Version 3 contains many improvements following suggestions
from the referee and others, and some additional materia
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