12,032 research outputs found
Loop Quantum Gravity a la Aharonov-Bohm
The state space of Loop Quantum Gravity admits a decomposition into
orthogonal subspaces associated to diffeomorphism equivalence classes of
spin-network graphs. In this paper I investigate the possibility of obtaining
this state space from the quantization of a topological field theory with many
degrees of freedom. The starting point is a 3-manifold with a network of
defect-lines. A locally-flat connection on this manifold can have non-trivial
holonomy around non-contractible loops. This is in fact the mathematical origin
of the Aharonov-Bohm effect. I quantize this theory using standard field
theoretical methods. The functional integral defining the scalar product is
shown to reduce to a finite dimensional integral over moduli space. A
non-trivial measure given by the Faddeev-Popov determinant is derived. I argue
that the scalar product obtained coincides with the one used in Loop Quantum
Gravity. I provide an explicit derivation in the case of a single defect-line,
corresponding to a single loop in Loop Quantum Gravity. Moreover, I discuss the
relation with spin-networks as used in the context of spin foam models.Comment: 19 pages, 1 figure; v2: corrected typos, section 4 expanded
Complex Networks and Symmetry I: A Review
In this review we establish various connections between complex networks and
symmetry. While special types of symmetries (e.g., automorphisms) are studied
in detail within discrete mathematics for particular classes of deterministic
graphs, the analysis of more general symmetries in real complex networks is far
less developed. We argue that real networks, as any entity characterized by
imperfections or errors, necessarily require a stochastic notion of invariance.
We therefore propose a definition of stochastic symmetry based on graph
ensembles and use it to review the main results of network theory from an
unusual perspective. The results discussed here and in a companion paper show
that stochastic symmetry highlights the most informative topological properties
of real networks, even in noisy situations unaccessible to exact techniques.Comment: Final accepted versio
Extended matter coupled to BF theory
Recently, a topological field theory of membrane-matter coupled to BF theory
in arbitrary spacetime dimensions was proposed [1]. In this paper, we discuss
various aspects of the four-dimensional theory. Firstly, we study classical
solutions leading to an interpretation of the theory in terms of strings
propagating on a flat spacetime. We also show that the general classical
solutions of the theory are in one-to-one correspondence with solutions of
Einstein's equations in the presence of distributional matter (cosmic strings).
Secondly, we quantize the theory and present, in particular, a prescription to
regularize the physical inner product of the canonical theory. We show how the
resulting transition amplitudes are dual to evaluations of Feynman diagrams
coupled to three-dimensional quantum gravity. Finally, we remove the regulator
by proving the topological invariance of the transition amplitudes.Comment: 27 pages, 7 figure
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
Topological Defects on the Lattice I: The Ising model
In this paper and its sequel, we construct topologically invariant defects in
two-dimensional classical lattice models and quantum spin chains. We show how
defect lines commute with the transfer matrix/Hamiltonian when they obey the
defect commutation relations, cousins of the Yang-Baxter equation. These
relations and their solutions can be extended to allow defect lines to branch
and fuse, again with properties depending only on topology. In this part I, we
focus on the simplest example, the Ising model. We define lattice spin-flip and
duality defects and their branching, and prove they are topological. One useful
consequence is a simple implementation of Kramers-Wannier duality on the torus
and higher genus surfaces by using the fusion of duality defects. We use these
topological defects to do simple calculations that yield exact properties of
the conformal field theory describing the continuum limit. For example, the
shift in momentum quantization with duality-twisted boundary conditions yields
the conformal spin 1/16 of the chiral spin field. Even more strikingly, we
derive the modular transformation matrices explicitly and exactly.Comment: 45 pages, 9 figure
Topological Defect Lines and Renormalization Group Flows in Two Dimensions
We consider topological defect lines (TDLs) in two-dimensional conformal
field theories. Generalizing and encompassing both global symmetries and
Verlinde lines, TDLs together with their attached defect operators provide
models of fusion categories without braiding. We study the crossing relations
of TDLs, discuss their relation to the 't Hooft anomaly, and use them to
constrain renormalization group flows to either conformal critical points or
topological quantum field theories (TQFTs). We show that if certain
non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a
non-degenerate gapped state. For various massive flows, we determine the
infrared TQFTs completely from the consideration of TDLs together with modular
invariance.Comment: 101 pages, 63 figures, 2 tables; v3: minor changes, added footnotes
and references, published versio
The Hilbert space of Chern-Simons theory on the cylinder. A Loop Quantum Gravity approach
As a laboratory for loop quantum gravity, we consider the canonical
quantization of the three-dimensional Chern-Simons theory on a noncompact space
with the topology of a cylinder. Working within the loop quantization
formalism, we define at the quantum level the constraints appearing in the
canonical approach and completely solve them, thus constructing a gauge and
diffeomorphism invariant physical Hilbert space for the theory. This space
turns out to be infinite dimensional, but separable.Comment: Minor changes and some references added. Latex, 16 pages, 1 figur
Topological Lattice Models in Four Dimensions
We define a lattice statistical model on a triangulated manifold in four
dimensions associated to a group . When , the statistical weight is
constructed from the -symbol as well as the -symbol for recombination
of angular momenta, and the model may be regarded as the four-dimensional
version of the Ponzano-Regge model. We show that the partition function of the
model is invariant under the Alexander moves of the simplicial complex, thus it
depends only on the piecewise linear topology of the manifold. For an
orientable manifold, the model is related to the so-called model. The
-analogue of the model is also constructed, and it is argued that its
partition function is invariant under the Alexander moves. It is discussed how
to realize the 't Hooft operator in these models associated to a closed surface
in four dimensions as well as the Wilson operator associated to a closed loop.
Correlation functions of these operators in the -deformed version of the
model would define a new type of invariants of knots and links in four
dimensions.Comment: 14 page
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