275 research outputs found
An analytic Approach to Turaev's Shadow Invariant
In the present paper we extend the "torus gauge fixing approach" by Blau and
Thompson (Nucl. Phys. B408(1):345--390, 1993) for Chern-Simons models with base
manifolds M of the form M= \Sigma x S^1 in a suitable way. We arrive at a
heuristic path integral formula for the Wilson loop observables associated to
general links in M. We then show that the right-hand side of this formula can
be evaluated explicitly in a non-perturbative way and that this evaluation
naturally leads to the face models in terms of which Turaev's shadow invariant
is defined.Comment: 44 pages, 2 figures. Changes have been made in Sec. 2.3, Sec 2.4,
Sec. 3.4, and Sec. 3.5. Appendix C is ne
Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation
The Turaev-Viro invariants are scalar topological invariants of compact,
orientable 3-manifolds. We give a quantum algorithm for additively
approximating Turaev-Viro invariants of a manifold presented by a Heegaard
splitting. The algorithm is motivated by the relationship between topological
quantum computers and (2+1)-D topological quantum field theories. Its accuracy
is shown to be nontrivial, as the same algorithm, after efficient classical
preprocessing, can solve any problem efficiently decidable by a quantum
computer. Thus approximating certain Turaev-Viro invariants of manifolds
presented by Heegaard splittings is a universal problem for quantum
computation. This establishes a novel relation between the task of
distinguishing non-homeomorphic 3-manifolds and the power of a general quantum
computer.Comment: 4 pages, 3 figure
Topological low-temperature limit of Z(2) spin-gauge theory in three dimensions
We study Z(2) lattice gauge theory on triangulations of a compact 3-manifold.
We reformulate the theory algebraically, describing it in terms of the
structure constants of a bidimensional vector space H equipped with algebra and
coalgebra structures, and prove that in the low-temperature limit H reduces to
a Hopf Algebra, in which case the theory becomes equivalent to a topological
field theory. The degeneracy of the ground state is shown to be a topological
invariant. This fact is used to compute the zeroth- and first-order terms in
the low-temperature expansion of Z for arbitrary triangulations. In finite
temperatures, the algebraic reformulation gives rise to new duality relations
among classical spin models, related to changes of basis of H.Comment: 10 pages, no figure
Towards the Theory of Non--Abelian Tensor Fields I
We present a triangulation--independent area--ordering prescription which
naturally generalizes the well known path ordering one. For such a prescription
it is natural that the two--form ``connection'' should carry three ``color''
indices rather than two as it is in the case of the ordinary one--form gauge
connection. To define the prescription in question we have to define how to
{\it exponentiate} a matrix with three indices. The definition uses the fusion
rule structure constants.Comment: 22 pages, 18 figure
Ground State Degeneracy in the Levin-Wen Model for Topological Phases
We study properties of topological phases by calculating the ground state
degeneracy (GSD) of the 2d Levin-Wen (LW) model. Here it is explicitly shown
that the GSD depends only on the spatial topology of the system. Then we show
that the ground state on a sphere is always non-degenerate. Moreover, we study
an example associated with a quantum group, and show that the GSD on a torus
agrees with that of the doubled Chern-Simons theory, consistent with the
conjectured equivalence between the LW model associated with a quantum group
and the doubled Chern-Simons theory.Comment: 8 pages, 2 figures. v2: reference added; v3: two appendices adde
From simplicial Chern-Simons theory to the shadow invariant II
This is the second of a series of papers in which we introduce and study a
rigorous "simplicial" realization of the non-Abelian Chern-Simons path integral
for manifolds M of the form M = Sigma x S1 and arbitrary simply-connected
compact structure groups G. More precisely, we introduce, for general links L
in M, a rigorous simplicial version WLO_{rig}(L) of the corresponding Wilson
loop observable WLO(L) in the so-called "torus gauge" by Blau and Thompson
(Nucl. Phys. B408(2):345-390, 1993). For a simple class of links L we then
evaluate WLO_{rig}(L) explicitly in a non-perturbative way, finding agreement
with Turaev's shadow invariant |L|.Comment: 53 pages, 1 figure. Some minor changes and corrections have been mad
A reason for fusion rules to be even
We show that certain tensor product multiplicities in semisimple braided
sovereign tensor categories must be even. The quantity governing this behavior
is the Frobenius-Schur indicator. The result applies in particular to the
representation categories of large classes of groups, Lie algebras, Hopf
algebras and vertex algebras.Comment: 6 pages, LaTe
Improved and Perfect Actions in Discrete Gravity
We consider the notion of improved and perfect actions within Regge calculus.
These actions are constructed in such a way that they - although being defined
on a triangulation - reproduce the continuum dynamics exactly, and therefore
capture the gauge symmetries of General Relativity. We construct the perfect
action in three dimensions with cosmological constant, and in four dimensions
for one simplex. We conclude with a discussion about Regge Calculus with curved
simplices, which arises naturally in this context.Comment: 28 pages, 2 figure
String-Net Models with Fusion Algebra
We study the Levin-Wen string-net model with a type fusion algebra.
Solutions of the local constraints of this model correspond to gauge
theory and double Chern-simons theories with quantum groups. For the first
time, we explicitly construct a spin- model with gauge symmetry
on a triangular lattice as an exact dual model of the string-net model with a
type fusion algebra on a honeycomb lattice. This exact duality exists
only when the spins are coupled to a gauge field living on the links of
the triangular lattice. The ungauged lattice spin models are a class of
quantum systems that bear symmetry-protected topological phases that may be
classified by the third cohomology group of . Our results
apply also to any case where the fusion algebra is identified with a finite
group algebra or a quantusm group algebra.Comment: 16 pages, 2 figures, publishe
A matrix solution to pentagon equation with anticommuting variables
We construct a solution to pentagon equation with anticommuting variables
living on two-dimensional faces of tetrahedra. In this solution, matrix
coordinates are ascribed to tetrahedron vertices. As matrix multiplication is
noncommutative, this provides a "more quantum" topological field theory than in
our previous works
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