221 research outputs found
Combinatorics, geometry and homology of non-crossing partition lattices for finite reflection groups
Non-crossing Partitions and Milnor Fibers
For a finite real reflection group, W, we use non-crossing partitions of type W to construct a finite cell complex with the homotopy type of the Milnor fiber of the associated W–discriminant, Δ_W, and another with the homotopy type of the Milnor fiber of the defining polynomial of the associated reflection arrangement. These complexes support natural cyclic group actions realizing the geometric monodromy. Using the shellability of the non-crossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of Δ_W
Three-dimensional color code thresholds via statistical-mechanical mapping
Three-dimensional (3D) color codes have advantages for fault-tolerant quantum
computing, such as protected quantum gates with relatively low overhead and
robustness against imperfect measurement of error syndromes. Here we
investigate the storage threshold error rates for bit-flip and phase-flip noise
in the 3D color code on the body-centererd cubic lattice, assuming perfect
syndrome measurements. In particular, by exploiting a connection between error
correction and statistical mechanics, we estimate the threshold for 1D
string-like and 2D sheet-like logical operators to be and . We obtain these
results by using parallel tempering Monte Carlo simulations to study the
disorder-temperature phase diagrams of two new 3D statistical-mechanical
models: the 4- and 6-body random coupling Ising models.Comment: 4+7 pages, 6 figures, 1 tabl
Weak Coupling, Degeneration and Log Calabi-Yau Spaces
We establish a new weak coupling limit in F-theory. The new limit may be
thought of as the process in which a local model bubbles off from the rest of
the Calabi-Yau. The construction comes with a small deformation parameter
such that computations in the local model become exact as . More
generally, we advocate a modular approach where compact Calabi-Yau geometries
are obtained by gluing together local pieces (log Calabi-Yau spaces) into a
normal crossing variety and smoothing, in analogy with a similar cutting and
gluing approach to topological field theories. We further argue for a
holographic relation between F-theory on a degenerate Calabi-Yau and a dual
theory on its boundary, which fits nicely with the gluing construction.Comment: 59 pp, 2 figs, LaTe
On the Defect Group of a 6D SCFT
We use the F-theory realization of 6D superconformal field theories (SCFTs)
to study the corresponding spectrum of stringlike, i.e. surface defects. On the
tensor branch, all of the stringlike excitations pick up a finite tension, and
there is a corresponding lattice of string charges, as well as a dual lattice
of charges for the surface defects. The defect group is data intrinsic to the
SCFT and measures the surface defect charges which are not screened by
dynamical strings. When non-trivial, it indicates that the associated theory
has a partition vector rather than a partition function. We compute the defect
group for all known 6D SCFTs, and find that it is just the abelianization of
the discrete subgroup of U(2) which appears in the classification of 6D SCFTs
realized in F-theory. We also explain how the defect group specifies defining
data in the compactification of a (1,0) SCFT.Comment: 24 page
Topological Color Codes and Two-Body Quantum Lattice Hamiltonians
Topological color codes are among the stabilizer codes with remarkable
properties from quantum information perspective. In this paper we construct a
four-valent lattice, the so called ruby lattice, governed by a 2-body
Hamiltonian. In a particular regime of coupling constants, degenerate
perturbation theory implies that the low energy spectrum of the model can be
described by a many-body effective Hamiltonian, which encodes the color code as
its ground state subspace. The gauge symmetry
of color code could already be realized by
identifying three distinct plaquette operators on the lattice. Plaquettes are
extended to closed strings or string-net structures. Non-contractible closed
strings winding the space commute with Hamiltonian but not always with each
other giving rise to exact topological degeneracy of the model. Connection to
2-colexes can be established at the non-perturbative level. The particular
structure of the 2-body Hamiltonian provides a fruitful interpretation in terms
of mapping to bosons coupled to effective spins. We show that high energy
excitations of the model have fermionic statistics. They form three families of
high energy excitations each of one color. Furthermore, we show that they
belong to a particular family of topological charges. Also, we use
Jordan-Wigner transformation in order to test the integrability of the model
via introducing of Majorana fermions. The four-valent structure of the lattice
prevents to reduce the fermionized Hamiltonian into a quadratic form due to
interacting gauge fields. We also propose another construction for 2-body
Hamiltonian based on the connection between color codes and cluster states. We
discuss this latter approach along the construction based on the ruby lattice.Comment: 56 pages, 16 figures, published version
Vortex Counting and Lagrangian 3-manifolds
To every 3-manifold M one can associate a two-dimensional N=(2,2)
supersymmetric field theory by compactifying five-dimensional N=2
super-Yang-Mills theory on M. This system naturally appears in the study of
half-BPS surface operators in four-dimensional N=2 gauge theories on one hand,
and in the geometric approach to knot homologies, on the other. We study the
relation between vortex counting in such two-dimensional N=(2,2) supersymmetric
field theories and the refined BPS invariants of the dual geometries. In
certain cases, this counting can be also mapped to the computation of
degenerate conformal blocks in two-dimensional CFT's. Degenerate limits of
vertex operators in CFT receive a simple interpretation via geometric
transitions in BPS counting.Comment: 70 pages, 29 figure
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