64,002 research outputs found
Refined descendant invariants of toric surfaces
We construct refined tropical enumerative genus zero invariants of toric
surfaces that specialize to the tropical descendant genus zero invariants
introduced by Markwig and Rau when the quantum parameter tends to . In the
case of trivalent tropical curves our invariants turn to be the
Goettsche-Schroeter refined broccoli invariants. We show that this is the only
possible refinement of the Markwig-Rau descendant invariants that generalizes
the Goettsche-Schroeter refined broccoli invariants. We discuss also the
computational aspect (a lattice path algorithm) and exhibit some examples.Comment: 30 pages, 7 figures; matches the published versio
Convexities related to path properties on graphs; a unified approach
Path properties, such as 'geodesic', 'induced', 'all paths' define a convexity on a connected graph. The general notion of path property, introduced in this paper, gives rise to a comprehensive survey of results obtained by different authors for a variety of path properties, together with a number of new results. We pay special attention to convexities defined by path properties on graph products and the classical convexity invariants, such as the Caratheodory, Helly and Radon numbers in relation with graph invariants, such as clique numbers and other graph properties.
Geometric local invariants and pure three-qubit states
We explore a geometric approach to generating local SU(2) and
invariants for a collection of qubits inspired by lattice
gauge theory. Each local invariant or 'gauge' invariant is associated to a
distinct closed path (or plaquette) joining some or all of the qubits. In
lattice gauge theory, the lattice points are the discrete space-time points,
the transformations between the points of the lattice are defined by parallel
transporters and the gauge invariant observable associated to a particular
closed path is given by the Wilson loop. In our approach the points of the
lattice are qubits, the link-transformations between the qubits are defined by
the correlations between them and the gauge invariant observable, the local
invariants associated to a particular closed path are also given by a Wilson
loop-like construction. The link transformations share many of the properties
of parallel transporters although they are not undone when one retraces one's
steps through the lattice. This feature is used to generate many of the
invariants. We consider a pure three qubit state as a test case and find we can
generate a complete set of algebraically independent local invariants in this
way, however the framework given here is applicable to mixed states composed of
any number of level quantum systems. We give an operational interpretation
of these invariants in terms of observables.Comment: 9 pages, 3 figure
On the Generalized Casson Invariant
The path integral generalization of the Casson invariant as developed by
Rozansky and Witten is investigated. The path integral for various three
manifolds is explicitly evaluated. A new class of topological observables is
introduced that may allow for more effective invariants. Finally it is shown
how the dimensional reduction of these theories corresponds to a generalization
of the topological B sigma model
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