3,563 research outputs found
Product Integral Representations of Wilson Lines and Wilson Loops, and Non-Abelian Stokes Theorem
We make use of product integrals to provide an unambiguous mathematical
representation of Wilson line and Wilson loop operators. Then, drawing upon
various properties of product integrals, we discuss such properties of these
operators as approximating them with partial sums, their convergence, and their
behavior under gauge transformations. We also obtain a surface product integral
representation for the Wilson loop operator. The result can be interpreted as
the non-abelian version of Stokes theorem.Comment: 20 pages, LaTe
Type IIB orientifolds with discrete torsion
We consider compact four-dimensional type IIB
orientifolds, for certain values of and . We allow the additional
feature of discrete torsion and discuss the modification of the consistency
conditions arising from tadpole cancellation. We point out the differences
between the cases with and without discrete torsion.Comment: 4 pages LaTeX. Write-up of talk at DPF2000, Columbus, OH, August 10,
2000. References adde
Quantum symmetries and exceptional collections
We study the interplay between discrete quantum symmetries at certain points
in the moduli space of Calabi-Yau compactifications, and the associated
identities that the geometric realization of D-brane monodromies must satisfy.
We show that in a wide class of examples, both local and compact, the monodromy
identities in question always follow from a single mathematical statement. One
of the simplest examples is the Z_5 symmetry at the Gepner point of the
quintic, and the associated D-brane monodromy identity
Generalized Fock spaces and the Stirling numbers
The Bargmann-Fock-Segal space plays an important role in mathematical
physics, and has been extended into a number of directions. In the present
paper we imbed this space into a Gelfand triple. The spaces forming the
Fr\'echet part (i.e. the space of test functions) of the triple are
characterized both in a geometric way and in terms of the adjoint of
multiplication by the complex variable, using the Stirling numbers of the
second kind. The dual of the space of test functions has a topological algebra
structure, of the kind introduced and studied by the first named author and G.
Salomon.Comment: revised versio
Geometric gauge potentials and forces in low-dimensional scattering systems
We introduce and analyze several low-dimensional scattering systems that
exhibit geometric phase phenomena. The systems are fully solvable and we
compare exact solutions of them with those obtained in a Born-Oppenheimer
projection approximation. We illustrate how geometric magnetism manifests in
them, and explore the relationship between solutions obtained in the diabatic
and adiabatic pictures. We provide an example, involving a neutral atom dressed
by an external field, in which the system mimics the behavior of a charged
particle that interacts with, and is scattered by, a ferromagnetic material. We
also introduce a similar system that exhibits Aharonov-Bohm scattering. We
propose some practical applications. We provide a theoretical approach that
underscores universality in the appearance of geometric gauge forces. We do not
insist on degeneracies in the adiabatic Hamiltonian, and we posit that the
emergence of geometric gauge forces is a consequence of symmetry breaking in
the latter.Comment: (Final version, published in Phy. Rev. A. 86, 042704 (2012
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Subtropical Real Root Finding
We describe a new incomplete but terminating method for real root finding for
large multivariate polynomials. We take an abstract view of the polynomial as
the set of exponent vectors associated with sign information on the
coefficients. Then we employ linear programming to heuristically find roots.
There is a specialized variant for roots with exclusively positive coordinates,
which is of considerable interest for applications in chemistry and systems
biology. An implementation of our method combining the computer algebra system
Reduce with the linear programming solver Gurobi has been successfully applied
to input data originating from established mathematical models used in these
areas. We have solved several hundred problems with up to more than 800000
monomials in up to 10 variables with degrees up to 12. Our method has failed
due to its incompleteness in less than 8 percent of the cases
Vertex-Coloring with Star-Defects
Defective coloring is a variant of traditional vertex-coloring, according to
which adjacent vertices are allowed to have the same color, as long as the
monochromatic components induced by the corresponding edges have a certain
structure. Due to its important applications, as for example in the
bipartisation of graphs, this type of coloring has been extensively studied,
mainly with respect to the size, degree, and acyclicity of the monochromatic
components.
In this paper we focus on defective colorings in which the monochromatic
components are acyclic and have small diameter, namely, they form stars. For
outerplanar graphs, we give a linear-time algorithm to decide if such a
defective coloring exists with two colors and, in the positive case, to
construct one. Also, we prove that an outerpath (i.e., an outerplanar graph
whose weak-dual is a path) always admits such a two-coloring. Finally, we
present NP-completeness results for non-planar and planar graphs of bounded
degree for the cases of two and three colors
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