1,432 research outputs found
Identifying all abelian periods of a string in quadratic time and relevant problems
Abelian periodicity of strings has been studied extensively over the last
years. In 2006 Constantinescu and Ilie defined the abelian period of a string
and several algorithms for the computation of all abelian periods of a string
were given. In contrast to the classical period of a word, its abelian version
is more flexible, factors of the word are considered the same under any
internal permutation of their letters. We show two O(|y|^2) algorithms for the
computation of all abelian periods of a string y. The first one maps each
letter to a suitable number such that each factor of the string can be
identified by the unique sum of the numbers corresponding to its letters and
hence abelian periods can be identified easily. The other one maps each letter
to a prime number such that each factor of the string can be identified by the
unique product of the numbers corresponding to its letters and so abelian
periods can be identified easily. We also define weak abelian periods on
strings and give an O(|y|log(|y|)) algorithm for their computation, together
with some other algorithms for more basic problems.Comment: Accepted in the "International Journal of foundations of Computer
Science
Quantization of anomaly coefficients in 6D supergravity
We obtain new constraints on the anomaly coefficients of 6D
supergravity theories using local and global anomaly
cancellation conditions. We show how these constraints can be strengthened if
we assume that the theory is well-defined on any spin space-time with an
arbitrary gauge bundle. We distinguish the constraints depending on the gauge
algebra only from those depending on the global structure of the gauge group.
Our main constraint states that the coefficients of the anomaly polynomial for
the gauge group should be an element of where is the unimodular string charge lattice. We show
that the constraints in their strongest form are realized in F-theory
compactifications. In the process, we identify the cocharacter lattice, which
determines the global structure of the gauge group, within the homology lattice
of the compactification manifold.Comment: 42 pages. v3: Some clarifications, typos correcte
A Note on Easy and Efficient Computation of Full Abelian Periods of a Word
Constantinescu and Ilie (Bulletin of the EATCS 89, 167-170, 2006) introduced
the idea of an Abelian period with head and tail of a finite word. An Abelian
period is called full if both the head and the tail are empty. We present a
simple and easy-to-implement -time algorithm for computing all
the full Abelian periods of a word of length over a constant-size alphabet.
Experiments show that our algorithm significantly outperforms the
algorithm proposed by Kociumaka et al. (Proc. of STACS, 245-256, 2013) for the
same problem.Comment: Accepted for publication in Discrete Applied Mathematic
Loop and surface operators in N=2 gauge theory and Liouville modular geometry
Recently, a duality between Liouville theory and four dimensional N=2 gauge
theory has been uncovered by some of the authors. We consider the role of
extended objects in gauge theory, surface operators and line operators, under
this correspondence. We map such objects to specific operators in Liouville
theory. We employ this connection to compute the expectation value of general
supersymmetric 't Hooft-Wilson line operators in a variety of N=2 gauge
theories.Comment: 60 pages, 11 figures; v3: further minor corrections, published
versio
Minimal-area metrics on the Swiss cross and punctured torus
The closed string field theory minimal-area problem asks for the conformal
metric of least area on a Riemann surface with the condition that all
non-contractible closed curves have length at least 2\pi. Through every point
in such a metric there is a geodesic that saturates the length condition, and
saturating geodesics in a given homotopy class form a band. The extremal metric
is unknown when bands of geodesics cross, as it happens for surfaces of
non-zero genus. We use recently proposed convex programs to numerically find
the minimal-area metric on the square torus with a square boundary, for various
sizes of the boundary. For large enough boundary the problem is equivalent to
the "Swiss cross" challenge posed by Strebel. We find that the metric is
positively curved in the two-band region and flat in the single-band regions.
For small boundary the metric develops a third band of geodesics wrapping
around it, and has both regions of positive and negative curvature. This
surface can be completed to provide the minimal-area metric on a once-punctured
torus, representing a closed-string tadpole diagram.Comment: 59 pages, 41 figures. v2: Minor edits and reference update
Second and Third Order Observables of the Two-Matrix Model
In this paper we complement our recent result on the explicit formula for the
planar limit of the free energy of the two-matrix model by computing the second
and third order observables of the model in terms of canonical structures of
the underlying genus g spectral curve. In particular we provide explicit
formulas for any three-loop correlator of the model. Some explicit examples are
worked out.Comment: 22 pages, v2 with added references and minor correction
Phases of Five-dimensional Theories, Monopole Walls, and Melting Crystals
Moduli spaces of doubly periodic monopoles, also called monopole walls or
monowalls, are hyperk\"ahler; thus, when four-dimensional, they are self-dual
gravitational instantons. We find all monowalls with lowest number of moduli.
Their moduli spaces can be identified, on the one hand, with Coulomb branches
of five-dimensional supersymmetric quantum field theories on
and, on the other hand, with moduli spaces of local
Calabi-Yau metrics on the canonical bundle of a del Pezzo surface. We explore
the asymptotic metric of these moduli spaces and compare our results with
Seiberg's low energy description of the five-dimensional quantum theories. We
also give a natural description of the phase structure of general monowall
moduli spaces in terms of triangulations of Newton polygons, secondary
polyhedra, and associahedral projections of secondary fans.Comment: 45 pages, 11 figure
- …