57,833 research outputs found
Permutation orbifolds of heterotic Gepner models
We study orbifolds by permutations of two identical N=2 minimal models within
the Gepner construction of four dimensional heterotic strings. This is done
using the new N=2 supersymmetric permutation orbifold building blocks we have
recently developed. We compare our results with the old method of modding out
the full string partition function. The overlap between these two approaches is
surprisingly small, but whenever a comparison can be made we find complete
agreement. The use of permutation building blocks allows us to use the complete
arsenal of simple current techniques that is available for standard Gepner
models, vastly extending what could previously be done for permutation
orbifolds. In particular, we consider (0,2) models, breaking of SO(10) to
subgroups, weight-lifting for the minimal models and B-L lifting. Some
previously observed phenomena, for example concerning family number
quantization, extend to this new class as well, and in the lifted models three
family models occur with abundance comparable to two or four.Comment: 49 pages, 4 figure
Bypasses for rectangular diagrams. Proof of Jones' conjecture and related questions
In the present paper a criteria for a rectangular diagram to admit a
simplification is given in terms of Legendrian knots. It is shown that there
are two types of simplifications which are mutually independent in a sense. A
new proof of the monotonic simplification theorem for the unknot is given. It
is shown that a minimal rectangular diagram maximizes the Thurston--Bennequin
number for the corresponding Legendrian links. Jones' conjecture about the
invariance of the algebraic number of intersections of a minimal braid
representing a fixed link type is proved.Comment: 50 pages, 62 Figures, numerous minor correction
The Brownian limit of separable permutations
We study random uniform permutations in an important class of
pattern-avoiding permutations: the separable permutations. We describe the
asymptotics of the number of occurrences of any fixed given pattern in such a
random permutation in terms of the Brownian excursion. In the recent
terminology of permutons, our work can be interpreted as the convergence of
uniform random separable permutations towards a "Brownian separable permuton".Comment: 45 pages, 14 figures, incorporating referee's suggestion
Grid classes and partial well order
We prove necessary and sufficient conditions on a family of (generalised)
gridding matrices to determine when the corresponding permutation classes are
partially well-ordered. One direction requires an application of Higman's
Theorem and relies on there being only finitely many simple permutations in the
only non-monotone cell of each component of the matrix. The other direction is
proved by a more general result that allows the construction of infinite
antichains in any grid class of a matrix whose graph has a component containing
two or more non-monotone-griddable cells. The construction uses a
generalisation of pin sequences to grid classes, together with a number of
symmetry operations on the rows and columns of a gridding.Comment: 22 pages, 7 figures. To appear in J. Comb. Theory Series
Properties of scattering forms and their relation to associahedra
We show that the half-integrands in the CHY representation of tree amplitudes
give rise to the definition of differential forms -- the scattering forms -- on
the moduli space of a Riemann sphere with marked points. These differential
forms have some remarkable properties. We show that all singularities are on
the divisor . Each
singularity is logarithmic and the residue factorises into two differential
forms of lower points. In order for this to work, we provide a threefold
generalisation of the CHY polarisation factor (also known as reduced Pfaffian)
towards off-shell momenta, unphysical polarisations and away from the solutions
of the scattering equations. We discuss explicitly the cases of bi-adjoint
scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes.Comment: 40 pages, version to be publishe
Attempted Bethe ansatz solution for one-dimensional directed polymers in random media
We study the statistical properties of one-dimensional directed polymers in a
short-range random potential by mapping the replicated problem to a many body
quantum boson system with attractive interactions. We find the full set of
eigenvalues and eigenfunctions of the many-body system and perform the
summation over the entire spectrum of excited states. The analytic continuation
of the obtained exact expression for the replica partition function from
integer to non-integer replica parameter N turns out to be ambiguous.
Performing the analytic continuation simply by assuming that the parameter N
can take arbitrary complex values, and going to the thermodynamic limit of the
original directed polymer problem, we obtain the explicit universal expression
for the probability distribution function of free energy fluctuations.Comment: 32 pages, 1 figur
- …