Free compact boson on branched coverings of the torus

We have studied the free compact boson on a $n$-sheeted covering of the torus gluing alone $m$ branch cuts. It is interesting because when the branched cuts are chosen to be real, the partition function is related to the $n$-th R\'enyi entanglement entropy of $m$ disjoint intervals in a finite system at finite temperature. After proposing a canonical homology basis and its dual basis of the covering surface, we find that the partition function can be written in terms of theta functions

No-go for tree-level R-symmetry breaking

We show that in gauge mediation models with tree-level R-symmetry breaking where supersymmetry and R-symmetries are broken by different fields, the gaugino mass either vanishes at one loop or finds a contribution from loop-level R-symmetry breaking. Thus tree-level R-symmetry breaking for phenomenology is either no-go or redundant in the simplest type of models. Including explicit messenger mass terms in the superpotential with a particular R-charge arrangement is helpful to bypass the no-go theorem, and the resulting gaugino mass is suppressed by the messenger mass scale.Comment: 8 pages, 7 figures; v2: discussion on Driac gauginos and references added; v3: a section on bypassing the no-go added, R-charge notation changed; v4: typos, EPJC pre-published versio

Two intervals Rényi entanglement entropy of compact free boson on torus

We compute the $N=2$ R\'enyi entanglement entropy of two intervals at equal time in a circle, for the theory of a 2d compact complex free scalar at finite temperature. This is carried out by performing functional integral on a genus 3 ramified cover of the torus, wherein the quantum part of the integral is captured by the four point function of twist fields on the worldsheet torus, and the classical piece is given by summing over winding modes of the genus 3 surface onto the target space torus. The final result is given in terms of a product of theta function and certain multi-dimensional theta function. We demonstrate the T-duality invariance of the result. We also study its low temperature limit. In the case in which the size of the intervals and of their separation are much smaller than the whole system, our result is in exact agreement with the known result for two intervals on an infinite system at zero temperature \cite{eeoftwo}. In the case in which the separation between the two intervals is much smaller than the interval length, the leading thermal corrections take the same universal form as proposed in \cite{Cardy:2014jwa,Chen:2015cna} for R\'enyi entanglement entropy of a single interval.Comment: 29 pages, 7 figure

Crossing Lilium Orientals of different ploidy creates Fusarium-resistant hybrid

Oriental hybrid lily is of great commercial value, but it is susceptible to Fusarium disease that causes a significant loss to the production. A diploid Oriental hybrid resistant to Fusarium, Cai-74, was diploidized from triploid obtained from the offspring of tetraploid (from ‘Star Fighter’) and diploid (‘Con Amore’, ‘Acapulco’) by screening the hybrids of different cross combinations following inoculating Fusarium oxysporum to the tissue cultured plantlets in a greenhouse. By analyzing saponins content in bulbs of a number of lily genotypes with a known Fusarium resistance, it was found that the mutant Cai-74 had a much higher content of saponin than its parents. Highly resistant wild _L. dauricum_ had the highest level (4.59mg/g), followed by the resistant Cai-74 with 4.01mg/g. The resistant OT cultivars ‘Conca d’or’ and ‘Robina’ had a higher saponins content (3.70 mg/g) and 2.83 mg/g, than the susceptible Oriental lily cultivars ‘Sorbonne’, ‘Siberia’ and ‘Tiber’. The hybrid Cai-74 had a different karyotype compared with the normal Lilium Oriental hybrid cultivars. The results suggested that Cai-74 carries a chromosomal variation correlated to Fusarium resistance. Cai-74 might be used as a genetic resource for breeding of Fusarium resistant cultivars of Oriental hybrid lilies

On Frobenius Formulas of Power Sequences

Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $g(A)$ is the greatest integer that can not be written as a nonnegative integer linear combination of the $a_i$'s. Recently we developed a combinatorial approach to $g(A)$ for $A=(a,a+B)=(a,a+b_1,\dots, a+b_k)$. The problem is reduced to solving an easier optimization problem $O_B(M)$. In this paper, we use this approach to solve the open problem of characterizing $g(A)$ for the square sequence $A=(a,a+1,a+2^2,..., a+k^2)$. Our point is that the Frobenius number $g(a,a+1^2,a+2^2,\dots)$ of infinite square sequence is easier to solve, since the corresponding $O_B(M)$ has a solution in Number Theory in terms of Lagrange's Four-Square Theorem and related results. This leads to a solution to the open problem by using Lagrange's Four-Square Theorem and generating functions. Moreover, the technique can be used for power sequence, i.e., the $b_i=i^s$ case where $s$ is a fixed positive integer.Comment: 24 pages 2 tables (Reorganize the contents

A Generalization of Mersenne and Thabit Numerical Semigroups

Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $F(A)$ is the largest integer not belonging to the numerical semigroup $\langle A\rangle$ generated by $A$. The genus $g(A)$ is the number of positive integer elements that are not in $\langle A\rangle$. The Frobenius problem is to find $F(A)$ and $g(A)$ for a given sequence $A$. In this paper, we study the Frobenius problem of $A=(a,2a+d,2^2a+3d,...,2^ka+(2^k-1)d)$ and obtain formulas for $F(A)$ and $g(A)$ when $a+d\geq k$. Our formulas simplifies further for some special cases, such as Mersenne and Thabit numerical semigroups. We obtain explicit formulas for generalized Mersenne and Thabit numerical semigroups and some more general numerical semigroups

The Frobenius Formula for A=(a,ha+d,ha+b2d,...,ha+bkd)A=(a,ha+d,ha+b_2d,...,ha+b_kd)

Given relative prime positive integers $A=(a_1, a_2, ..., a_n)$, the Frobenius number $g(A)$ is the largest integer not representable as a linear combination of the $a_i$'s with nonnegative integer coefficients. We find the ``Stable" property introduced for the square sequence $A=(a,a+1,a+2^2,\dots, a+k^2)$ naturally extends for $A(a)=(a,ha+d,ha+b_2d,...,ha+b_kd)$. This gives a parallel characterization of $g(A(a))$ as a ``congruence class function" modulo $b_k$ when $a$ is large enough. For orderly sequence $B=(1,b_2,\dots,b_k)$, we find good bound for $a$. In particular we calculate $g(a,ha+dB)$ for $B=(1,2,b,b+1,2b)$, $B=(1,b,2b-1)$ and $B=(1,2,...,k,K)$