scholarly journals The EFT stringy viewpoint on large distances

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Stefano Lanza ◽  
Fernando Marchesano ◽  
Luca Martucci ◽  
Irene Valenzuela
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
String Tension ◽  
Energy Scale ◽  
Direct Relation ◽  
Field Range ◽  
Large Classes ◽  
String Compactifications ◽  
Distance Limit ◽  
Maximal Field

Abstract We observe a direct relation between the existence of fundamental axionic strings, dubbed EFT strings, and infinite distance limits in 4d $$ \mathcal{N} $$ N = 1 EFTs coupled to gravity. The backreaction of EFT strings can be interpreted as RG flow of their couplings, and allows one to probe different regimes within the field space of the theory. We propose that any 4d EFT infinite distance limit can be realised as an EFT string flow. We show that along such limits the EFT string becomes asymptotically tensionless, and so the EFT eventually breaks down. This provides an upper bound for the maximal field range of an EFT with a finite cut-off, and reproduces the Swampland Distance Conjecture from a bottom-up perspective. Even if there are typically other towers of particles becoming light, we propose that the mass of the leading tower scales as m2 ∼ $$ \mathcal{T} $$ T w in Planck units, with $$ \mathcal{T} $$ T the EFT string tension and w a positive integer. Our results hold even in the presence of a non-trivial potential, as long as its energy scale remains well below the cut-off. We check both proposals for large classes of 4d $$ \mathcal{N} $$ N = 1 string compactifications, finding that only the values w = 1, 2, 3 are realised.

scholarly journals Chern-Simons invariants and heterotic superpotentials

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Lara B. Anderson ◽  
James Gray ◽  
Andre Lukas ◽  
Juntao Wang
Keyword(s):  
Heterotic String ◽  
New Techniques ◽  
Vacuum Stability ◽  
Large Classes ◽  
Moduli Stabilization ◽  
String Compactifications ◽  
Key Features ◽  
Effective Theories ◽  
Chern Simons ◽  
Tangent Bundles

Abstract The superpotential in four-dimensional heterotic effective theories contains terms arising from holomorphic Chern-Simons invariants associated to the gauge and tangent bundles of the compactification geometry. These effects are crucial for a number of key features of the theory, including vacuum stability and moduli stabilization. Despite their importance, few tools exist in the literature to compute such effects in a given heterotic vacuum. In this work we present new techniques to explicitly determine holomorphic Chern-Simons invariants in heterotic string compactifications. The key technical ingredient in our computations are real bundle morphisms between the gauge and tangent bundles. We find that there are large classes of examples, beyond the standard embedding, where the Chern-Simons superpotential vanishes. We also provide explicit examples for non-flat bundles where it is non-vanishing and non-integer quantized, generalizing previous results for Wilson lines.

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals The Exponent of the Homotopy Groups of Moore Spectra and the Stable Hurewicz Homomorphism

1996 ◽  
Vol 48 (3) ◽  
pp. 483-495 ◽  
Author(s):  
Dominique Arlettaz
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Homology Theory ◽  
Homotopy Groups ◽  
Integral Homology ◽  
Ring Spectrum ◽  
Bounded Below ◽  
Torsion Free ◽  
Ordinary Integral

AbstractThis paper shows that for the Moore spectrum MG associated with any abelian group G, and for any positive integer n, the order of the Postnikov k-invariant kn+1(MG) is equal to the exponent of the homotopy group πnMG. In the case of the sphere spectrum S, this implies that the exponents of the homotopy groups of S provide a universal estimate for the exponent of the kernel of the stable Hurewicz homomorphism hn: πnX → En(X) for the homology theory E*(—) corresponding to any connective ring spectrum E such that π0E is torsion-free and for any bounded below spectrum X. Moreover, an upper bound for the exponent of the cokernel of the generalized Hurewicz homomorphism hn: En(X) → Hn(X; π0E), induced by the 0-th Postnikov section of E, is obtained for any connective spectrum E. An application of these results enables us to approximate in a universal way both kernel and cokernel of the unstable Hurewicz homomorphism between the algebraic K-theory of any ring and the ordinary integral homology of its linear group.

scholarly journals Universal upper bound on the inflationary energy scale from the trans-Planckian censorship conjecture

2020 ◽  
Vol 102 (2) ◽  
Author(s):  
Shuntaro Mizuno ◽  
Shinji Mukohyama ◽  
Shi Pi ◽  
Yun-Long Zhang
Keyword(s):  
Upper Bound ◽  
Energy Scale

scholarly journals A lower bound for a remainder term associated with the sum of digits function

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bound ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

scholarly journals THE DISTRIBUTION OF r-FREE NUMBERS IN ARITHMETIC PROGRESSIONS

2014 ◽  
Vol 10 (03) ◽  
pp. 559-563 ◽  
Author(s):  
JASON GIBSON
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Arithmetic Progressions

Let r ≥ 2. A positive integer n is called r-free if n is not divisible by the r th power of a prime. Generalizing an earlier work of Orr, we provide an upper bound of Bombieri–Vinogradov type for the r-free numbers in arithmetic progressions.

scholarly journals A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions

2018 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Rational Solution ◽  
Infinitely Many Solutions ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions

Let f ( 1 ) = 1 , and let f ( n + 1 ) = 2 2 f ( n ) for every positive integer n. We consider the following hypothesis: if a system S ⊆ {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies x1, . . . , xn ≤ f (2n). We prove:   (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, then M is computable.

scholarly journals Coloring the Cube with Rainbow Cycles

10.37236/2957 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Dhruv Mubayi ◽  
Randall Stading
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Sidon Sets ◽  
Dimensional Cube

For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of the $k$-cycle $C_k$ receive $k$ distinct colors. Faudree, Gyárfás, Lesniak and Schelp proved that $f(n,4)=n$ for $n=4$ or $n>5$. We consider larger $k$ and prove that if $k \equiv 0$ (mod 4), then there are positive constants $c_1, c_2$ depending only on $k$ such that$$c_1n^{k/4} < f(n,k) < c_2 n^{k/4}.$$Our upper bound uses an old construction of Bose and Chowla of generalized Sidon sets. For $k \equiv 2$ (mod 4), the situation seems more complicated. For the smallest case $k=6$ we show that $$3n-2 \le f(n, 6) < n^{1+o(1)}$$ with the lower bound holding for $n \ge 3$. The upper bound is obtained from Behrend's construction of a subset of integers with no three term arithmetic progression.

scholarly journals The Solutions of Generalized Euler Function Equation φ_2(n-φ_2 (n))=2ω(n)

2021 ◽  
pp. 15-22
Author(s):  
Xu Yifan ◽  
Shen Zhongyan
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Euler Function ◽  
Prime Factors ◽  
Integer Solutions

By using the properties of Euler function, an upper bound of solutions of Euler function equation  is given, where  is a positive integer. By using the classification discussion and the upper bound we obtained, all positive integer solutions of the generalized Euler function equation  are given, where is the number of distinct prime factors of n.

scholarly journals Large Equiangular Sets of Lines in Euclidean Space

10.37236/1533 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
D. De Caen
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Euclidean Space ◽  
Upper Bound ◽  
Equiangular Lines

A construction is given of ${{2}\over {9}} (d+1)^2$ equiangular lines in Euclidean $d$-space, when $d = 3 \cdot 2^{2t-1}-1$ with $t$ any positive integer. This compares with the well known "absolute" upper bound of ${{1}\over {2}} d(d+1)$ lines in any equiangular set; it is the first known constructive lower bound of order $d^2$ .

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