3,784 research outputs found
Elliptic Reciprocity
The paper introduces the notions of an elliptic pair, an elliptic cycle and
an elliptic list over a square free positive integer d. These concepts are
related to the notions of amicable pairs of primes and aliquot cycles that were
introduced by Silverman and Stange. Settling a matter left open by Silverman
and Stange it is shown that for d=3 there are elliptic cycles of length 6. For
d not equal to 3 the question of the existence of proper elliptic lists of
length n over d is reduced to the the theory of prime producing quadratic
polynomials. For d=163 a proper elliptic list of length 40 is exhibited. It is
shown that for each d there is an upper bound on the length of a proper
elliptic list over d. The final section of the paper contains heuristic
arguments supporting conjectured asymptotics for the number of elliptic pairs
below integer X. Finally, for d congruent to 3 modulo 8 the existence of
infinitely many anomalous prime numbers is derived from Bunyakowski's
Conjecture for quadratic polynomials.Comment: 17 pages, including one figure and two table
Matching Higher Conserved Charges for Strings and Spins
We demonstrate that the recently found agreement between one-loop scaling
dimensions of large dimension operators in N=4 gauge theory and energies of
spinning strings on AdS_5 x S^5 extends to the eigenvalues of an infinite
number of hidden higher commuting charges. This dynamical agreement is of a
mathematically highly intricate and non-trivial nature. In particular, on the
gauge side the generating function for the commuting charges is obtained by
integrable quantum spin chain techniques from the thermodynamic density
distribution function of Bethe roots. On the string side the generating
function, containing information to arbitrary loop order, is constructed by
solving exactly the Backlund equations of the integrable classical string sigma
model. Our finding should be an important step towards matching the integrable
structures on the string and gauge side of the AdS/CFT correspondence.Comment: Latex, 33 pages, v2: new section added (completing the analytic proof
that the entire infinite towers of commuting gauge and string charges match);
references adde
Thirty Years of Turnstiles and Transport
To characterize transport in a deterministic dynamical system is to compute
exit time distributions from regions or transition time distributions between
regions in phase space. This paper surveys the considerable progress on this
problem over the past thirty years. Primary measures of transport for
volume-preserving maps include the exiting and incoming fluxes to a region. For
area-preserving maps, transport is impeded by curves formed from invariant
manifolds that form partial barriers, e.g., stable and unstable manifolds
bounding a resonance zone or cantori, the remnants of destroyed invariant tori.
When the map is exact volume preserving, a Lagrangian differential form can be
used to reduce the computation of fluxes to finding a difference between the
action of certain key orbits, such as homoclinic orbits to a saddle or to a
cantorus. Given a partition of phase space into regions bounded by partial
barriers, a Markov tree model of transport explains key observations, such as
the algebraic decay of exit and recurrence distributions.Comment: Updated and corrected versio
Quasiclassical Geometry and Integrability of AdS/CFT Correspondence
We discuss the quasiclassical geometry and integrable systems related to the
gauge/string duality. The analysis of quasiclassical solutions to the Bethe
anzatz equations arising in the context of the AdS/CFT correspondence is
performed, compare to stationary phase equations for the matrix integrals. We
demonstrate how the underlying geometry is related to the integrable
sigma-models of dual string theory, and investigate some details of this
correspondence.Comment: Based on talks at the conferences "Classical and quantum integrable
systems", January 2004, Dubna, and "Quarks-2004", May 2004, Pushkinskie Gory,
Russia; LaTeX, 17 pp, 3 figures; references adde
Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions
We analyse the Weak Gravity Conjecture for chiral four-dimensional F-theory
compactifications with N=1 supersymmetry. Extending our previous work on nearly
tensionless heterotic strings in six dimensions, we show that under certain
assumptions a tower of asymptotically massless states arises in the limit of
vanishing coupling of a U(1) gauge symmetry coupled to gravity. This tower
contains super-extremal states whose charge-to-mass ratios are larger than
those of certain extremal dilatonic Reissner-Nordstrom black holes, precisely
as required by the Weak Gravity Conjecture. Unlike in six dimensions, the tower
of super-extremal states does not always populate a charge sub-lattice. The
main tool for our analysis is the elliptic genus of the emergent heterotic
string in the chiral N=1 supersymmetric effective theories. This also governs
situations where the heterotic string is non-perturbative. We show how it can
be computed in terms of BPS invariants on elliptic four-folds, by making use of
various dualities and mirror symmetry. Compared to six dimensions, the geometry
of the relevant elliptically fibered four-folds is substantially richer than
that of the three-folds, and we classify the possibilities for obtaining
critical, nearly tensionless heterotic strings. We find that the
(quasi-)modular properties of the elliptic genus crucially depend on the choice
of flux background. Our general results are illustrated in a detailed example.Comment: 72 pages, 2 figure
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