32,101 research outputs found
Explicit bounds for generators of the class group
Assuming Generalized Riemann's Hypothesis, Bach proved that the class group SICK of a number field K may be generated using prime ideals whose norm is bounded by 121og(2)delta(K), and by (4 + o(l)) log(2) delta(K) asymptotically, where delta(K) is the absolute value of the discriminant of K. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates SICK and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that SICK is generated by prime ideals whose norm is bounded by the minimum of 4.01 log(2) delta(K), 4(l + (2 pi e(gamma))N-_(K))(2) log(2) delta(k) and 4( log delta(k) + log log delta(K) - (gamma + log 2 pi)N-K + 1 + (N-K + 1) log(7log delta(K)/log delta(K))(2). Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedma's algorithms, confirming that it has size SIC (log delta(K) log log delta(K))(2). In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than log(2) delta(K) except for 7 out of the 31292 fields we tested
Multigraded Factorial Rings and Fano varieties with torus action
In a first result, we describe all finitely generated factorial algebras over
an algebraically closed field of characteristic zero that come with an
effective multigrading of complexity one by means of generators and relations.
This enables us to construct systematically varieties with free divisor class
group and a complexity one torus action via their Cox rings. For the Fano
varieties of this type that have a free divisor class group of rank one, we
provide explicit bounds for the number of possible deformation types depending
on the dimension and the index of the Picard group in the divisor class group.
As a consequence, one can produce classification lists for fixed dimension and
Picard index. We carry this out expemplarily in the following cases. There are
15 non-toric surfaces with Picard index at most six. Moreover, there are 116
non-toric threefolds with Picard index at most two; nine of them are locally
factorial, i.e. of Picard index one, and among these one is smooth, six have
canonical singularities and two have non-canonical singularities. Finally,
there are 67 non-toric locally factorial fourfolds and two one-dimensional
families of non-toric locally factorial fourfolds. In all cases, we list the
Cox rings explicitly.Comment: several corrections, 30 pages, to appear in Doc. Mat
On finiteness of Type IIB compactifications: Magnetized branes on elliptic Calabi-Yau threefolds
The string landscape satisfies interesting finiteness properties imposed by
supersymmetry and string-theoretical consistency conditions. We study N=1
supersymmetric compactifications of Type IIB string theory on smooth
elliptically fibered Calabi-Yau threefolds at large volume with magnetized
D9-branes and D5-branes. We prove that supersymmetry and tadpole cancellation
conditions imply that there is a finite number of such configurations. In
particular, we derive an explicitly computable bound on the number of magnetic
flux quanta, as well as the number of D5-branes, which is independent of the
continuous moduli of the setup. The proof applies if a number of easy to check
geometric conditions of the twofold base are met. We show that these geometric
conditions are satisfied for the almost Fano twofold bases given by each toric
variety associated to a reflexive two-dimensional polytope as well as by the
generic del Pezzo surfaces dP_n with n=0,...,8. Physically, this finiteness
proof shows that there exist a finite collection of four-dimensional gauge
groups and chiral matter spectra in the 4D supergravity theories realized by
these compactifications. As a by-product we explicitly construct all generators
of the Kaehler cones of dP_n and work out their relation to representation
theory.Comment: 49 pages, 1 figure, 5 tables. Minor corrections and improvements, a
version to appear in JHE
Asymptotic structure of supergravity in 3D: extended super-BMS and nonlinear energy bounds
The asymptotically flat structure of supergravity in
three spacetime dimensions is explored. The asymptotic symmetries are spanned
by an extension of the super-BMS algebra, with two independent
currents of electric and magnetic type. These currents are associated to
fields being even and odd under parity, respectively. Remarkably, although the
fields do not generate a backreaction on the metric, they provide
nontrivial Sugawara-like contributions to the BMS generators, and hence to
the energy and the angular momentum. The entropy of flat cosmological
spacetimes with fields then acquires a nontrivial dependence on the
charges. If the spin structure is odd, the ground state
corresponds to Minkowski spacetime, and although the anticommutator of the
canonical supercharges is linear in the energy and in the electric-like
charge, the energy becomes bounded from below by the energy of the
ground state shifted by the square of the electric-like charge. If
the spin structure is even, the same bound for the energy generically holds,
unless the absolute value of the electric-like charge is less than minus the
mass of Minkowski spacetime in vacuum, so that the energy has to be
nonnegative. The explicit form of the Killing spinors is found for a wide class
of configurations that fulfills our boundary conditions, and they exist
precisely when the corresponding bounds are saturated. It is also shown that
the spectra with periodic or antiperiodic boundary conditions for the fermionic
fields are related by spectral flow, in a similar way as it occurs for the
super-Virasoro algebra. Indeed, our super-BMS algebra can
be recovered from the flat limit of the superconformal algebra with
, truncating the fermionic generators of the right copy.Comment: 32 pages, no figures. Talk given at the ESI Programme and Workshop
"Quantum Physics and Gravity" hosted by ESI, Vienna, June 2017. V3: minor
changes and typos corrected. Matches published versio
Hypersymmetry bounds and three-dimensional higher-spin black holes
We investigate the hypersymmetry bounds on the higher spin black hole
parameters that follow from the asymptotic symmetry superalgebra in higher-spin
anti-de Sitter gravity in three spacetime dimensions. We consider anti-de
Sitter hypergravity for which the analysis is most transparent. This is a
Chern-Simons theory which contains,
besides a spin- field, a spin- field and a spin- field. The
asymptotic symmetry superalgebra is then the direct sum of two-copies of the
hypersymmetric extension of , which contains
fermionic generators of conformal weight and bosonic generators of
conformal weight in addition to the Virasoro generators. Following standard
methods, we derive bounds on the conserved charges from the anticommutator of
the hypersymmetry generators. The hypersymmetry bounds are nonlinear and are
saturated by the hypersymmetric black holes, which turn out to possess
-hypersymmetry and to be "extreme", where extremality can be defined in
terms of the entropy: extreme black holes are those that fulfill the
extremality bounds beyond which the entropy ceases to be a real function of the
black hole parameters. We also extend the analysis to other -solitonic
solutions which are maximally (hyper)symmetric.Comment: 26 page
Complexities of 3-manifolds from triangulations, Heegaard splittings, and surgery presentations
We study complexities of 3-manifolds defined from triangulations, Heegaard
splittings, and surgery presentations. We show that these complexities are
related by linear inequalities, by presenting explicit geometric constructions.
We also show that our linear inequalities are asymptotically optimal. Our
results are used in [arXiv:1405.1805] to estimate Cheeger-Gromov
-invariants in terms of geometric group theoretic and knot theoretic
data.Comment: 16 pages, 9 figures. An error in the triangulation argument found by
a referee has been fixed. Constants in Theorems A and B have been improved.
Minor remaining typos have been fixed. To appear in the Quarterly Journal of
Mathematic
Pseudorandomness via the discrete Fourier transform
We present a new approach to constructing unconditional pseudorandom
generators against classes of functions that involve computing a linear
function of the inputs. We give an explicit construction of a pseudorandom
generator that fools the discrete Fourier transforms of linear functions with
seed-length that is nearly logarithmic (up to polyloglog factors) in the input
size and the desired error parameter. Our result gives a single pseudorandom
generator that fools several important classes of tests computable in logspace
that have been considered in the literature, including halfspaces (over general
domains), modular tests and combinatorial shapes. For all these classes, our
generator is the first that achieves near logarithmic seed-length in both the
input length and the error parameter. Getting such a seed-length is a natural
challenge in its own right, which needs to be overcome in order to derandomize
RL - a central question in complexity theory.
Our construction combines ideas from a large body of prior work, ranging from
a classical construction of [NN93] to the recent gradually increasing
independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some
novel analytic machinery which might find other applications
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