487 research outputs found
Formation of Compressed Flat Electron Beams with High Transverse-Emittance Ratios
Flat beams -- beams with asymmetric transverse emittances -- have important
applications in novel light-source concepts, advanced-acceleration schemes and
could possibly alleviate the need for damping rings in lepton colliders. Over
the last decade, a flat-beam-generation technique based on the conversion of an
angular-momentum-dominated beam was proposed and experimentally tested. In this
paper we explore the production of compressed flat beams. We especially
investigate and optimize the flat-beam transformation for beams with
substantial fractional energy spread. We use as a simulation example the
photoinjector of the Fermilab's Advanced Superconducting Test Accelerator
(ASTA). The optimizations of the flat beam generation and compression at ASTA
were done via start-to-end numerical simulations for bunch charges of 3.2 nC,
1.0 nC and 20 pC at ~37 MeV. The optimized emittances of flat beams with
different bunch charges were found to be 0.25 {\mu}m (emittance ratio is ~400),
0.13 {\mu}m, 15 nm before compression, and 0.41 {\mu}m, 0.20 {\mu}m, 16 nm
after full compression, respectively with peak currents as high as 5.5 kA for a
3.2-nC flat beam. These parameters are consistent with requirements needed to
excite wakefields in asymmetric dielectric-lined waveguides or produce
significant photon flux using small-gap micro-undulators.Comment: 17
Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or
noncompact varieties the classical total homology Chern class of the tangent
bundle of a smooth compact complex manifold. The theory of CSM classes has been
extended to the equivariant setting by Ohmoto. We prove that for an arbitrary
complex projective manifold , the homogenized, torus equivariant CSM class
of a constructible function is the restriction of the characteristic
cycle of via the zero section of the cotangent bundle of . This
extends to the equivariant setting results of Ginzburg and Sabbah. We
specialize to be a (generalized) flag manifold . In this case CSM
classes are determined by a Demazure-Lusztig (DL) operator. We prove a `Hecke
orthogonality' of CSM classes, determined by the DL operator and its
Poincar{\'e} adjoint. We further use the theory of holonomic
-modules to show that the characteristic cycle of a Verma
module, restricted to the zero section, gives the CSM class of the
corresponding Schubert cell. Since the Verma characteristic cycles naturally
identify with the Maulik and Okounkov's stable envelopes, we establish an
equivalence between CSM classes and stable envelopes; this reproves results of
Rim{\'a}nyi and Varchenko. As an application, we obtain a Segre type formula
for CSM classes. In the non-equivariant case this formula is manifestly
positive, showing that the expansion in the Schubert basis of the CSM class of
a Schubert cell is effective. This proves a previous conjecture by Aluffi and
Mihalcea, and it extends previous positivity results by J. Huh in the Grassmann
manifold case. Finally, we generalize all of this to partial flag manifolds
.Comment: 40 pages; main changes in v2: removed some unnecessary compactness
hypotheses; added remarks 7.2 and 9.6 explaining how orthogonality of
characteristic cycles for transversal Schubert cell stratifications leads to
orthogonality of stable envelopes and that of CSM classe
Finiteness of cominuscule quantum K-theory
The product of two Schubert classes in the quantum K-theory ring of a
homogeneous space X = G/P is a formal power series with coefficients in the
Grothendieck ring of algebraic vector bundles on X. We show that if X is
cominuscule, then this power series has only finitely many non-zero terms. The
proof is based on a geometric study of boundary Gromov-Witten varieties in the
Kontsevich moduli space, consisting of stable maps to X that take the marked
points to general Schubert varieties and whose domains are reducible curves of
genus zero. We show that all such varieties have rational singularities, and
that boundary Gromov-Witten varieties defined by two Schubert varieties are
either empty or unirational. We also prove a relative Kleiman-Bertini theorem
for rational singularities, which is of independent interest. A key result is
that when X is cominuscule, all boundary Gromov-Witten varieties defined by
three single points in X are rationally connected.Comment: 16 pages; proofs slightly improved; explicit multiplications in
QK(Cayley plane) from v1 no longer necessar
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