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 $X$, the homogenized, torus equivariant CSM class of a constructible function $\varphi$ is the restriction of the characteristic cycle of $\varphi$ via the zero section of the cotangent bundle of $X$. This extends to the equivariant setting results of Ginzburg and Sabbah. We specialize $X$ to be a (generalized) flag manifold $G/B$. 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 $\mathcal{D}_X$-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 $G/P$.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