We prove a quantum version of the localization formula of Witten that relates invariants of a git quotient with the equivariant invariants of the action. Using the formula we prove a quantum version of an abelianization formula of S. Martin relating
invariants of geometric invariant theory quotients by a group and its maximal torus, conjectured by Bertram, Ciocan-Fontanine, and Kim. By similar techniques we prove a quantum Lefschetz principle for holomorphic symplectic reductions. As an application, we give a formula for the fundamental solution to the quantum differential equation (qde) for the moduli space of points on the projective line and for the smoothed moduli space of framed sheaves on the projective plane (a Nakajima quiver variety).
We prove a gluing theorem for a symplectic vortex on a compact complex curve and a collection of holomorphic sphere bubbles. Using the theorem we show that the moduli space of regular stable symplectic vortices on a fixed curve with varying markings
has the structure of a stratified-smooth topological orbifold. In addition, we show that the moduli space has a non-canonical $C^1$-orbifold structure.
The ISO/LWS far-infrared spectrum of the ultraluminous galaxy Mkn 231 shows OH and H_2O lines in absorption from energy levels up to 300 K above the ground state, and emission in the [O I] 63 micron and [C II] 158 micron lines. Our analysis shows tha
t OH and H_2O are radiatively pumped by the far-infrared continuum emission of the galaxy. The absorptions in the high-excitation lines require high far-infrared radiation densities, allowing us to constrain the properties of the underlying continuum source. The bulk of the far-infrared continuum arises from a warm (T_dust=70-100 K), optically thick (tau_100micron=1-2) medium of effective diameter 200-400 pc. In our best-fit model of total luminosity L_IR, the observed OH and H2O high-lying lines arise from a luminous (L/L_IR~0.56) region with radius ~100 pc. The high surface brightness of this component suggests that its infrared emission is dominated by the AGN. The derived column densities N(OH)>~10^{17} cm^{-2} and N(H_2O)>~6x10^{16} cm^{-2} may indicate XDR chemistry, although significant starburst chemistry cannot be ruled out. The lower-lying OH, [C II] 158 micron, and [O I] 63 micron lines arise from a more extended (~350 pc) starburst region. We show that the [C II] deficit in Mkn 231 is compatible with a high average abundance of C+ because of an extreme overall luminosity to gas mass ratio. Therefore, a [C II] deficit may indicate a significant contribution to the luminosity by an AGN, and/or by extremely efficient star formation.
We have modeled the emission of H2O rotational lines from the extreme C-rich star IRC+10216. Our treatment of the excitation of H2O emissions takes into account the excitation of H2O both through collisions, and through the pumping of the nu2 and nu3
vibrational states by dust emission and subsequent decay to the ground state. Regardless of the spatial distribution of the water molecules, the H2O 1_{10}-1_{01} line at 557 GHz observed by the Submillimeter Wave Astronomy Satellite (SWAS) is found to be pumped primarily through the absorption of dust-emitted photons at 6 $mu$m in the nu2 band. As noted by previous authors, the inclusion of radiative pumping lowers the ortho-H2O abundance required to account for the 557 GHz emission, which is found to be (0.5-1)x10^{-7} if the presence of H2O is a consequence of vaporization of orbiting comets or Fischer-Tropsch catalysis. Predictions for other submillimeter H2O lines that can be observed by the Herschel Space Observatory (HSO) are reported. Multitransition HSO observations promise to reveal the spatial distribution of the circumstellar water vapor, discriminating among the several hypotheses that have been proposed for the origin of the H2O vapor in the envelope of IRC+10216. We also show that, for observations with HSO, the H2O 1_{10}-1_{01} 557 GHz line affords the greatest sensitivity in searching for H2O in other C-rich AGB stars.
