Quantum integrable systems and representations of Lie algebras

In this paper the quantum integrals of the Hamiltonian of the quantum many-body problem with the interaction potential K/sinh^2(x) (Sutherland operator) are constructed as images of higher Casimirs of the Lie algebra gl(N) under a certain homomorphism from the center of U(gl(N)) to the algebra of differential operators in N variables. A similar construction applied to the affine gl(N) at the critical level k=-N defines a correspondence between higher Sugawara operators and quantum integrals of the Hamiltonian of the quantum many-body problem with the potential equal to constant times the Weierstrass function. This allows one to give a new proof of the Olshanetsky-Perelomov theorem stating that this Hamiltonian defines a completely integrable quantum system. We also give a new expression for eigenfunctions of the quantum integrals of the Sutherland operator as traces of intertwining operators between certain representations of gl(N).Comment: 17 pages, no figure

MCMC with Strings and Branes: The Suburban Algorithm (Extended Version)

Motivated by the physics of strings and branes, we develop a class of Markov chain Monte Carlo (MCMC) algorithms involving extended objects. Starting from a collection of parallel Metropolis-Hastings (MH) samplers, we place them on an auxiliary grid, and couple them together via nearest neighbor interactions. This leads to a class of "suburban samplers" (i.e., spread out Metropolis). Coupling the samplers in this way modifies the mixing rate and speed of convergence for the Markov chain, and can in many cases allow a sampler to more easily overcome free energy barriers in a target distribution. We test these general theoretical considerations by performing several numerical experiments. For suburban samplers with a fluctuating grid topology, performance is strongly correlated with the average number of neighbors. Increasing the average number of neighbors above zero initially leads to an increase in performance, though there is a critical connectivity with effective dimension d_eff ~ 1, above which "groupthink" takes over, and the performance of the sampler declines.Comment: v2: 55 pages, 13 figures, references and clarifications added. Published version. This article is an extended version of "MCMC with Strings and Branes: The Suburban Algorithm

Extended trigonometric Cherednik algebras and nonstationary Schr\"odinger equations with delta-potentials

We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schr\"odinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schr\"odinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations in their spectral parameter. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with pairwise delta-function interactions is indicated.Comment: 23 pages; Version 2: expanded introduction and misprints correcte

Separate ways: The Mass-Metallicity Relation does not strongly correlate with Star Formation Rate in SDSS-IV MaNGA galaxies

We present the integrated stellar mass-metallicity relation (MZR) for more than 1700 galaxies included in the integral field area SDSS-IV MaNGA survey. The spatially resolved data allow us to determine the metallicity at the same physical scale (effective radius in arcsecs, $\mathrm{R_{eff}}$ ) using a heterogeneous set of ten abundance calibrators. Besides scale factors, the shape of the MZR is similar for all calibrators, consistent with those reported previously using single-fiber and integral field spectroscopy. We compare the residuals of this relation against the star formation rate (SFR) and specific SFR (sSFR). We do not find a strong secondary relation of the MZR with either SFR or the sSFR for any of the calibrators, in contrast with previous single-fiber spectroscopic studies. Our results agree with an scenario in which metal enrichment happens at local scales, with global outflows playing a secondary role in shaping the chemistry of galaxies and cold-gas inflows regulating the stellar formation.Comment: 10 pages, 9 Figures. Accepted for publication in Ap

Local Starbursts in a Cosmological Context

In this contribution I introduce some of the major issues that motivate the conference, with an emphasis on how starbursts fit into the ``big picture''. I begin by defining starbursts in several different ways, and discuss the merits and limitations of these definitions. I will argue that the most physically useful definition of a starburst is its ``intensity'' (star formation rate per unit area). This is the most natural parameter to compare local starbursts with physically similar galaxies at high redshift, and indeed I will argue that local starbursts are unique laboratories to study the processes at work in the early universe. I will describe how NASA's GALEX mission has uncovered a rare population of close analogs to Lyman Break Galaxies in the local universe. I will then compare local starbursts to the Lyman-Break and sub-mm galaxies high redshift populations, and speculate that the multidimensional ``manifold'' of starbursts near and far can be understood largely in terms of the Schmidt/Kennicutt law and galaxy mass-metallicity relation. I will briefly summarize he properties of starburst-driven galactic superwinds and their possible implications for the evolution of galaxies and the IGM. These complex multiphase flows are best studied in nearby starbursts, where we can study the the hot X-ray gas that contains the bulk of the energy and newly produced metals.Comment: Proceedings of the Conference "Starbursts: Fropm 30 Doradus to Lyman Break Galaxies

Active Galactic Nuclei in Void Regions

We present a comprehensive study of accretion activity in the most underdense environments in the universe, the voids, based on the SDSS DR2 data. Based on investigations of multiple void regions, we show that AGN's occurrence rate and properties differ from those in walls. AGN are more common in voids than in walls, but only among moderately luminous and massive galaxies (M_r < -20, log M_*/M_sun < 10.5), and this enhancement is more pronounced for the weakly accreting systems (i.e., L_[O III] < 10^39 erg/s). Void AGN hosted by moderately massive and luminous galaxies are accreting at equal or lower rates than their wall counterparts, show less obscuration than in walls, and similarly aged stellar populations. The very few void AGN in massive bright hosts accrete more strongly, are more obscured, and are associated with younger stellar emission than wall AGN. Thus, accretion strength is probably connected to the availability of fuel supply, and accretion and star-formation co-evolve and rely on the same source of fuel. Nearest neighbor statistics indicate that the weak accretion activity (LINER-like) is not influenced by the local environment. However, H IIs, Seyferts, and Transition objects prefer more grouped small scale structures, indicating that the rate at which galaxies interact with each other affects their activity. These trends support a potential H II -> Seyfert/Transition Object -> LINER evolutionary sequence that we show is apparent in many properties of actively line-emitting galaxies, in both voids and walls. The subtle differences between void and wall AGN might be explained by a longer, less disturbed duty cycle of these systems in voids.Comment: 19 pages, 7 figures (1 color); to appear in ApJ, submitted on May 11, 200

Uncovering Spiral Structure in Flocculent Galaxies

We present K'(2.1 micron) observations of four nearby flocculent spirals, which clearly show low-level spiral structure and suggest that kiloparsec-scale spiral structure is more prevalent in flocculent spirals than previously supposed. In particular, the prototypical flocculent spiral NGC 5055 is shown to have regular, two-arm spiral structure to a radius of 4 kpc in the near infrared, with an arm-interarm contrast of 1.3. The spiral structure in all four galaxies is weaker than that in grand design galaxies. Taken in unbarred galaxies with no large, nearby companions, these data are consistent with the modal theory of spiral density waves, which maintains that density waves are intrinsic to the disk. As an alternative, mechanisms for driving spiral structure with non-axisymmetric perturbers are also discussed. These observations highlight the importance of near infrared imaging for exploring the range of physical environments in which large-scale dynamical processes, such as density waves, are important.Comment: 12 pages AASTeX; 3 compressed PS figures can be retrieved from ftp://ftp.astro.umd.edu/pub/michele as file thornley.tar (1.6Mbytes). Accepted to Ap.J. Letters.(Figures now also available here, and from ftp://ftp.astro.umd.edu/pub/michele , in GIF format.