Central extensions of current groups in two dimensions

In this paper we generalize some of these results for loop algebras and groups as well as for the Virasoro algebra to the two-dimensional case. We define and study a class of infinite dimensional complex Lie groups which are central extensions of the group of smooth maps from a two dimensional orientable surface without boundary to a simple complex Lie group G. These extensions naturally correspond to complex curves. The kernel of such an extension is the Jacobian of the curve. The study of the coadjoint action shows that its orbits are labelled by moduli of holomorphic principal G-bundles over the curve and can be described in the language of partial differential equations. In genus one it is also possible to describe the orbits as conjugacy classes of the twisted loop group, which leads to consideration of difference equations for holomorphic functions. This gives rise to a hope that the described groups should possess a counterpart of the rich representation theory that has been developed for loop groups. We also define a two-dimensional analogue of the Virasoro algebra associated with a complex curve. In genus one, a study of a complex analogue of Hill's operator yields a description of invariants of the coadjoint action of this Lie algebra. The answer turns out to be the same as in dimension one: the invariants coincide with those for the extended algebra of currents in sl(2).Comment: 17 page

Higgs Bundles, Gauge Theories and Quantum Groups

The appearance of the Bethe Ansatz equation for the Nonlinear Schr\"{o}dinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schr\"{o}dinger equation in the $N$-particle sector. This implies the full equivalence between the above gauge theory and the $N$-particle sub-sector of the quantum theory of Nonlinear Schr\"{o}dinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of degenerate double affine Hecke algebra. We propose similar construction based on the $G/G$ gauged WZW model leading to the representation theory of the double affine Hecke algebra. The relation with the Nahm transform and the geometric Langlands correspondence is briefly discussed.Comment: 48 pages, typos corrected, one reference adde

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant.Comment: v2: added due credits to the work of Burger, Iozzi and Wienhard. v3: corrected count of connected components for G=SU(p,q) (p \neq q); added due credits to the work of Xia and Markman-Xia; minor corrections and clarifications. 31 page

On Global Aspects Of Gauged Wess-Zumino-Witten Model

This is a thesis for Rigaku-Hakushi($\simeq$ Ph. D.). It clarifies the geometric meaning and field theoretical consequences of the spectral flows acting on the space of states of the `$G/H$ coset model'. As suggested by Moore and Seiberg, the spectral flow is realized as the response of states to certain change of background gauge field together with the gauge transformation on a circle. Applied to the boundary circle of a disc with field insertion, such a realization leads to a certain relation among correlators of the gauged WZW model for various principal $H$-bundles. In the course of derivation, we find an expression of a (dressed) gauge invariant field as an integral over the flag manifold of $H$ and an expression of a correlator as an integral over a certain moduli space of holomorphic $H_{\bf C}$-bundles with quasi-flag structure at the insertion point. We also find that the gauge transformation on the circle corresponding to the spectral flow determines a bijection of the set of isomorphism classes of holomorphic $H_{\bf C}$-bundles with quasi-flag structure of one topological type to that of another. As an application, it is pointed out that problems arising from the field identification fixed points may be resolved by taking into account of all principal $H$-bundles.Comment: (Thesis) 125 pages, UT-Komaba/94-3 (Latex errors are corrected

Historical atmospheric pollution trends in Southeast Asia inferred from lake sediment records

Fossil fuel combustion leads to increased levels of air pollution, which negatively affects human health as well as the environment. Documented data for Southeast Asia (SEA) show a strong increase in fossil fuel consumption since 1980, but information on coal and oil combustion before 1980 is not widely available. Spheroidal carbonaceous particles (SCPs) and heavy metals, such as mercury (Hg), are emitted as by-products of fossil fuel combustion and may accumulate in sediments following atmospheric fallout. Here we use sediment SCP and Hg records from several freshwater lentic ecosystems in SEA (Malaysia, Philippines, Singapore) to reconstruct long-term, region-wide variations in levels of these two key atmospheric pollution indicators. The age-depth models of Philippine sediment cores do not reach back far enough to date first SCP presence, but single SCP occurrences are first observed between 1925 and 1950 for a Malaysian site. Increasing SCP flux is observed at our sites from 1960 onward, although individual sites show minor differences in trends. SCP fluxes show a general decline after 2000 at each of our study sites. While the records show broadly similar temporal trends across SEA, absolute SCP fluxes differ between sites, with a record from Malaysia showing SCP fluxes that are two orders of magnitude lower than records from the Philippines. Similar trends in records from China and Japan represent the emergence of atmospheric pollution as a broadly-based inter-region environmental problem during the 20th century. Hg fluxes were relatively stable from the second half of the 20th century onward. As catchment soils are also contaminated with atmospheric Hg, future soil erosion can be expected to lead to enhanced Hg flux into surface waters

Co-TUD-1 catalysed aerobic oxidation of cyclohexane

Co-TUD-1, an amorphous sponge-like mesoporous cobalt-containing silicate, was shown to be a very active and selective catalyst for the aerobic oxidation of cyclohexane. In addition, the decomposition of cyclohexyl hydroperoxide (CHHP) was carried out over Co-TUD-1 and it catalysed complete conversion of CHHP with excellent selectivity towards cyclohexanol and cyclohexanone. A stable activity over a period of one week was established for Co-TUD-1 in our rigorous leaching experiments during the continuous decomposition of CHHP. No leaching of the metal could be detected proving the catalyst to be truly heterogeneous. (C) 2008 Elsevier B.V. All rights reserved

Optimization of machining parameters at high speed drilling of carbon fiber reinforced plastic (CFRP) laminates

International audienceHigh speed machining is now acknowledged to be one of the key manufacturing technologies to ensure high productivity and throughput. Drilling of CFRP,

Energy gap and band alignment for (HfO2)x(Al 2O3)1-x on (100) Si

10.1063/1.1492024Applied Physics Letters812376-378APPL