Equivariant volumes of non-compact quotients and instanton counting

Motivated by Nekrasov's instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-K\"ahler geometry by means of the Jeffrey-Kirwan residue-formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on $\R^{4}$ and rederive the formulas for the equivariant volumes obtained earlier by Nekrasov-Shadchin, expressing these volumes as iterated residues of a single rational function.Comment: 34 pages, 2 figures; minor typos corrected, to appear in Comm. Math. Phy

Toric Structures on the Moduli Space of Flat Connections on a Riemann Surface: Volumes and the Moment Map

AbstractIn earlier papers we constructed a Hamiltonian torus action on an open dense set in the moduli space of flat SU(2) connections on a compact Riemann surface, where the dimension of the torus is half the dimension of the moduli space. This torus action shows that this set can be viewed symplectically as a (noncompact) toric variety. The number of integral points of the moment map for the torus action turns out to be identical to the Verlinde dimension D(g, k). As an application, we furnish a new proof of the relation between the large-k limit of D(g, k) and the volume of the moduli space. From our point of view, this relation follows from the equality between the symplectic volume of a toric variety and the Euclidean volume of the image of the moment map. Similar considerations are shown to give rise to the volumes of moduli spaces of parabolic bundles on a Riemann surface. Knowledge of these volumes has been shown to allow a proof of the Verlinde formula for the dimension of the space of holomorphic sections of line bundles on this space

Morse theory of the moment map for representations of quivers

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-H\"older filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts.Comment: 48 pages, small revisions from previous version based on referee's comments. To appear in Geometriae Dedicat

Level Sets of the Takagi Function: Local Level Sets

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a "generic" full Lebesgue measure set of ordinates y, the level sets are finite sets. Here it is shown for a "generic" full Lebesgue measure set of abscissas x, the level set L(\tau(x)) is uncountable. An interesting singular monotone function is constructed, associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly 3/2.Comment: 32 pages, 2 figures, 1 table. Latest version has updated equation numbering. The final publication will soon be available at springerlink.co

Chern-Simons States at Genus One

We present a rigorous analysis of the Schr\"{o}dinger picture quantization for the $SU(2)$ Chern-Simons theory on 3-manifold torus$\times$line, with insertions of Wilson lines. The quantum states, defined as gauge covariant holomorphic functionals of smooth $su(2)$-connections on the torus, are expressed by degree $2k$ theta-functions satisfying additional conditions. The conditions are obtained by splitting the space of semistable $su(2)$-connections into nine submanifolds and by analyzing the behavior of states at four codimension $1$ strata. We construct the Knizhnik-Zamolodchikov-Bernard connection allowing to compare the states for different complex structures of the torus and different positions of the Wilson lines. By letting two Wilson lines come together, we prove a recursion relation for the dimensions of the spaces of states which, together with the (unproven) absence of states for spins\s>{_1\over^2}level implies the Verlinde dimension formula.Comment: 33 pages, IHES/P

Localization for Yang-Mills Theory on the Fuzzy Sphere

We present a new model for Yang-Mills theory on the fuzzy sphere in which the configuration space of gauge fields is given by a coadjoint orbit. In the classical limit it reduces to ordinary Yang-Mills theory on the sphere. We find all classical solutions of the gauge theory and use nonabelian localization techniques to write the partition function entirely as a sum over local contributions from critical points of the action, which are evaluated explicitly. The partition function of ordinary Yang-Mills theory on the sphere is recovered in the classical limit as a sum over instantons. We also apply abelian localization techniques and the geometry of symmetric spaces to derive an explicit combinatorial expression for the partition function, and compare the two approaches. These extend the standard techniques for solving gauge theory on the sphere to the fuzzy case in a rigorous framework.Comment: 55 pages. V2: references added; V3: minor corrections, reference added; Final version to be published in Communications in Mathematical Physic

On the Quantum Invariant for the Spherical Seifert Manifold

We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold $S^3/\Gamma$ where $\Gamma$ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of the modular forms with half-integral weight, and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the Eichler integral. We further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to $\Gamma$.Comment: 36 page

Greenhouse gas emissions from inland waters: A perspective and research agenda for the tropics and subtropics

peer reviewedStrong consensus indicates that inland waters emit globally significant quantities of greenhouse gases such as carbon dioxide, methane, and nitrous oxide. Tropical inland waters are often considered major contributors to higher greenhouse gas fluxes, yet accurate estimates of aquatic greenhouse gas fluxes are limited for the tropics. We provide a historical perspective on research carried out across low latitudes since the 1980s, synthesize current understanding of the sources and drivers of greenhouse gas emissions, and highlight priority research areas for future tropical inland water greenhouse gas research. We show that much of the focus has been on the humid tropics while the wet-dry, (semi)arid, and mountainous regions remain underrepresented in global datasets. Consistent and reliable greenhouse gas emission estimates will require (1) addressing the observational mismatch with new data from understudied ecoregions, (2) favoring direct and high-resolution carbon dioxide measurements over indirect estimates based on water chemistry parameters, (3) developing approaches that cross boundaries between ecosystem types and scales, and (4) sharing and publishing data more systematically