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484
Lectures on 2D YangMills Theory, Equivariant Cohomology and Topological Field Theories
, 1996
"... These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying ..."
Abstract

Cited by 99 (7 self)
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These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.
Finite element exterior calculus, homological techniques, and applications
 ACTA NUMERICA
, 2006
"... ..."
Transverse measures, the modular class, and a cohomology pairing for Lie algebroids
, 1996
"... We show that every Lie algebroid A over a manifold P has a natural representation on the line bundle QA = ∧ top A ⊗ ∧ top T ∗ P. The line bundle QA may be viewed as the Lie algebroid analog of the orientation bundle in topology, and sections of QA may be viewed as transverse measures to A. As a co ..."
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Cited by 81 (10 self)
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We show that every Lie algebroid A over a manifold P has a natural representation on the line bundle QA = ∧ top A ⊗ ∧ top T ∗ P. The line bundle QA may be viewed as the Lie algebroid analog of the orientation bundle in topology, and sections of QA may be viewed as transverse measures to A. As a consequence, there is a welldefined class in the first Lie algebroid cohomology H 1 (A) called the modular class of the Lie algebroid A. This is the same as the one introduced earlier by Weinstein using the Poisson structure on A ∗. We show that there is a natural pairing between the Lie algebroid cohomology spaces of A with trivial coefficients and with coefficients in QA. This generalizes the pairing used in the Poincare duality of finitedimensional Lie algebra cohomology. The case of holomorphic Lie algebroids is also discussed, where the existence of the modular
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly ..."
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Cited by 71 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly
K3 surfaces and string duality
"... The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial sp ..."
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Cited by 65 (14 self)
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The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review “old string theory ” on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E8 × E8 heterotic string, and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface
SasakiEinstein manifolds and volume minimisation
, 2006
"... We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian ..."
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Cited by 58 (2 self)
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We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone M, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat– Heckman formula and also to a limit of a certain equivariant index on M that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of any Sasaki–Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a–maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki
Differentiable and algebroid cohomology, van Est . . . classes
, 2000
"... In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and ..."
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Cited by 57 (17 self)
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In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by WeinsteinXu [47]). As a second application we extend van Est’s argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately gives a slight improvement of HectorDazord’s integrability criterion [12]. In the third section we describe the relevant characteristic classes of representations, living in algebroid cohomology, as well as their relation to the van Est map. This extends EvensLuWeinstein’s characteristic class θL [17] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles
Coverage and Holedetection in Sensor Networks via Homology
 Fourth International Conference on Information Processing in Sensor Networks (IPSN’05), UCLA
, 2005
"... We consider coverage problems in sensor networks of stationary nodes with minimal geometric data. In particular, there are no coordinates and no localization of nodes. We introduce a new technique for detecting holes in coverage by means of homology, an algebraic topological invariant. The impetus f ..."
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Cited by 54 (5 self)
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We consider coverage problems in sensor networks of stationary nodes with minimal geometric data. In particular, there are no coordinates and no localization of nodes. We introduce a new technique for detecting holes in coverage by means of homology, an algebraic topological invariant. The impetus for these techniques is a completion of network communication graphs to two types of simplicial complexes: the nerve complex and the Rips complex. The former gives information about coverage intersection of individual sensor nodes, and is very difficult to compute. The latter captures connectivity in terms of internode communication: it is easy to compute but does not in itself yield coverage data. We obtain coverage data by using persistence of homology classes for Rips complexes. These homological invariants are computable: we provide simulation results. I.
Quantum Quivers and Hall/Hole Halos
, 2002
"... Two pictures of BPS bound states in CalabiYau compactifications of type II string theory exist, one as a set of particles at equilibrium separations from each other, the other as a fusion of Dbranes at a single point of space. We show how stringy quiver quantum mechanics smoothly interpolates be ..."
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Cited by 51 (5 self)
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Two pictures of BPS bound states in CalabiYau compactifications of type II string theory exist, one as a set of particles at equilibrium separations from each other, the other as a fusion of Dbranes at a single point of space. We show how stringy quiver quantum mechanics smoothly interpolates between the two, and use this, together with recent mathematical results on the cohomology of quiver varieties, to solve some nontrivial ground state counting problems in multiparticle quantum mechanics, including one arising in the setup of the spherical quantum Hall effect. A crucial ingredient is a nonrenormalization theorem in N = 4 quantum mechanics for the first order part of the Lagrangian in an expansion in powers of velocity.
Gromov invariants for holomorphic maps from Riemann surfaces to Grassmanians
, 1993
"... Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothend ..."
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Cited by 51 (4 self)
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Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as “Gromov invariants”) on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one.