Results 1  10
of
731
Conjugation spaces
, 2004
"... There are classical examples of spaces X with an involution τ whose mod2comhomology ring resembles that of their fixed point set X τ: there is a ring isomorphism κ: H 2 ∗ (X) ≈ H ∗ (X τ). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ..."
Abstract

Cited by 192 (2 self)
 Add to MetaCart
There are classical examples of spaces X with an involution τ whose mod2comhomology ring resembles that of their fixed point set X τ: there is a ring isomorphism κ: H 2 ∗ (X) ≈ H ∗ (X τ). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism κ is part of an interesting structure in equivariant cohomology called an H ∗frame. An H ∗ frame, if it exists, is natural and unique. A space with involution admitting an H ∗ frame is called a conjugation space. Many examples of conjugation spaces are constructed, for instance by successive adjunctions of cells homeomorphic to a disk in C k with the complex conjugation. A compact symplectic manifold, with an antisymplectic involution compatible with a Hamiltonian action of a torus T, is a conjugation space, provided X T is itself a conjugation space. This includes the coadjoint orbits of any semisimple compact Lie group, equipped with the Chevalley involution. We also study conjugateequivariant complex vector bundles (“real bundles ” in the sense of Atiyah) over a conjugation space and show that the isomorphism κ maps the Chern classes onto the StiefelWhitney classes of the fixed bundle.
Topological Gauge Theories and Group Cohomology
, 1989
"... We show that three dimensional ChernSimons gauge theories with a compact gauge group G (not necessarily connected or simply connected) can be classified by the integer cohomology group H 4 (BG, Z). In a similar way, possible WessZumino interactions of such a group G are classified by H 3 (G, Z). ..."
Abstract

Cited by 171 (2 self)
 Add to MetaCart
We show that three dimensional ChernSimons gauge theories with a compact gauge group G (not necessarily connected or simply connected) can be classified by the integer cohomology group H 4 (BG, Z). In a similar way, possible WessZumino interactions of such a group G are classified by H 3 (G, Z). The relation between three dimensional ChernSimons gauge theory and two dimensional sigma models involves a certain natural map from H 4 (BG, Z) to H 3 (G, Z). We generalize this correspondence to topological ‘spin ’ theories, which are defined on three manifolds with spin structure, and are related to what might be called Z2 graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.
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 141 (11 self)
 Add to MetaCart
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.
Asymptotically flat selfdual solutions to Euclidean gravity, Phys
 Lett. B74
, 1978
"... In an attempt to find gravitational analogs of Yang Mills pseudoparticles, we obtain two classes of selfdual solutions to the eucfidean Einstein equations. These metrics are free from singularities and approach a flat metric at infinity. The discovery of pseudoparticle solutions to the euclidean SU ..."
Abstract

Cited by 129 (0 self)
 Add to MetaCart
In an attempt to find gravitational analogs of Yang Mills pseudoparticles, we obtain two classes of selfdual solutions to the eucfidean Einstein equations. These metrics are free from singularities and approach a flat metric at infinity. The discovery of pseudoparticle solutions to the euclidean SU(2) YangMills theory [1] has suggested the possibility that analogous solutions might occur in Einstein's theory of gravitation. The existence of such solutions would have a profound effect on the quantum theory of gravitation [2,3]. Since fire YangMills pseudoparticles possess selfdual field strengths, one likely possibility is that gravitational pseudoparticles are characterized by selfdual curvature. In fact it has been pointed out by Hawking [3] that the TaubNUT metric [4], when appropriately continued to euclidean spacetime, produces a selfdual curvature and hence is a possible candidate for a gravitational pseudoparticle. He has also given a generalized multiTaubNUT metric. However, these metrics do not approach a fiat metric at infinity [5]. To see this, let us write the euclidean TaubNUT solution as (ds) 2 = [(R +m)/(R m)l dR 2 +4(R2m2){o2 x + 02 +(2m/(R +m))2o2}, where ex, Oy, o z form a standard Caftan basis,
E(8) Gauge Theory and a Derivation of Ktheory from Mtheory
"... The partition function of RamondRamond pform fields in Type IIA supergravity on a tenmanifold X contains subtle phase factors that are associated with Tduality, selfduality, and the relation of the RR fields to Ktheory. The analogous partition function of Mtheory on X × S 1 contains subtle pha ..."
Abstract

Cited by 110 (9 self)
 Add to MetaCart
(Show Context)
The partition function of RamondRamond pform fields in Type IIA supergravity on a tenmanifold X contains subtle phase factors that are associated with Tduality, selfduality, and the relation of the RR fields to Ktheory. The analogous partition function of Mtheory on X × S 1 contains subtle phases that are similarly associated with E8 gauge theory. We analyze the detailed phase factors on the two sides and show that they agree, thereby testing Mtheory/Type IIA duality as well as the Ktheory formalism in an interesting way. We also show that certain Dbrane states wrapped on nontrivial homology cycles are actually unstable, that (−1) FL symmetry in Type IIA superstring theory depends in general on a cancellation between a fermion anomaly and an anomaly of RR fields, and that Type IIA superstring theory with no wrapped branes is welldefined only on a spacetime with W7 = 0. On leave from Institute for Advanced Study, Princeton, NJ 08540.
A survey of foliations and operator algebras
 Proc. Sympos. Pure
, 1982
"... 1 Transverse measure for flows 4 2 Transverse measure for foliations 6 ..."
Abstract

Cited by 82 (6 self)
 Add to MetaCart
(Show Context)
1 Transverse measure for flows 4 2 Transverse measure for foliations 6
Spin Geometry and Seiberg–Witten Invariants
, 1996
"... Over the last year remarkable new developments have no less than revolutionized the subject of 4manifold topology. When Seiberg and Witten discovered their monopole equations in October 1994 it was soon realized by Kronheimer, Mrowka, Taubes, and others that these new invariants led to remarkably s ..."
Abstract

Cited by 80 (13 self)
 Add to MetaCart
Over the last year remarkable new developments have no less than revolutionized the subject of 4manifold topology. When Seiberg and Witten discovered their monopole equations in October 1994 it was soon realized by Kronheimer, Mrowka, Taubes, and others that these new invariants led to remarkably simpler proofs of many of Donaldson’s theorems and gave rise to new interconnections between Riemannian geometry, 4manifolds, and symplectic topology. For example, manifolds with nontrivial invariants do not admit metrics of positive scalar curvature, Kronheimer and Mrowka finally settled the Thom conjecture, and Taubes proved that symplectic 4manifolds have nontrivial invariants, thus settling a longstanding conjecture related to the existence of symplectic structures. One of the deepest and most striking new results in this circle of ideas is Taubes ’ theorem about the relation between the SeibergWitten and the Gromov invariants in the symplectic case. This can be interpreted as an existence theorem for Jholomorphic curves and it gave rise to a number of new theorems about
Lectures on Dbranes
, 1998
"... This is an introduction to the physics of Dbranes. Topics covered include Polchinski’s original calculation, a critical assessment of some duality checks, Dbrane scattering, and effective worldvolume actions. Based on lectures given in 1997 at the Isaac Newton ..."
Abstract

Cited by 69 (5 self)
 Add to MetaCart
This is an introduction to the physics of Dbranes. Topics covered include Polchinski’s original calculation, a critical assessment of some duality checks, Dbrane scattering, and effective worldvolume actions. Based on lectures given in 1997 at the Isaac Newton
The Cohomology Ring of Polygon Spaces
 Ann. Inst. Fourier
, 1998
"... We compute the integer cohomology rings of the "polygon spaces" introduced in [Kl, KM]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we calculate its kernel using ideas from the theory of Grobner bases. Since we do not inver ..."
Abstract

Cited by 63 (9 self)
 Add to MetaCart
(Show Context)
We compute the integer cohomology rings of the "polygon spaces" introduced in [Kl, KM]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we calculate its kernel using ideas from the theory of Grobner bases. Since we do not invert the prime 2, we can tensor with Z 2 ; halving all degrees we show this produces the Z 2 cohomology rings of the planar polygon spaces. In the equilateral case, where there is an action of the symmetric group permuting the edges, we show that the induced action on the integer cohomology is not the standard one, despite it being so on the rational cohomology [Kl]. Finally, our formulae for the Poincar'e polynomials are more computationally effective than those known [Kl]. Introduction A "polygon space" Pol (ff 1 ; ff 2 ; :::ff m ); ff i 2 R+ can be seen to arise in several ways: 1. the family of piecewise linear paths in R 3 , whose ith step (which is of length ff i ) can proceed in any direction s...