Showing papers on "Equivariant map published in 2008"

Sasaki-Einstein manifolds and volume minimisation

Abstract: 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 X, 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 X 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 a 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 invariant of the transverse space, the latter being an obstruction to the existence of a Kahler–Einstein metric.

Renormalization and blow up for charge one equivariant critical wave maps

Abstract: We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ2+1→S 2 of the form $u(t,r)=Q(\lambda(t)r)+\mathcal{R}(t,r)$ where u is the polar angle on the sphere, $Q(r)=2\arctan r$ is the ground state harmonic map, λ(t)=t -1-ν, and $\mathcal{R}(t,r)$ is a radiative error with local energy going to zero as t→0. The number $ u>\frac{1}{2}$ can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis.

Refined BPS state counting from Nekrasov's formula and Macdonald functions

Abstract: It has been argued that the Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi-Yau spaces. We show that a refined version of the topological vertex we proposed before (hep-th/0502061) is a building block of the Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal-Kozcaz-Vafa (hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on C^2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang-Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.

Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds

Abstract: We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Marino, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.

Twisted equivariant K-theory with complex coefficients

Abstract: Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact group acting on itself by conjugation and relate the result to the Verlinde algebra and to the Kac numerator at q=1. Verlinde's formula is also discussed in this context.

Orbifolds as stacks

Abstract: The first goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps has to be thought of as a 2-category. Compare this with the classical definition of Satake and Thurston of orbifolds as a 1-category of sets with extra structure and/or with the "modern" definition of orbifolds as proper etale Lie groupoids up to Morita equivalence. The second goal is to describe two complementary ways of thinking of orbifolds as a 2-category: 1. the weak 2-category of foliation Lie groupoids, bibundles and equivariant maps between bibundles and 2. the strict 2-category of Deligne-Mumford stacks over the category of smooth manifolds.

The orbifold quantum cohomology of C^2/Z_3 and Hurwitz-Hodge integrals

Abstract: Let Z_3 act on C^2 by non-trivial opposite characters. Let X =[C^2/Z_3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials of X and Y are equal after a change of variables -- verifying the Crepant Resolution Conjecture for the pair (X,Y). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of independent interest. In a self contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.

Gluing Affine Torus Actions Via Divisorial Fans

Abstract: Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a “proper polyhedral divisor” introduced in earlier work, we develop the concept of a “divisorial fan” and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like $ \mathbb{C} $ *-surfaces and projectivizations of (nonsplit) vector bundles over toric varieties.

Homological algebra in bivariant K-theory and other triangulated categories. II

Abstract: Bivariant (equivariant) K-theory is the standard setting for non- commutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from homological algebra: phantom maps, exact chain complexes, projective resolutions, and derived functors. We introduce these notions and apply them to examples from bivariant K-theory. An important observation of Beligiannis is that we can approximate our category by an Abelian category in a canonical way, such that our homolog- ical concepts reduce to the corresponding ones in this Abelian category. We compute this Abelian approximation in several interesting examples, where it turns out to be very concrete and tractable. The derived functors comprise the second page of a spectral sequence that, in favourable cases, converges towards Kasparov groups and other interesting objects. This mechanism is the common basis for many different spectral sequences. Here we only discuss the very simple 1-dimensional case, where the spectral sequences reduce to short exact sequences.

Embeddings of Discrete Groups and the Speed of Random Walks

Abstract: Let G be a group generated by a finite set S and equipped with the associated left-invariant word metric d G . For a Banach space X, let α * X (G) (respectively, α / # X (G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively, an equivariant mapping) f:G → X and c > 0 such that for all x, y ∈ G we have ‖ f(x) − f(y)‖ ≥ c · d G (x, y) α . In particular, the Hilbert compression exponent (respectively, the equivariant Hilbert compression exponent) of G is (respectively, ). We show that if X has modulus of smoothness of power type p, then . Here β * (G) is the largest β ≥ 0 for which there exists a set of generators S of G and c > 0, such that for all we have , where { W t } ∞ t=0 is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X = L p , generalizes a theorem of Guentner and Kaminker (20), and answers a question posed by Tessera (37). We also show that, if then . This improves the previous bound due to Stalder and Valette (36). We deduce that if we write and then and use this result to answer a question posed by Tessera in (37) on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C 2 ≀ C n embed into L 1 with uniformly bounded distortion, answering a question posed by Lee, Naor, and Peres in (26). Finally, we use these results to show that edge Markov type need not imply Enflo type.

Congruence Subgroups and Generalized Frobenius-Schur Indicators

Abstract: We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C, an equivariant indicator of an object in C is defined as a functional on the Grothendieck algebra of the quantum double Z(C) via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholzer. In addition, we obtain two formulae for the generalized indicators, one of them a generalization of Bantay's second indicator formula for a rational conformal field theory. This formula implies a conjecture of Pradisi-Sagnotti-Stanev, as well as a conjecture of Borisov-Halpern-Schweigert.

Piecewise polynomials, Minkowski weights, and localization on toric varieties

Abstract: We use localization to describe the restriction map from equivariant Chow cohomology to ordinary Chow cohomology for complete toric varieties in terms of piecewise polynomial functions and Minkowski weights. We compute examples showing that this map is not surjective in general, and that its kernel is not always generated in degree one. We prove a localization formula for mixed volumes of lattice polytopes and, more generally, a Bott residue formula for toric vector bundles.

SU(3)-equivariant quiver gauge theories and nonabelian vortices

Abstract: We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory on Kahler manifolds of the form M × SU(3)/H, with H = SU(2) × U(1) or H = U(1) × U(1). The induced rank two quiver gauge theories on M are worked out in detail for representations of H which descend from a generic irreducible SU(3)-representation. The reduction of the Donaldson-Uhlenbeck-Yau equations on these spaces induces nonabelian quiver vortex equations on M, which we write down explicitly. When M is a noncommutative deformation of the space d, we construct explicit BPS and non-BPS solutions of finite energy for all cases. We compute their topological charges in three different ways and propose a novel interpretation of the configurations as states of D-branes. Our methods and results generalize from SU(3) to any compact Lie group.

Equivariant covers for hyperbolic groups

Abstract: Recall that a cover U is of dimension N if every x 2 X is contained in no more then N C 1 members of U . The asymptotic dimension of a finitely generated group is its asymptotic dimension as a metric space with respect to any word metric. An important result of Yu [19] asserts that the Novikov conjecture holds for groups of finite asymptotic dimension. This can be viewed as an injectivity result for the assembly map in L–theory (after inverting 2). Further injectivity results for assembly maps for groups with finite asymptotic dimension can be found in Bartels [1], Carlsson and Goldfarb [6] and Bartels and Rosenthal [4]. On the other hand no surjectivity statement of assembly maps is known for all groups of finite asymptotic dimension and this is very much related to the absence of any equivariance condition for the cover U as above.

L'isomorphisme entre les tours de Lubin-Tate et de Drinfeld

Abstract: Ce livre contient une demonstration detaillee et complete de l'existence d'un isomorphisme equivariant entre les tours p-adiques de Lubin-Tate et de Drinfeld. Le resultat est etabli en egales et inegales caracteristiques. Il y est egalement donne comme application une demonstration du fait que les cohomologies equivariantes de ces deux tours sont isomorphes, un resultat qui a des applications a l'etude de la correspondance de Langlands locale. Au cours de la preuve des rappels et des complements sont donnes sur la structure des deux espaces de modules precedents, les groupes formels p-divisibles et la geometrie analytique rigide p-adique. This book gives a complete and thorough proof of the existence of an equivariant isomorphism between Lubin-Tate and Drinfeld towers in infinite level. The result is established in equal and inequal characteristics. Moreover, the book contains as an application the proof of the equality between the equivariant cohomology of both towers, a result that has applications to the local Langlands correspondence. Along the proof background and complements are given on the structure of both moduli spaces, p-divisible formal groups and p-adic rigid analytic geometry."

Invariant submanifold flows

Abstract: Given a Lie group acting on a manifold, our aim is to analyze the evolution of differential invariants under invariant submanifold flows. The constructions are based on the equivariant method of moving frames and the induced invariant variational bicomplex. Applications to integrable soliton dynamics, and to the evolution of differential invariant signatures, used in equivalence problems and object recognition and symmetry detection in images, are discussed.

Universal Abelian covers of rational surface singularities and multi-index filtrations

Abstract: In previous papers, the authors computed the Poincare series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincare series were expressed as the integer parts of certain fractional power series, whose interpretation was not given. In this paper, we show that, up to a simple change of variables, these fractional power series are reductions of the equivariant Poincare series for filtrations on the ring of germs of functions on the universal Abelian cover of the surface. We compute these equivariant Poincare series.

Twisted cyclic theory, equivariant KK theory and KMS States

Abstract: Recently, examples of an index theory for KMS states of circle actions were discovered, \cite{CPR2,CRT}. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C*-algebra A to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in \cite{CPR2,CRT} as special cases. Next we use the Araki-Woods III_\lambda representations of the Fermion algebra to show that there are examples which are not Cuntz-Krieger systems.

Asymptotic stability of harmonic maps under the Schrödinger flow

Abstract: For Schr ¨ odinger maps from R 2 × R + to the 2-sphere S 2 , it is not known if finite energy solutions can form singularities (blow up) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispersive sense, i.e., scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the generalized Hasimoto transform, and Strichartz (dispersive) estimates for a certain two space–dimensional linear Schr¨ odinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length scale of a nearby harmonic map.

Equivariant Hopf bifurcation for neutral functional differential equations

Abstract: In this paper we employ an equivariant Lyapunov-Schmidt procedure to give a clearer understanding of the one-to-one correspondence of the periodic solutions of a system of neutral functional differential equations with the zeros of the reduced bifurcation map, and then set up equivariant Hopf bifurcation theory. In the process we derive criteria for the existence and direction of branches of bifurcating periodic solutions in terms of the original system, avoiding the process of center manifold reduction.

Pattern equivariant functions, deformations and equivalence of tiling spaces

Abstract: We reinvestigate the theory of deformations of tilings using P -equivariant cohomology. In particular we relate the notion of asymptotically negligible shape functions introduced by Clark and Sadun to weakly P -equivariant forms. We then investigate more closely the relation between deformations of patterns and homeomorphism or topological conjugacy of pattern spaces.

Moduli of toric vector bundles

Abstract: We give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy’s law, in the sense of Vakil. The preliminary sections of the paper give a selfcontained introduction to Klyachko’s classification of toric vector bundles.

Principal noncommutative torus bundles

Abstract: In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T-duals.

Yang-Mills connections on nonorientable surfaces

Abstract: Let G be a compact, connected Lie group. In [1], Atiyah and Bott identified the affine space A of connections on a principal G-bundle P over a Riemann surface with the affine space C of holomorphic structures on P = P ×G G, where G is the complexification of G. The identification A ∼= C is an isomorphism of affine spaces, thus a diffeomorphism. It was conjectured in [1] that under this identification the Morse stratification of the Yang–Mills functional on A exists and coincides with the stratification of C from algebraic geometry [14, 25]. The conjecture was proved by Daskalopoulos in [6] (see also [24] by Rade). The top stratum Css of C consists of semi-stable holomorphic structures on P. Atiyah and Bott showed that the stratification of C is G-perfect, where G = Aut(P). It has strong implications on the topology of the moduli space M(P ) of S-equivalence classes of semi-stable holomorphic structures on P. When M(P ) is smooth, Atiyah and Bott found a complete set of generators of the cohomology groups H∗(M(P );Q) and recursive relations which determine the Poincare polynomial Pt(M(P );Q). WhenM(P ) is singular, their results give generators of the equivariant cohomology groups H∗ GC(Css;Q) and formula for the equivariant Poincare series P G C t (Css;Q). Under the isomorphism A ∼= C, the top stratum Css corresponds to Ass which is the stable manifold of Nss, the set of central Yang–Mills connections, where the Yang–Mills functional achieves its absolute minimum [1,6].

The cohomology of line bundles of splice-quotient singularities

Abstract: We provide several results on splice-quotient singularities: a combinatorial expression of the dimension of the first cohomology of all `natural' line bundles, an equivariant Campillo-Delgado-Gusein-Zade type formula about the dimension of relative sections of line bundles (proving that the equivariant, divisorial multi-variable Hilbert series is topological), a combinatorial description of divisors of analytic function-germs, and an expression for the multiplicity of the singularity from its resolution graph. Additional, we establish a new formula for the Seiberg-Witten invariants of any rational homology sphere singularity link.