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Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.


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TL;DR: In this article, a Kuranishi structure with corners was constructed and the Euler number was defined for the special case where the expected dimension of the moduli space is zero, and there is an $S^1$ action on the pair $(X,L)$ which preserves $J$ and acts freely on $L$.
Abstract: Let $(X,\omega)$ be a symplectic manifold, $J$ be an $\omega$-tame almost complex structure, and $L$ be a Lagrangian submanifold. The stable compactification of the moduli space of parametrized $J$-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^\infty$-topology. We construct a Kuranishi structure with corners in the sense of Fukaya and Ono. This Kuranishi structure is orientable if $L$ is spin. In the special case where the expected dimension of the moduli space is zero, and there is an $S^1$ action on the pair $(X,L)$ which preserves $J$ and acts freely on $L$, we define the Euler number for this $S^1$ equivariant pair and the prescribed topological data. We conjecture that this rational number is the one computed by localization techniques using the given $S^1$ action.

137 citations

Journal ArticleDOI
TL;DR: In this article, De Concini and Procesi have constructed an equivariant compactification X which has a number of remarkable properties, some of them being: i) The boundary is the union of divisors D1,..., Dr.
Abstract: Let G be a complex semisimple group and let H C G be the group of fixed points of an involutive automorphism of G. Then X = G/H is called a symmetric variety. In [CP], De Concini and Procesi have constructed an equivariant compactification X which has a number of remarkable properties, some of them being: i) The boundary is the union of divisors D1,... , Dr. ii) There are exactly 2r orbits. Their closures are the intersections Di1 n .nDi (even schematically). In particular, there is only one closed orbit. iii) In case G is of adjoint type, all orbit closures are smooth. It is called the wonderful embedding of X or a complete symmetric variety and is the foundation for most deeper results about X. Independently, Luna and Vust developed in [LV] a general theory of equivariant compactifications of homogeneous varieties under a connected reductive group G. In particular, they realized the reason which makes symmetric varieties behave so nicely: A Borel subgroup B has an open dense orbit in G/H. Varieties with this property are called spherical. Luna and Vust were able to describe all equivariant compactifications of them in terms of combinatorial data, very similar to torus embeddings which are actually a special case. They obtained in particular that every spherical embedding has only finitely many orbits. Nevertheless, the reason for the existence of a compactification with properties i)-iii) remained mysterious. Then Brion and Pauer established a relation with the automorphism group. They proved in [BP]: A spherical variety X = G/H possesses an equivariant compactification with exactly one closed orbit if and only if AutG X = NG (H)/H is finite. In this case there is a unique one which dominates all others: the wonderful compactification X. They also showed that the orbits of X correspond to the faces of a strictly convex polyhedral cone Z. Then properties i) and ii) above are equivalent to Z being simplicial. This fact is much deeper and was proved by Brion in [Brl]. In fact he showed much more. Let F be the set of characters of B which are the characters of a rational B-eigenfunction on X. This is a finitely generated free abelian group. Then the cone Z is a subset of the real vector space Hom(F, R). Brion showed that there is a finite reflection group Wx acting on F such that Z is one of its Weyl chambers. In case of a symmetric variety, Wx is its little Weyl group.

136 citations

Journal ArticleDOI
01 Feb 2003-K-theory
TL;DR: In this article, the authors characterized all equivariant odd spectral triples for the quantum SU(2) group acting on its L2-space and having a nontrivial Chern character.
Abstract: We characterize all equivariant odd spectral triples for the quantum SU(2) group acting on its L2-space and having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and given any element of the K-homology group of SUq(2), there is an equivariant odd spectral triple of dimension 3 inducing that element. The method employed to get equivariant spectral triples in the quantum case is then used for classical SU(2), and we prove that for p<4, there does not exist any equivariant spectral triple with nontrivial K-homology class and dimension p acting on the L2-space.

134 citations

Journal ArticleDOI
TL;DR: In this article, a 3+summable spectral triple Open Image in new window over the quantum group SUq(2) which is equivariant with respect to a left and a right action was constructed.
Abstract: We construct a 3+-summable spectral triple Open image in new window over the quantum group SUq(2) which is equivariant with respect to a left and a right action of Open image in new window The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.

134 citations

Journal ArticleDOI
TL;DR: In this article, the authors define topological crystalline materials rigorously on the basis of a mathematical theory, known as twisted equivariant K-theory, and explain the Mayer-Vietoris sequence and module structure in terms of band theory.
Abstract: Topological crystalline materials are emergent topological phases due to crystalline space group symmetry. They are either gapful or gapless in the bulk, while hosting topological states at the boundary. Here, the authors define topological crystalline materials rigorously on the basis of a mathematical theory, known as twisted equivariant K-theory. Abstract mathematical ideas, such as the Mayer-Vietoris sequence and module structure, are explained in terms of band theory. The formulation is applicable to bulk gapful topological crystalline insulators/superconductors and their gapless boundary and defect states as well as to bulk gapless topological materials, such as Weyl and Dirac semimetals or nodal superconductors. The authors present a complete classification of topological crystalline surface states and band insulators protected by 17 wallpaper groups in the absence of time-reversal invariance, which may support topological states beyond simple Dirac fermions.

134 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
2023463
2022888
2021630
2020658
2019526