Equivariant map

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

Quiver varieties and tensor products

Abstract: In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety &?tilde; in a quiver variety, and show the following results: (1) The homology group of &?tilde; is a representation of a symmetric Kac-Moody Lie algebra ?, isomorphic to the tensor product V(λ1)⊗...⊗V(λ N ) of integrable highest weight modules. (2) The set of irreducible components of &?tilde; has a structure of a crystal, isomorphic to that of the q-analogue of V(λ1)⊗...⊗V(λ N ). (3) The equivariant K-homology group of &?tilde; is isomorphic to the tensor product of universal standard modules of the quantum loop algebra U q (L?), when ? is of type ADE. We also give a purely combinatorial description of the crystal of (2). This result is new even when N=1.

Dirac Operator on the Standard Podles Quantum Sphere

Abstract: Using principles of quantum symmetries we derive the algebraic part of the real spectral triple data for the standard Podle\'s quantum sphere: equivariant representation, chiral grading $\gamma$, reality structure $J$ and the Dirac operator $D$, which has bounded commutators with the elements of the algebra and satisfies the first order condition.

The Reduced Genus-One Gromov-Witten Invariants of Calabi-Yau Hypersurfaces

Abstract: We compute the reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces. As a consequence, we confirm the 1993 Bershadsky-Cecotti Ooguri-Vafa (BCOV) prediction for the standard genus 1 GW-invariants of a quintic threefold. We combine constructions from a series of previous papers with the classical localization theorem to relate the reduced genus 1 invariants of a CY-hypersurface to previously computed integrals on moduli spaces of stable genus 0 maps into projective space. The resulting, rather unwieldy, expressions for a genus 1 equivariant generating function simplify drastically, using a regularity property of a genus 0 equivariant generating function in half of the cases. Finally, by disregarding terms that cannot effect the non-equivariant part of the former, we relate the answer to an explicit hypergeometric series in a simple way. The approach described in this paper is systematic. It is directly applicable to computing reduced genus 1 GW-invariants of other complete intersections and should apply to higher-genus localization computations.

Computer algebra methods for equivariant dynamical systems

Abstract: Grobner bases.- Algorithms for the computation of invariants and equivariants.- Symmetric bifurcation theory.- 4. Orbit space reduction.

The equivariant Dehn's lemma and loop theorem

Abstract: In [4] the authors observed that the topological methods in the theory of three-dimensional manifolds can be modified to settle some old problems in the classical theory of minimal surfaces in euclidean space (see also [1], [12]). In [4] and [5] we found that we could use the theory of minimal surfaces to extend the theorems of Papakriakopoulous, Whitehead and Shapiro, Stalling and Epstein on the Dehn's lemma, loop theorem and sphere theorem. The key point to our approach to these topological theorems is the following: Given a certain family of maps of the disk or sphere into our three-dimensional manifold M, we minimize the area of the maps (with respect to the pulled back metric) in this family and prove the existence of the minimal map. Then by using the area minimizing property of the map and the tower construction in topology, we prove that any area minimizing map in the family is an embedding. In this way, we realize the solutions to the above topological theorems by minimal surfaces. In [4] and [5] we used the above area minimizing solutions to prove equivariant versions of the loop and the sphere theorem, and we applied these new theorems to the classification of compact group actions on R 3 in [11]. In this paper we generalize some of the theorems in [4] and [5] to compact planar domains by proving the existence of embedded planar domains of least area of a given genus and by proving a certain disjointness property for planar domains of least area. We then use this disjointness property to prove the equivariant Dehn's lemma for planar domains. On the other hand, we use a different variation approach to get a geodesic version of the loop theorem. More precisely, we prove the following: suppose that the induced map i.:~rl(OM)--* \"rr~(M) of the inclusion of the boundary has nontrivial kernel K. Then for any metric on OM, any nontrivial geodesic of least length in K is embedded and any two such geodesics are equal or disjoint. This geodesic loop theorem coupled with the above equivariant Dehn's lemma yields a new version of the equivariant loop theorem in [5]. As the placement of curves on a surface is easier to understand this new equivariant loop theorem is easier to