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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 paper, the authors evaluate the equivariant vertex for stable pairs on toric 3-folds in terms of weighted box counting and show that the conjectural equivalence with the DT vertex predicts remarkable identities.
Abstract: The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3‐folds. We evaluate the equivariant vertex for stable pairs on toric 3‐folds in terms of weighted box counting. In the toric Calabi‐Yau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities. The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent theory for stable pairs. 14N35; 14M25, 14D20, 14J30
82 citations
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TL;DR: In this paper, it was shown that the positive part of $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined.
Abstract: We present three equivalent definitions of $S^1$-equivariant symplectic homology. We show that, using rational coefficients, the positive part of $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, as well as a rigorous definition of cylindrical/linearized contact homology based on an $S^1$-equivariant construction.
82 citations
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TL;DR: In this article, a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties was proposed, based on the notion of a polyhedral divisor.
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.
81 citations
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TL;DR: In this paper, the authors study complex Chern-Simons theory on a Seifert manifold M_3 by embedding it into string theory and show that it is equivalent to a topologically twisted supersymmetric theory and its partition function can naturally regularize by turning on a mass parameter.
Abstract: We study complex Chern–Simons theory on a Seifert manifold M_3 by embedding it into string theory. We show that complex Chern–Simons theory on M_3 is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between (1) the Verlinde algebra, (2) quantum cohomology of the Grassmannian, (3) Chern–Simons theory on Σ×S^1 and (4) index of a spin^c Dirac operator on the moduli space of flat connections to a new set of relations between (1) the “equivariant Verlinde algebra” for a complex group, (2) the equivariant quantum K-theory of the vortex moduli space, (3) complex Chern–Simons theory on Σ×S^1 and (4) the equivariant index of a spin^c Dirac operator on the moduli space of Higgs bundles.
81 citations
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TL;DR: In this paper, the authors established analogous results for the category of A_d-modules, for any d. Modules over A-d are closely related to the structures used by the authors in previous works studying syzygies of Segre and Veronese embeddings, and they hope the results of this paper will eventually lead to improvements on those works.
Abstract: Twisted commutative algebras (tca's) have played an important role in the nascent field of representation stability. Let A_d be the complex tca freely generated by d indeterminates of degree 1. In a previous paper, we determined the structure of the category of A_1-modules (which is equivalent to the category of FI-modules). In this paper, we establish analogous results for the category of A_d-modules, for any d. Modules over A_d are closely related to the structures used by the authors in previous works studying syzygies of Segre and Veronese embeddings, and we hope the results of this paper will eventually lead to improvements on those works. Our results also have implications in asymptotic commutative algebra.
81 citations