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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, it was shown that a quantum field theory living on the line and having a group G of inner symmetries gives rise to a category G-Loc A of twisted representations.
Abstract: The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G–Loc A of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT Open image in new window modular category Open image in new window 3-manifold invariant.
34 citations
TL;DR: In this article, a 1-dimensional variational approach to the analytical construction of equivariant biharmonic maps is described, which enables analysts to compute directly the analytical conditions which guarantee biharmonicity in the presence of suitable symmetries.
Abstract: In this paper we describe a 1-dimensional variational approach to the analytical construction of equivariant biharmonic maps. Our goal is to provide a direct method which enables analysts to compute directly the analytical conditions which guarantee biharmonicity in the presence of suitable symmetries. In the second part of our work, we illustrate and discuss some examples. In particular, we obtain a 1-dimensional stability result, and also show that biharmonic maps do not satisfy the classical maximum principle proved by Sampson for harmonic maps.
34 citations
TL;DR: In this article, it was shown that the Beilinson Bernstein localization theorem holds at the root of the unity case of the quantum flag variety, and that one can understand the representation theory of quantum groups at roots of unity through the (nowcommutative) geometry of the Springer fibers.
Abstract: Let C be the field of complex numbers and fix q G C*. Let g be a semi-simple Lie algebra over C and let G be the corresponding simply connected algebraic group. Let Uq be the quantized enveloping algebra of q. Let Oq be the algebra of quantized functions on G. Let Oq(B) be the quotient Hopf algebra of Oq corresponding to a Borel subgroup B of G. In the paper [BK] we defined categories of equivariant quantum O^-modules and Pg-modules on the quantum flag variety of G. We proved that the Beilinson Bernstein localization theorem holds at a generic q. Namely, the global section functor gives an equivalence between categories of [/^-modules and Vq-iaod\?es on the quantum flag variety. Thus one can translate questions about the represen tation theory of quantum groups to the study of the (noncommutative) geometry of the quantum flag variety. In this paper we prove that a derived version of this theorem holds at the root of unity case. Using this equivalence, we get that one can understand the representation theory of quantum groups at roots of unity through the (now-commutative) geometry of the Springer fibers. We now recall the main results in [BK]. The constructions given there are crucial for the present paper and a fairly detailed survey of the material there is given in the next section. We defined an equivariant sheaf of quasi-coherent modules over the quantum flag variety to be a left Oq-mod\?e equipped with a right ?q(B)-comod\ile structure satisfying certain compatibility conditions. Such objects form a category denoted Ai?q(Gq). It contains certain line bundles Oq(\) for ? in the weight lattice. We proved that Oq(X) is ample for ? >> 0 holds for every q. This implies that the category M.Bq(Gq) is a Proj-category in the sense of Serre.
33 citations
TL;DR: In this article, a saddle point approach is employed to study the Nekrasov-Shatashvili limit of the gauge theory, leading to define β-deformed, or quantized, Seiberg-Witten curve and differential form.
Abstract: The AGT conjecture relates $ \mathcal{N} $
= 2 4d SUSY gauge theories to 2d CFTs. Matrix model techniques can be used to investigate both sides of this relation. The large N limit refers here to the size of Young tableaux in the expression of the gauge theory partition function. It corresponds to the vanishing of Ω-background equivariant deformation parameters, and should not be confused with the t’Hooft expansion at large number of colors. In this paper, a saddle point approach is employed to study the Nekrasov-Shatashvili limit of the gauge theory, leading to define β-deformed, or quantized, Seiberg-Witten curve and differential form. Then this formalism is compared to the large N limit of the Dijkgraaf-Vafa β-ensemble. A transformation law relating the wave functions appearing at both sides of the conjecture is proposed. It implies a transformation of the Seiberg-Witten 1-form in agreement with the definition specified earlier. As a side result, a remarkable property of $ \mathcal{N} $
= 2 theories emerged: the instanton contribution to the partition function can be determined from the perturbative term analysis.
33 citations
TL;DR: In this paper, an equivariant cohomology theory for CW-complexes has been constructed, provided that certain conditions about the coefficients of the theory are satisfied, such as the coefficients having a Mackey structure.
Abstract: We construct for an equivariant cohomology theory for proper equivariant CW-complexes an equivariant Chern character, provided that certain conditions about the coefficients are satisfied. These conditions are fulfilled if the coefficients of the equivariant cohomology theory possess a Mackey structure. Such a structure is present in many interesting examples.
33 citations