Equivariant map

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

\( \mathcal{N}=2 \) supersymmetric gauge theories on S2 × S2 and Liouville Gravity

Abstract: We consider $$ \mathcal{N}=2 $$ supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry. This is used to explicitly compute the supersymmetric path integral on S2 × S2 via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity.

A metric Kan–Thurston theorem

Abstract: For every simplicial complex X we construct a locally CAT(0) cubical complex T(X), a cellular isometric involution tau on T(X) and a map t from T(X) to X with the following properties: t is equivariant for tau; t is a homology isomorphism; the induced map from the quotient space T(X)/tau to X is a homotopy equivalence; the induced map from the tau-fixed point set in T(X) to X is a homology isomorphism. The construction is functorial in X. One corollary is an equivariant Kan-Thurston theorem: every connected proper G-CW-complex has the same equivariant homology as the classifying space for proper actions of some other group. From this we obtain extensions of a theorem of Quillen on the spectrum of a (Borel) equivariant cohomology ring and of a result of Block concerning assembly conjectures. Another corollary of our main result is that there can be no algorithm to decide whether a CAT(0) cubical group is generated by torsion. In appendices we prove some foundational results concerning cubical complexes, notably in the infinite-dimensional case. We characterize the cubical complexes for which the natural metric is complete; we establish Gromov's criterion for infinite-dimensional cubical complexes; we show that every CAT(0) cube complex is cubical; we deduce that the second cubical subdivision of any locally CAT(0) cube complex is cubical. [A version of this paper was submitted in September 2010. This is a revised version I made in April 2011 (improvements to some material in the appendices).]

Projectively Equivariant Quantization and Symbol Calculus: Noncommutative Hypergeometric Functions

Abstract: We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on S3.

Character sheaves on unipotent groups in positive characteristic: foundations

Abstract: In this article we formulate and prove the main theorems of the theory of character sheaves on unipotent groups over an algebraically closed field of characteristic $$p>0$$ . In particular, we show that every admissible pair for such a group $$G$$ gives rise to an $$\mathbb{L }$$ -packet of character sheaves on $$G$$ and that conversely, every $$\mathbb{L }$$ -packet of character sheaves on $$G$$ arises from a (nonunique) admissible pair. In the Appendices we discuss two abstract category theory patterns related to the study of character sheaves. The first Appendix sketches a theory of duality for monoidal categories, which generalizes the notion of a rigid monoidal category and is close in spirit to the Grothendieck–Verdier duality theory. In the second one we use a topological field theory approach to define the canonical braided monoidal structure and twist on the equivariant derived category of constructible sheaves on an algebraic group; moreover, we show that this category carries an action of the surface operad. The third Appendix proves that the “naive” definition of the equivariant $$\ell $$ -adic derived category with respect to a unipotent algebraic group is equivalent to the “correct” one.

Asymptotic decomposition for semilinear wave and equivariant wave map equations

Abstract: In this paper we give a unified proof to the soliton resolution conjecture along a sequence of times, for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations, including the four dimensional radial Yang Mills equation, without using outer energy type inequalities. Such inequalities have played a crucial role in previous works with similar results. Roughly speaking, we prove that along a sequence of times $t_n\to T_+$ (the maximal time of existence), the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space, plus a free radiation term in the global case or a regular part in the finite time blow up case. The main difficulty is that in general (for instance for the radial four dimensional Yang Mills case and the radial six dimensional semilinear wave case) we do not have a favorable outer energy inequality for the associated linear wave equations. Our main new input is the simultaneous use of two virial identities.