The localization theorem for framed motivic spaces

We prove the analog of the Morel–Voevodsky localization theorem for framed motivic spaces. We deduce that framed motivic spectra are equivalent to motivic spectra over arbitrary schemes, and we give a new construction of the motivic cohomology of arbitrary schemes.

DERIVED NON-ARCHIMEDEAN ANALYTIC HILBERT SPACE

In this short paper, we combine the representability theorem introduced in [Porta and Yu, Representability theorem in derived analytic geometry, preprint, 2017, arXiv:1704.01683; Porta and Yu, Derived Hom spaces in rigid analytic geometry, preprint, 2018, arXiv:1801.07730] with the theory of derived formal models introduced in [António, $p$ -adic derived formal geometry and derived Raynaud localization theorem, preprint, 2018, arXiv:1805.03302] to prove the existence representability of the derived Hilbert space $\mathbf{R}\text{Hilb}(X)$ for a separated $k$ -analytic space $X$ . Such representability results rely on a localization theorem stating that if $\mathfrak{X}$ is a quasi-compact and quasi-separated formal scheme, then the $\infty$ -category $\text{Coh}^{-}(\mathfrak{X}^{\text{rig}})$ of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the $\infty$ -category $\text{Coh}^{-}(\mathfrak{X})$ . Along the way, we prove several results concerning the $\infty$ -categories of formal models for almost perfect modules on derived $k$ -analytic spaces.

Quiver matrix model of ADHM type and BPS state counting in diverse dimensions

We review the problem of Bogomol’nyi–Prasad–Sommerfield (BPS) state counting described by the generalized quiver matrix model of Atiyah–Drinfield–Hitchin–Manin type. In four dimensions the generating function of the counting gives the Nekrasov partition function, and we obtain a generalization in higher dimensions. By the localization theorem, the partition function is given by the sum of contributions from the fixed points of the torus action, which are labeled by partitions, plane partitions and solid partitions. The measure or the Boltzmann weight of the path integral can take the form of the plethystic exponential. Remarkably, after integration the partition function or the vacuum expectation value is again expressed in plethystic form. We regard it as a characteristic property of the BPS state counting problem, which is closely related to the integrability.

Borel Localization for a Circle Action

This chapter explores Borel localization for a circle action. For a circle action, the Borel localization theorem says that up to torsion, the equivariant cohomology of an S1-manifold is concentrated on its fixed point set and that the isomorphism in localized equivariant cohomology of the manifold and its fixed point set is a ring isomorphism. This is clearly an important result in its own right. Moreover, since the fixed point set is a regular submanifold and is usually simpler than the manifold, the Borel localization theorem sometimes allows one to obtain the ring structure of the equivariant cohomology of an S1-manifold from that of its fixed point set. The chapter demonstrates this method with the example of S1 acting on S2 by rotations.

Introductory Lectures on Equivariant Cohomology

Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into K-theory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah–Bott and Berline–Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, the book begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localization theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.

Derived Equivalences for Symplectic Reflection Algebras

In this paper we study derived equivalences for symplectic reflection algebras. We establish a version of the derived localization theorem between categories of modules over these algebras and categories of coherent sheaves over quantizations of $\mathbb{Q}$-factorial terminalizations of the symplectic quotient singularities. To do this we construct a Procesi sheaf on the terminalization and show that the quantizations of the terminalization are simple sheaves of algebras. We will also sketch some applications to the generalized Bernstein inequality and to perversity of wall crossing functors.