Related papers: Forms over fields and Witt's lemma

Bott periodicity plays an important role in topological K-theory. The purpose of this paper is to extend the periodicity theorem in a discrete context, where all classical groups are involved and not just the general linear group. The…

Let G be a p-adic reductive group, and R an algebraically closed field. Let us consider a smooth representation of G on an R-vector space V. Fix an open compact subgroup K of G and a smooth irreducible representation of K on a…

Let k be a commutative ring in which 2 is invertible. We prove that the Hermitian K-theory of quadric hypersurfaces over k admits fibration sequences relating it to the base ring and to Clifford algebras equipped with various duality…

It was recently argued that quantum field theories possess one-form and higher-form symmetries, labelled `generalized global symmetries.' In this paper, we describe how those higher-form symmetries can be understood mathematically as…

We prove that the hermitian Gersten-Witt complex is exact for Azumaya algebras with involution of the first- or second kind over a regular local ring, which is essentially smooth over a field, or over a discrete valuation ring.

We introduce a cohomology theory for a class of projective varieties over a finite field coming from the canonical trace on a C*-algebra attached to the variety. Using the cohomology, we prove the rationality, functional equation and the…

In our work we investigate Witt equivalence of general function fields over global fields. It is proven that for any two such fields K and L the Witt equivalence induces a canonical bijection between Abhyankar valuations on K and L having…

We explore complex Riemannian geometry and Hermitian metrics on complex algebraic varieties and analytic spaces, respectively. In particular, we introduce Hermitian metrics on holomorphic Lie algebroids and examine the associated…

We show the existence of formal equivalences between reversible and Hamiltonian vector fields. The main tool we employ is the normal form theory.

Let $V$ be an algebraic variety over $\mathbb R$. The purpose of this paper is to compare its algebraic Witt group $W(V)$ with a new topological invariant $WR(V_{\mathbb C})$, based on symmetric forms on Real vector bundles (in the sense of…

Although there is no natural internal product for hermitian forms over an algebra with involution of the first kind, we describe how to multiply two $\varepsilon$-hermitian forms to obtain a quadratic form over the base field. This allows…

Using Morita type stratifications, we establish a one-to-one correspondence between geometric vector fields on a separated differentiable stack and stratified vector fields on its orbit space. This correspondence enables us to derive a…

Let k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack \hat G over k, the metaplectic extension of the Greenberg realization of Sp_{2n}(R). We also…

We give a complete theta expression of a pair of Hermitian modular forms as an inverse period mapping of lattice polarized $K3$ surfaces. Our result gives a non-trivial relation among moduli of $K3$ surfaces, theta functions and the finite…

Let F be a field of characteristic two. We determine all non-hyperbolic quadratic forms over F that are Witt equivalent to a second trace form.

We define Weil spaces, Weil manifolds, Weil varieties and Weil Lie groups over an arbitrary commutative base ring K (in particular, over discrete rings such as the integers), and we develop the basic theory of such spaces, leading up the…

We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary $W$-algebra. We show that for a minimal simple $W$-algebra…

We provide a practical technique to obtain plenty of algebraic relations for theta functions on the bounded symmetric domains of type $I$. In our framework, each theta relation is controlled by combinatorial properties of a pair $(T,P)$ of…

We begin with the construction of Mathai-Quillen's Thom form. We also study the case with group actions, with a review of equivariant cohomology and then Mathai-Quillen's construction in this setting. Next, we show that much of the above…

In these notes we construct a quantization functor, associating an Hilbert space H(V) to a finite dimensional symplectic vector space V over a finite field F_q. As a result, we obtain a canonical model for the Weil representation of the…