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Geometric States

Benoit Dherin, Alan Weinstein

Abstract

We introduce a special family of distributional alpha-densities and give a transversality criterion stating when their product is defined, closely related to Hormander's criterion for general distributions. Moreover, we show that for the subspace of distributional half-densities in this family the distribution product naturally yields a pairing that extends the usual one on smooth half-densities.

Geometric States

Abstract

We introduce a special family of distributional alpha-densities and give a transversality criterion stating when their product is defined, closely related to Hormander's criterion for general distributions. Moreover, we show that for the subspace of distributional half-densities in this family the distribution product naturally yields a pairing that extends the usual one on smooth half-densities.
Paper Structure (5 sections, 5 theorems, 23 equations)

This paper contains 5 sections, 5 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Geometric States
  3. Geometric states as distributions
  4. Product of geometric distributions
  5. Pairing of geometric distributions

Key Result

Proposition 5

Let $C$ be a smooth submanifold of $X$. Consider the conormal bundle $N^{*}C$ to $C$ in $X$. Then the restriction of a half-density in $|\Omega|^{\frac{1}{2}}(N^{*}C)$ to the zero section of $N^{*}C\rightarrow C$ can be identified with an element of $|\Omega|^{\frac{1}{2}}(C)\otimes\Gamma(|N^{*}C|^{

Theorems & Definitions (14)

  • Definition 1
  • Example 2
  • Example 3
  • Example 4
  • Proposition 5
  • proof
  • Example 6
  • Lemma 7
  • proof
  • Proposition 8
  • ...and 4 more