Contact reduction and groupoid actions

Abstract

We introduce a new method to perform reduction of contact manifolds that extends Willett’s and Albert’s results. To carry out our reduction procedure all we need is a complete Jacobi map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J colon upper M right-arrow normal upper Gamma 0"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>:</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">J:M \rightarrow \Gamma _0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 0"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\Gamma _0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and we show that the quotients of fibers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Superscript negative 1 Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">J^{-1}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by suitable Lie subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript x"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Gamma _x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are either contact or locally conformal symplectic manifolds with structures induced by the one on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that Willett’s reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett’s reduction can be performed only at distinguished points. As an application we obtain Kostant’s prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.