Anti–de Sitter flag superspace

This work aims to develop a global formulation for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="script">N</a:mi> <a:mo>=</a:mo> <a:mn>2</a:mn> </a:math> harmonic/projective anti–de Sitter (AdS) superspace <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msup> <d:mi>AdS</d:mi> <d:mrow> <d:mn>4</d:mn> <d:mo stretchy="false">|</d:mo> <d:mn>8</d:mn> </d:mrow> </d:msup> <d:mo>×</d:mo> <d:msup> <d:mi>S</d:mi> <d:mn>2</d:mn> </d:msup> <d:mo>≃</d:mo> <d:msup> <d:mi>AdS</d:mi> <d:mrow> <d:mn>4</d:mn> <d:mo stretchy="false">|</d:mo> <d:mn>8</d:mn> </d:mrow> </d:msup> <d:mo>×</d:mo> <d:mi mathvariant="double-struck">C</d:mi> <d:msup> <d:mi>P</d:mi> <d:mn>1</d:mn> </d:msup> </d:math> that allows for a simple action of superconformal (and hence AdS isometry) transformations. First of all, we provide an alternative supertwistor description of the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi mathvariant="script">N</i:mi> </i:math> -extended AdS superspace in four dimensions, <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:msup> <l:mi>AdS</l:mi> <l:mrow> <l:mn>4</l:mn> <l:mo stretchy="false">|</l:mo> <l:mn>4</l:mn> <l:mi mathvariant="script">N</l:mi> </l:mrow> </l:msup> </l:math> , which corresponds to a realization of the connected component <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mrow> <p:msub> <p:mrow> <p:mi mathvariant="sans-serif">O</p:mi> <p:mi mathvariant="sans-serif">S</p:mi> <p:mi mathvariant="sans-serif">p</p:mi> </p:mrow> <p:mrow> <p:mn>0</p:mn> </p:mrow> </p:msub> <p:mo stretchy="false">(</p:mo> <p:mi mathvariant="script">N</p:mi> <p:mo stretchy="false">|</p:mo> <p:mn>4</p:mn> <p:mo>;</p:mo> <p:mi mathvariant="double-struck">R</p:mi> <p:mo stretchy="false">)</p:mo> </p:mrow> </p:math> of the AdS isometry supergroup as <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mrow> <z:mrow> <z:mi mathvariant="sans-serif">S</z:mi> <z:mi mathvariant="sans-serif">U</z:mi> </z:mrow> <z:mo stretchy="false">(</z:mo> <z:mn>2</z:mn> <z:mo>,</z:mo> <z:mn>2</z:mn> <z:mo stretchy="false">|</z:mo> <z:mi mathvariant="script">N</z:mi> <z:mo stretchy="false">)</z:mo> <z:mo>∩</z:mo> <z:mrow> <z:mi mathvariant="sans-serif">O</z:mi> <z:mi mathvariant="sans-serif">S</z:mi> <z:mi mathvariant="sans-serif">p</z:mi> </z:mrow> <z:mo stretchy="false">(</z:mo> <z:mi mathvariant="script">N</z:mi> <z:mo stretchy="false">|</z:mo> <z:mn>4</z:mn> <z:mo>;</z:mo> <z:mi mathvariant="double-struck">C</z:mi> <z:mo stretchy="false">)</z:mo> </z:mrow> </z:math> . The proposed realization yields the following properties: (i) <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"> <pb:msup> <pb:mi>AdS</pb:mi> <pb:mrow> <pb:mn>4</pb:mn> <pb:mo stretchy="false">|</pb:mo> <pb:mn>4</pb:mn> <pb:mi mathvariant="script">N</pb:mi> </pb:mrow> </pb:msup> </pb:math> is an open domain of the compactified <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"> <tb:mi mathvariant="script">N</tb:mi> </tb:math> -extended Minkowski superspace, <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"> <wb:msup> <wb:mover accent="true"> <wb:mi mathvariant="double-struck">M</wb:mi> <wb:mo stretchy="false">¯</wb:mo> </wb:mover> <wb:mrow> <wb:mn>4</wb:mn> <wb:mo stretchy="false">|</wb:mo> <wb:mn>4</wb:mn> <wb:mi mathvariant="script">N</wb:mi> </wb:mrow> </wb:msup> </wb:math> ; (ii) the infinitesimal <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML" display="inline"> <dc:mi mathvariant="script">N</dc:mi> </dc:math> -extended superconformal transformations naturally act on <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"> <gc:msup> <gc:mi>AdS</gc:mi> <gc:mrow> <gc:mn>4</gc:mn> <gc:mo stretchy="false">|</gc:mo> <gc:mn>4</gc:mn> <gc:mi mathvariant="script">N</gc:mi> </gc:mrow> </gc:msup> </gc:math> ; and (iii) the isometry transformations of <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"> <kc:msup> <kc:mi>AdS</kc:mi> <kc:mrow> <kc:mn>4</kc:mn> <kc:mo stretchy="false">|</kc:mo> <kc:mn>4</kc:mn> <kc:mi mathvariant="script">N</kc:mi> </kc:mrow> </kc:msup> </kc:math> are described by those superconformal transformations which obey a certain constraint. The obtained results for <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" display="inline"> <oc:msup> <oc:mi>AdS</oc:mi> <oc:mrow> <oc:mn>4</oc:mn> <oc:mo stretchy="false">|</oc:mo> <oc:mn>4</oc:mn> <oc:mi mathvariant="script">N</oc:mi> </oc:mrow> </oc:msup> </oc:math> are then applied to develop a supertwistor formulation for an AdS flag superspace <sc:math xmlns:sc="http://www.w3.org/1998/Math/MathML" display="inline"> <sc:msup> <sc:mi>AdS</sc:mi> <sc:mrow> <sc:mn>4</sc:mn> <sc:mo stretchy="false">|</sc:mo> <sc:mn>8</sc:mn> </sc:mrow> </sc:msup> <sc:mo>×</sc:mo> <sc:msub> <sc:mi mathvariant="double-struck">F</sc:mi> <sc:mn>1</sc:mn> </sc:msub> <sc:mo stretchy="false">(</sc:mo> <sc:mn>2</sc:mn> <sc:mo stretchy="false">)</sc:mo> </sc:math> that we identify with the <yc:math xmlns:yc="http://www.w3.org/1998/Math/MathML" display="inline"> <yc:mi mathvariant="script">N</yc:mi> <yc:mo>=</yc:mo> <yc:mn>2</yc:mn> </yc:math> harmonic/projective AdS superspace. This construction makes it p