Nested holography

Recently, we introduced a symmetry on the structure of angular momentum that interchanges internal and external degrees of freedom. The spin-orbit duality is a holographic map that projects a massive theory in four-dimensional flat spacetime onto the three-dimensional <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi mathvariant="double-struck">S</a:mi><a:mn>2</a:mn></a:msup><a:mo>×</a:mo><a:mi mathvariant="double-struck">R</a:mi></a:math> null infinity. This cylinder has radius <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>R</e:mi><e:mo>∼</e:mo><e:mn>1</e:mn><e:mo>/</e:mo><e:mi>m</e:mi></e:math> and, quantum mechanically, its vacuum state is a fuzzy sphere. Progress shows that, first, this duality realizes the Hopf map, a fact manifest on the superparticle. Second, the bulk Poincarè group transforms into the conformal group on the cylinder. In fact, the general version of the duality yields that the dual symmetries include the Bondi-Metzner-Sachs group, as is appropriate at null infinity. As an example, the Landau levels in <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi mathvariant="double-struck">R</g:mi><g:mn>3</g:mn></g:msup></g:math> are shown to match those of a Dirac monopole on the dual <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msup><j:mi mathvariant="double-struck">S</j:mi><j:mn>2</j:mn></j:msup></j:math>, in the thermodynamic limit. This dual system is actually identified with a three-dimensional critical Ising model. The map is then realized on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>N</m:mi><m:mi>f</m:mi></m:msub></m:math> massive fermions in flat space which, indeed, are the hologram of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mn>2</o:mn><o:msub><o:mi>N</o:mi><o:mi>f</o:mi></o:msub></o:math> massless fermions on the cylinder. However, the dual space is really the conformal class of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi mathvariant="double-struck">S</q:mi><q:mn>2</q:mn></q:msup><q:mo>×</q:mo><q:mi mathvariant="double-struck">R</q:mi></q:math>, naturally enclosing the universal cover of a conformally compactified four-dimensional anti–de Sitter (<u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:msub><u:mrow><u:mi>AdS</u:mi></u:mrow><u:mrow><u:mn>4</u:mn></u:mrow></u:msub></u:mrow></u:math>) spacetime. We argue that, in the absence of interactions, the massless fermions on the conformal boundary are, in turn, dual to <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msub><w:mi>N</w:mi><w:mi>f</w:mi></w:msub></w:math> massive fermions in <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mrow><y:msub><y:mrow><y:mi>AdS</y:mi></y:mrow><y:mrow><y:mn>4</y:mn></y:mrow></y:msub></y:mrow></y:math>. For free fermions, all path integrals—the ones in <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msup><ab:mi mathvariant="double-struck">R</ab:mi><ab:mn>4</ab:mn></ab:msup></ab:math> and <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:msup><db:mi mathvariant="double-struck">S</db:mi><db:mn>2</db:mn></db:msup><db:mo>×</db:mo><db:mi mathvariant="double-struck">R</db:mi></db:math> and <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mrow><hb:msub><hb:mrow><hb:mi>AdS</hb:mi></hb:mrow><hb:mrow><hb:mn>4</hb:mn></hb:mrow></hb:msub></hb:mrow></hb:math>—are shown to match. Hence, AdS/CFT duality emerges into a larger context, where one holography nests inside the other, suggesting a complete holographic bridge between fields in flat space and the AdS superstring.