<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>CFT</mml:mi></mml:mrow><mml:mi>D</mml:mi></mml:msub></mml:math> from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:

We show that the path integral of conformal field theories in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>D</a:mi></a:math> dimensions (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>CFT</c:mi><c:mi>D</c:mi></c:msub></c:math>) can be constructed by solving for eigenstates of a renormalization group (RG) operator following from the Turaev-Viro formulation of a topological field theory (topological quantum field theory) (TQFT) in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>D</e:mi><e:mo>+</e:mo><e:mn>1</e:mn></e:math> dimensions (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mrow><g:mi>TQFT</g:mi></g:mrow><g:mrow><g:mi>D</g:mi><g:mo>+</g:mo><g:mn>1</g:mn></g:mrow></g:msub></g:math>), explicitly realizing the holographic sandwich relation between a symmetric theory and a TQFT. Generically, exact eigenstates corresponding to symmetric <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mrow><i:mi>TQFT</i:mi></i:mrow><i:mi>D</i:mi></i:msub></i:math> follow from Frobenius algebra in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mrow><k:mi>TQFT</k:mi></k:mrow><k:mrow><k:mi>D</k:mi><k:mo>+</k:mo><k:mn>1</k:mn></k:mrow></k:msub></k:math>. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>D</m:mi><m:mo>=</m:mo><m:mn>2</m:mn></m:math>, we construct eigenstates that produce 2D rational CFT path integrals exactly, which curiously connect a continuous-field theoretic path integral with the Turaev-Viro state sum. We also devise and illustrate numerical methods for <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>D</o:mi><o:mo>=</o:mo><o:mn>2</o:mn></o:math>, 3 to search for <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>CFT</q:mi><q:mi>D</q:mi></q:msub></q:math> as phase transition points between symmetric <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mrow><s:mi>TQFT</s:mi></s:mrow><s:mi>D</s:mi></s:msub></s:math>. Finally, since the RG operator is in fact an exact analytic holographic tensor network, we compute “bulk-boundary” correlators and compare them with the AdS/CFT dictionary at <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>D</u:mi><u:mo>=</u:mo><u:mn>2</u:mn></u:math>. Promisingly, they are numerically compatible given our accuracy, although further works will be needed to explore the precise connection to the AdS/CFT correspondence. Published by the American Physical Society 2024