Modular-Invariant Random Matrix Theory and AdS3 Wormholes

We develop a nonperturbative definition of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>RMT</a:mi></a:mrow><a:mrow><a:mn>2</a:mn></a:mrow></a:msub></a:mrow></a:math>: a generalization of random matrix theory that is compatible with the symmetries of two-dimensional conformal field theory. Given any random matrix ensemble, its <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>n</c:mi></c:math>-point spectral correlations admit a prescribed modular-invariant lift to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msub><e:mrow><e:mi>RMT</e:mi></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:msub></e:mrow></e:math>, which moreover reduce to the original random matrix correlators in a near-extremal limit. Central to the prescription is a presentation of random matrix theory in Mellin space, which lifts to two dimensions via the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>SL</g:mi><g:mo stretchy="false">(</g:mo><g:mn>2</g:mn><g:mo>,</g:mo><g:mi mathvariant="double-struck">Z</g:mi><g:mo stretchy="false">)</g:mo></g:math> spectral decomposition employed in previous work. As a demonstration we perform the explicit <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mrow><l:msub><l:mrow><l:mi>RMT</l:mi></l:mrow><l:mrow><l:mn>2</l:mn></l:mrow></l:msub></l:mrow></l:math> lift of two-point correlations of the GUE Airy model. We propose that in <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mrow><n:msub><n:mrow><n:mi>AdS</n:mi></n:mrow><n:mrow><n:mn>3</n:mn></n:mrow></n:msub></n:mrow></n:math> pure gravity, semiclassical amplitudes for off-shell <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>n</p:mi></p:math>-boundary torus wormholes with topology <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mrow><r:msub><r:mrow><r:mi mathvariant="normal">Σ</r:mi></r:mrow><r:mrow><r:mn>0</r:mn><r:mo>,</r:mo><r:mi>n</r:mi></r:mrow></r:msub><r:mo>×</r:mo><r:msup><r:mrow><r:mi>S</r:mi></r:mrow><r:mrow><r:mn>1</r:mn></r:mrow></r:msup></r:mrow></r:math> are given by the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:msub><u:mrow><u:mi>RMT</u:mi></u:mrow><u:mrow><u:mn>2</u:mn></u:mrow></u:msub></u:mrow></u:math> lift of JT gravity wormhole amplitudes. For the three-boundary case, we identify a gravity calculation which matches the <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mrow><w:msub><w:mrow><w:mi>RMT</w:mi></w:mrow><w:mrow><w:mn>2</w:mn></w:mrow></w:msub></w:mrow></w:math> result.