Deforming the double Liouville string

We consider a generalization of the double Liouville theory, which can be thought of as a two-parameter family of deformations of the so-called Virasoro minimal string (VMS). The latter consists of a timelike (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>c</a:mi></a:mrow><a:mrow><a:mo>−</a:mo></a:mrow></a:msub><a:mo>&lt;</a:mo><a:mn>1</a:mn></a:mrow></a:math>) and a spacelike (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>c</c:mi><c:mo>+</c:mo></c:msub><c:mo>&gt;</c:mo><c:mn>25</c:mn></c:math>) Liouville field theory formulated on a fluctuating Riemann surface. For the deformed theory, we compute the sphere partition function exactly in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>1</e:mn><e:mo>/</e:mo><e:msub><e:mi>c</e:mi><e:mo>±</e:mo></e:msub></e:math> and at third order in the coupling constant (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>λ</g:mi></g:math>) that controls the deformation. We also discuss the analogous computation in the case of the complex Liouville string (CLS) theory, which is defined as two spacelike Liouville theories with complex central charges <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:msub><i:mrow><i:mi>c</i:mi></i:mrow><i:mrow><i:mo>±</i:mo></i:mrow></i:msub><i:mo>=</i:mo><i:mn>13</i:mn><i:mo>±</i:mo><i:mi>i</i:mi><i:msub><i:mrow><i:mi mathvariant="double-struck">R</i:mi></i:mrow><i:mrow><i:mo>&gt;</i:mo><i:mn>0</i:mn></i:mrow></i:msub></i:mrow></i:math>. We show that the partition functions of VMS and CLS differ at leading order in <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>λ</l:mi></l:math> due to the presence of elliptic functions in the observables of the latter. Both VMS and CLS theories have recently been studied in relation to many interesting models, including the double-scaled Sachdev-Ye-Kitaev model, matrix models, and de Sitter gravity in two and three dimensions. We comment on the interpretation of the marginal deformation in some of these contexts.