Classical Liouville action and uniformization of orbifold Riemann surfaces

We study the classical Liouville field theory on Riemann surfaces of genus <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>g</a:mi><a:mo>></a:mo><a:mn>1</a:mn></a:math> in the presence of vertex operators associated with branch points of orders <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>m</c:mi><c:mi>i</c:mi></c:msub><c:mo>></c:mo><c:mn>1</c:mn></c:math>. In order to do so, we will consider the generalized Schottky space <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi mathvariant="fraktur">S</e:mi><e:mrow><e:mi>g</e:mi><e:mo>,</e:mo><e:mi>n</e:mi></e:mrow></e:msub><e:mo stretchy="false">(</e:mo><e:mi mathvariant="bold-italic">m</e:mi><e:mo stretchy="false">)</e:mo></e:math> obtained as a holomorphic fibration over the Schottky space <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi mathvariant="fraktur">S</k:mi><k:mi>g</k:mi></k:msub></k:math> of the (compactified) underlying Riemann surface. The fibers of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msub><n:mi mathvariant="fraktur">S</n:mi><n:mrow><n:mi>g</n:mi><n:mo>,</n:mo><n:mi>n</n:mi></n:mrow></n:msub><n:mo stretchy="false">(</n:mo><n:mi mathvariant="bold-italic">m</n:mi><n:mo stretchy="false">)</n:mo><n:mo stretchy="false">→</n:mo><n:msub><n:mi mathvariant="fraktur">S</n:mi><n:mi>g</n:mi></n:msub></n:math> correspond to configuration spaces of <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>n</v:mi></v:math> orbifold points of orders <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi mathvariant="bold-italic">m</x:mi><x:mo>=</x:mo><x:mo stretchy="false">(</x:mo><x:msub><x:mi>m</x:mi><x:mn>1</x:mn></x:msub><x:mo>,</x:mo><x:mo>…</x:mo><x:mo>,</x:mo><x:msub><x:mi>m</x:mi><x:mi>n</x:mi></x:msub><x:mo stretchy="false">)</x:mo></x:math>. Drawing on the previous work of Park [] as well as Takhtajan and Zograf [; L. A. Takhtajan and P. Zograf], we define Hermitian metrics <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi mathvariant="sans-serif">h</cb:mi><cb:mi>i</cb:mi></cb:msub></cb:math> for tautological line bundles <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:msub><fb:mi>ℒ</fb:mi><fb:mi>i</fb:mi></fb:msub></fb:math> over <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:msub><hb:mi mathvariant="fraktur">S</hb:mi><hb:mrow><hb:mi>g</hb:mi><hb:mo>,</hb:mo><hb:mi>n</hb:mi></hb:mrow></hb:msub><hb:mo stretchy="false">(</hb:mo><hb:mi mathvariant="bold-italic">m</hb:mi><hb:mo stretchy="false">)</hb:mo></hb:math>. These metrics are expressed in terms of the first coefficient of the expansion of covering map <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"><nb:mi>J</nb:mi></nb:math> near each singular point on the Schottky domain. Additionally, we define the regularized classical Liouville action <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:msub><pb:mi>S</pb:mi><pb:mi mathvariant="bold-italic">m</pb:mi></pb:msub></pb:math> using Schottky global coordinates on Riemann orbisurfaces with genus <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:mi>g</sb:mi><sb:mo>></sb:mo><sb:mn>1</sb:mn></sb:math>. We demonstrate that <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mi>exp</ub:mi><ub:mo stretchy="false">[</ub:mo><ub:msub><ub:mi>S</ub:mi><ub:mi mathvariant="bold-italic">m</ub:mi></ub:msub><ub:mo>/</ub:mo><ub:mi>π</ub:mi><ub:mo stretchy="false">]</ub:mo></ub:math> serves as a Hermitian metric in the holomorphic <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" display="inline"><zb:mi mathvariant="double-struck">Q</zb:mi></zb:math>-line bundle <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:mrow><cc:mi>ℒ</cc:mi><cc:mo>=</cc:mo><cc:msubsup><cc:mrow><cc:mo stretchy="false">⊗</cc:mo></cc:mrow><cc:mrow><cc:mi>i</cc:mi><cc:mo>=</cc:mo><cc:mn>1</cc:mn></cc:mrow><cc:mrow><cc:mi>n</cc:mi></cc:mrow></cc:msubsup><cc:msubsup><cc:mrow><cc:mi>ℒ</cc:mi></cc:mrow><cc:mrow><cc:mi>i</cc:mi></cc:mrow><cc:mrow><cc:mo stretchy="false">⊗</cc:mo><cc:mo stretchy="false">(</cc:mo><cc:mn>1</cc:mn><cc:mo>−</cc:mo><cc:mn>1</cc:mn><cc:mo>/</cc:mo><cc:msubsup><cc:mrow><cc:mi>m</cc:mi></cc:mrow><cc:mrow><cc:mi>i</cc:mi></cc:mrow><cc:mrow><cc:mn>2</cc:mn></cc:mrow></cc:msubsup><cc:mo stretchy="false">)</cc:mo></cc:mrow></cc:msubsup></cc:mrow></cc:math> over <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"><ic:msub><ic:mi mathvariant="fraktur">S</ic:mi><ic:mrow><ic:mi>g</ic:mi><ic:mo>,</ic:mo><ic:mi>n</ic:mi></ic:mrow></ic:msub><ic:mo stretchy="false">(</ic:mo><ic:mi mathvariant="bold-italic">m</ic:mi><ic:mo stretchy="false">)</ic:mo></ic:math>. Furthermore, we explicitly compute the first and second variations of the smooth real-valued function <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" display="inline"><oc:msub><oc:mi>𝒮</oc:mi><oc:mi mathvariant="bold-italic">m</oc:mi></oc:msub><oc:mo>=</oc:mo><oc:msub><oc:mi>S</oc:mi><oc:mi mathvariant="bold-italic">m</oc:mi></oc:msub><oc:mo>−</oc:mo><oc:mi>π</oc:mi><oc:msubsup><oc:mo>∑</oc:mo><oc:mrow><oc:mi>i</oc:mi><oc:mo>=</oc:mo><oc:mn>1</oc:mn></oc:mrow><oc:mi>n</oc:mi></oc:msubsup><oc:mo stretchy="false">(</oc:mo><oc:msub><oc:mi>m</oc:mi><oc:mi>i</oc:mi></oc:msub><oc:mo>−</oc:mo><oc:mfrac><oc:mn>1</oc:mn><oc:msub><oc:mi>m</oc:mi><oc:mi>i</oc:mi></oc:msub></oc:mfrac><oc:mo stretchy="false">)</oc:mo><oc:mi>log</oc:mi><oc:msub><oc:mi mathvariant="sans-serif">h</oc:mi><oc:mi>i</oc:mi></oc:msub></oc:math> on the Schottky deformation space <vc:math xmlns:vc="http://www.w3.org/1998/Math/MathML" display="inline"><vc:msub><vc:mi mathvariant="fraktur">S</vc:mi><vc:mrow><vc:mi>g</vc:mi><vc:mo>,</vc:mo><vc:mi>n</vc:mi></vc:mrow></vc:msub><vc:mo stretchy="false">(</vc:mo><vc:mi mathvariant="bold-italic">m</vc:mi><vc:mo stretchy="false">)</vc:mo></vc:math>. We establish two key results: (i) <bd:math xmlns:bd="http://www.w3.org/1998/Math/MathML" display="inline"><bd:msub><bd:mi>𝒮</bd:mi><bd:mi mathvariant="bold-italic">m</bd:mi></bd:msub></bd:math> generates a combination of accessory and auxiliary parameters, and (ii) <ed:math xmlns:ed="http://www.w3.org/1998/Math/MathML" display="inline"><ed:mo>−</ed:mo><ed:msub><ed:mi>𝒮</ed:mi><ed:mi mathvariant="bold-italic">m</ed:mi></ed:msub></ed:math> acts as a Kähler potential for a specific combination of Weil-Petersson and Takhtajan-Zograf metrics that appear in the local index theorem for orbifold Riemann surfaces [Takhtajan and Zograf, ]. The obtained results can then be interpreted in terms of the complex geometry of the Hodge line bundle equipped with Quillen’s metric over the moduli space <hd:math xmlns:hd="http://www.w3.org/1998/Math/MathML" display="inline"><hd:msub><hd:mi mathvariant="fraktur">M</hd:mi><hd:mrow><hd:mi>g</hd:mi><hd:mo>,</hd:mo><hd:mi>n</hd:mi></hd:mrow></hd:msub><hd:mo stretchy="false">(</hd:mo><hd:mi mathvariant="bold-italic">m</hd:mi><hd:mo stretchy="false">)</hd:mo></hd:math> of Riemann orbisurfaces and the tree-level approximation of conformal Ward identities associated with quantum Liouville theory. Published by the American Physical Society 2024