“School of Particles And Accelerator”
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| Paper IPM / Particles And Accelerator / 17985 |
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| Abstract: | |||||||
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In [1], two of the authors studied the function Sm = Sm â Ï Pn
i=1(mi â
1
mi ) log hi for orbifold Riemann surfaces of signature (g; m1, ..., mne ; np) on the generalized
Schottky space Sg,n(m). In this paper, we prove the holographic duality between Sm and
the renormalized hyperbolic volume Vren of the corresponding Schottky 3-orbifolds with lines
of conical singularity that reach the conformal boundary. In case of the classical Liouville
action on Sg and Sg,n(â), the holography principle was proved in [2] and [3], respectively.
Our result implies that Vren acts as a Kähler potential for a particular combination of
the WeilâPetersson and TakhtajanâZograf metrics that appears in the local index theorem
for orbifold Riemann surfaces [4]. Moreover, we demonstrate that under the conformal
transformations, the change of function Sm is equivalent to the Polyakov anomaly, which
indicates that the function Sm is a consistent height function with a unique hyperbolic
solution. Consequently, the associated renormalized hyperbolic volume Vren also admits a
Polyakov anomaly formula. The method we used to establish this equivalence may provide
an alternative approach to derive the renormalized Polyakov anomaly for Riemann surfaces
with punctures (cusps), as described in [5].
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