Abstract
Studies of modular linear differential equations (MLDE) for the classification of rational CFT characters have been limited to the case where the coefficient functions (in monic form) have no poles, or poles at special points of moduli space. Here we initiate an exploration of the vast territory of MLDEs with two characters and any number of poles at arbitrary points of moduli space. We show how to parametrise the most general equation precisely and count its parameters. Eliminating logarithmic singularities at all the poles provides constraint equations for the accessory parameters. By taking suitable limits, we find recursion relations between solutions for different numbers of poles. The cases of one and two movable poles are examined in detail and compared with predictions based on quasi-characters to find complete agreement. We also comment on the limit of coincident poles. Finally we show that there exist genuine CFT corresponding to many of the newly-studied cases. We emphasise that the modular data is an output, rather than an input, of our approach.
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Acknowledgments
AD would like to thank Jishu Das and Naveen Balaji Umasankar for useful discussions on modular forms and MLDEs. He would also like to thank Sigma Samhita for helpful discussions regarding SageMath. CNG thanks Iosif Bena and gratefully acknowledges the hospitality of CEA Saclay where some part of this work was done. CNG also thanks Bobby Acharya, Paolo Creminelli, Atish Dabholkar and gratefully acknowledges the hospitality of the High-Energy section of the ICTP where some part of this work was done. SM would like to thank the Department of Theoretical Physics at CERN, Geneva for its warm hospitality. JS would like to thank Suresh Govindarajan for valuable discussions. He gratefully acknowledges the hospitality of the School of Physical Sciences at NISER, Bhubaneswar. He would also like to acknowledge support from the Institute Postdoctoral fund of IIT Madras.
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ArXiv ePrint: 2308.00069
Adjunct Professor, ICTS-TIFR, Bengaluru, and Honorary Emeritus Professor, IISER Pune. (Sunil Mukhi)
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Das, A., Gowdigere, C.N., Mukhi, S. et al. Modular differential equations with movable poles and admissible RCFT characters. J. High Energ. Phys. 2023, 143 (2023). https://doi.org/10.1007/JHEP12(2023)143
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DOI: https://doi.org/10.1007/JHEP12(2023)143