2025 Winter: Semiclassical conformal field theory, QFT, and gravitational physics

https://www.math.uni-hamburg.de/home/holstein/zmp2526

Results from the semiclassical (large central charge) limits of conformal field theory have recently found some striking applications to quantum field theory and gravitational physics.
The mathematical basis for these applications is the observation that several ordinary differential equations on Riemann surfaces such as the Heun equation, in the following called "oper equations", are obtained as limits of the Belavin-Polyakov-Zamolodchikov (BPZ) equations satisfied by the Virasoro conformal blocks. Methods and ideas from conformal field theory can therefore shed new light on classical problems related to the solution theory of the oper equations such as the Riemann-Hilbert problem.
It has turned out, on the other hand, that there are several important problems of mathematical physics which require a detailed understanding of the solutions to the oper equations. Two types of problems have recently attracted particular interest.
The first type of applications is motivated or inspired by the relations between supersymmetric quantum field theories, conformal field theory, and opers equations discovered by Alday, Gaiotto and Tachikawa, as well as Nekrasov, Rosly and Shatashvili. A second type of applications concerns black hole physics. It is based on the fact that the Heun-equation, an ordinary differential equation on a punctured Riemann sphere, arises in gravitational physics in the study of perturbations of the Kerr black hole solution. The fact that the Heun equation can not be solved exactly represents an obstacle for many directions of research in black hole physics, and many potential applications related to the quasi normal modes, as well as tidal response of black holes and Love numbers. The relations between the Heun equation and conformal field theory have inspired new computational methods for these problems.
The goal of the ZMP seminar would be to introduce both into the mathematical background from conformal field theory and the Riemann-Hilbert problem, and into some of the above-mentioned applications.

Literature

[AGH] G. Aminov, A. Grassi, Y. Hatsuda, Black hole quasi normal modes and Seiberg-Witten theory, Ann. Henri Poincaré 23, (2022), 1951–1977; arXiv:2006.06111 [hep-th].

[AGT] L.F. Alday, D.Gaiotto, Y.Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91, (2010), 167–197; arXiv:0906.3219 [hep-th].

[BBITZ] Y.F. Bautista, G. Bonelli, C. Iossa, A. Tanzini, Z. Zhou, Black Hole Perturbation Theory Meets CFT2: Kerr Compton Amplitudes from Nekrasov-Shatashvili Functions, arXiv:2312.05965

[BIPT21] G. Bonelli, C. Iossa, D. Panea Lichtig, A. Tanzini, Exact solution of Kerr black hole perturbations via CFT2 and instanton counting. Greybody factor, quasinormal modes and Love numbers, Phys. Rev. D105, (2022), 044047; arXiv:2105.04483 [hep-th]. (Key reference on Kerr-Heun-CFT-connection)

[BIPT22] G. Bonelli, C. Iossa, D. Panea Lichtig, A.Tanzini, Irregular Liouville correlators and connection formulae for Heun functions, arXiv:2201.04491 [hep-th]. (Original reference for "Trieste formula")

[CC21] B. Carneiro da Cunha, J.P. Cavalcante, Confluent conformal blocks and the Teukolsky master equation, Phys. Rev. D102, (2020), 105013; arXiv:1906.10638 [hep-th].

[DGILZ] M. Dodelson, A.Grassi, C. Iossa, D.Panea Lichtig, A. Zhiboedov, Holographic thermal correlators from supersymmetric instantons, arXiv:2206.07720 [hep-th].

[HK] L. Hollands, O. Kidwai Higher length-twist coordinates, generalized Heun's opers, and twisted superpotentials, arXiv:1710.04438

[ILT] N. Iorgov, O. Lisovyy, J. Teschner, Isomonodromic tau-functions from Liouville conformal blocks, Comm. Math. Phys. 336, (2015), 671–694; arXiv:1401.6104 [hep-th].

[LN22] O. Lisovyy, A. Naidiuk Perturbative connection formulas for Heun equations arXiv:2208.01604 (Simplified approach to Heun-CFT-connection and "Trieste formula" + rigorous checks)