Abstract
In two dimensional conformal field theories the limit of large central charge plays the role of a semi-classical limit. Certain universal observables, such as conformal blocks involving the exchange of the identity operator, can be expanded around this classical limit in powers of the central charge c. This expansion is an asymptotic series, so — via the same resurgence analysis familiar from quantum mechanics — necessitates the existence of non-perturbative effects. In the case of identity conformal blocks, these new effects have a simple interpretation: the CFT must possess new primary operators with dimension of order the central charge. This constrains the data of CFTs with large central charge in a way that is similar to (but distinct from) the conformal bootstrap. We study this phenomenon in three ways: numerically, analytically using Zamolodchikov’s recursion relations, and by considering non-unitary minimal models with large (negative) central charge. In the holographic dual to a CFT2, the expansion in powers of c is the perturbative loop expansion in powers of ћ. So our results imply that the graviton loop expansion is an asymptotic series, whose cure requires the inclusion of new saddle points in the gravitational path integral. In certain cases these saddle points have a simple interpretation: they are conical excesses, particle-like states with negative mass which are not in the physical spectrum but nevertheless appear as non-manifold saddle points that control the asymptotic behaviour of the loop expansion. This phenomenon also has an interpretation in SL(2, ℝ) Chern-Simons theory, where the non-perturbative effects are associated with the non-Teichmüller component of the moduli space of flat connections.
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Acknowledgments
We are very grateful to S. Caron-Huot, K. Dasgupta, L. Eberhardt, A. L. Fitzpatrick, M. Mariño, H. Maxfield, S. Shenker, E. Verlinde, H. Verlinde, and E. Witten for useful conversations. We thank A. L. Fitzpatrick for very helpful comments on a draft. A.M. is supported in part by the Simons Foundation Grant No. 385602 and the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number SAPIN/00047-2020. N.B. is supported by the Sherman Fairchild Foundation and the U.S. Department of Energy, Office of Science, Office of High Energy Physics Award Number DE-SC0011632. The work of S.C. is supported by the Sam B. Treiman fellowship at the Princeton Center for Theoretical Science. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.
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Benjamin, N., Collier, S., Maloney, A. et al. Resurgence, conformal blocks, and the sum over geometries in quantum gravity. J. High Energ. Phys. 2023, 166 (2023). https://doi.org/10.1007/JHEP05(2023)166
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DOI: https://doi.org/10.1007/JHEP05(2023)166