## Abstract

We study *d*-dimensional Conformal Field Theories (CFTs) on the cylinder, Sd−1×R, and its deformations. In *d* = 2 the Casimir energy (i.e. the vacuum energy) is universal and is related to the central charge *c*. In *d* = 4 the vacuum energy depends on the regularization scheme and has no intrinsic value. We show that this property extends to infinitesimally deformed cylinders and support this conclusion with a holographic check. However, for N=1supersymmetric CFTs, a natural analog of the Casimir energy turns out to be *scheme independent* and thus intrinsic. We give two proofs of this result. We compute the Casimir energy for such theories by reducing to a problem in supersymmetric quantum mechanics. For the round cylinder the vacuum energy is proportional to *a* + 3*c*. We also compute the dependence of the Casimir energy on the squashing parameter of the cylinder. Finally, we revisit the problem of supersymmetric regularization of the path integral on Hopf surfaces.

Original language | English |
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Article number | 43 |

Pages (from-to) | 1-43 |

Journal | Journal of High Energy Physics |

Volume | 2015 |

Issue number | 7 |

Early online date | 29 Jul 2015 |

DOIs | |

Publication status | Published - Jul 2015 |

## Keywords

- Anomalies in Field and String Theories
- Supersymmetric Effective Theories
- Supersymmetric gauge theory