# Cynthia Yan

Harvey Mudd College Mathematics 2017

Midyear Report: | Mathematics of Emergent Gravity Based on Quantum Entanglement |
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Thesis Advisor: | Prof. Weiqing Gu |

Second Reader: | Prof. Vatche Sahakian |

E-Mail: | xyan@g.hmc.edu |

## Abstract

Quantum Mechanics suggests that gravity is some long distance approximation of a quantum observable of some collective degrees of freedom. There are many ways proposed to resolve gravity at microscopic scale. In this thesis, we follow the notation that gravity comes from Von Neumann entanglement entropy and focus on understanding the relationship between gravity and quantum entanglement entropy in Matrix theory. More concretely, we consider a noncommutative spherical membrane in light-cone M theory, stabilized by external force with two massless probes far away from the source. We compute the Lagrangian of this system under BFSS theory which is a dimensional reduction of 10d super Yang-Mills. Intuitively, if there is trace of gravity in the entanglement entropy, it should be a tidal force between the two probes. This thesis calculates the exact expression of the relation between this entropy and gravity. We gain a deeper understanding of the problem of emergent gravity from a mathematical prospective.