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Laplacian determinants and random surfaces

Banff International Research Station for Mathematical Innovation and Discovery

I will discuss how dimer models and other statistical physics models are related to Laplacian determinants, both on the discrete level and on the continuum level. In particular, I will recall the geometric meaning of the so-called zeta-regularized determinant of the Laplacian, as it is defined on a compact surface, with or without boundary. Using an appropriate regularization, we find that a Brownian loop soup of intensity c has a partition function described by the (-c/2)th power of the determinant of the Laplacian. In a certain sense, this means that decorating a random surface by a Brownian loop soup of intensity c corresponds to weighting the law of the surface by the (-c/2)th power of the determinant of the Laplacian. I will then introduce a method of regularizing a unit area LQG sphere, and show that weighting the law of this random surface by the (-c'/2)th power of the Laplacian determinant has precisely the effect of changing the matter central charge from c to c'. Taken together with the earlier results, this provides a way of interpreting an LQG surface of matter central charge c as a pure LQG surface decorated by a Brownian loop soup of intensity c. This is based on joint work with Morris Ang, Minjae Park, and Joshua Pfeffer.

Mathematics; Probability Theory And Stochastic Processes

video/mp4

Author affiliation: Massachusetts Institute of Technology

2020-05-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/74550

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/