Abstract
We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw–Sommerfeld’s heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.
Highlights
A Polyakov Formula for SectorsCitation for the original published paper (version of record): Aldana, C., Rowlett, J
Polyakov’s formula expresses a difference of zeta-regularized determinants of Laplace operators, an anomaly of global quantities, in terms of simple localquantities
Rowlett main applications of Polyakov’s formula are in differential geometry and mathematical physics. This formula arose in the study of the quantum theory of strings [37] and has been used in connection to conformal quantum field theory [6] and Feynman path integrals [18]
Summary
Citation for the original published paper (version of record): Aldana, C., Rowlett, J. N.B. When citing this work, cite the original published paper. Research.chalmers.se offers the possibility of retrieving research publications produced at Chalmers University of Technology. It covers all kind of research output: articles, dissertations, conference papers, reports etc. Research.chalmers.se is administrated and maintained by Chalmers Library (article starts on page) It covers all kind of research output: articles, dissertations, conference papers, reports etc. since 2004. research.chalmers.se is administrated and maintained by Chalmers Library (article starts on page)
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