Path integral of free fields and the determinant of Laplacian in warped space-time
Abstract We revisit the problem of computing the determinant of Klein-Gordon operator ∆ = −∇2 + M 2 on Euclideanized AdS 3 with the Euclideanized time coordinate compactified with period β, H 3/Z, by explicitly computing its eigenvalues and computing their product. Upon assuming that eigenfunctions...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2024-12-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP12(2024)143 |
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| Summary: | Abstract We revisit the problem of computing the determinant of Klein-Gordon operator ∆ = −∇2 + M 2 on Euclideanized AdS 3 with the Euclideanized time coordinate compactified with period β, H 3/Z, by explicitly computing its eigenvalues and computing their product. Upon assuming that eigenfunctions are normalizable on H 3/Z, we found that there are no such eigenfunctions. Upon closer examination, we discover that the intuition that H 3/Z is like a box with normalizable eigenfunctions was false, and that there is, instead, a set of eigenfunctions which forms a continuum. Somewhat to our surprise, we find that there is a different operator ∆ ~ $$ \overset{\sim }{\Delta } $$ = r 2∆, which has the property that (1) the determinant of ∆ and the determinant of r 2∆ have the same dependence on β, and that (2) the Green’s function of ∆ can be spectrally decomposed into eigenfunctions of ∆ ~ $$ \overset{\sim }{\Delta } $$ . We identify the ∆ ~ $$ \overset{\sim }{\Delta } $$ operator as the “weighted Laplacian” in the context of warped compactifications, and comment on possible applications. |
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| ISSN: | 1029-8479 |