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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| Format: | Article |
| Language: | English |
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SpringerOpen
2024-12-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP12(2024)143 |
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| author | Soumangsu Chakraborty Akikazu Hashimoto Horatiu Nastase |
| author_facet | Soumangsu Chakraborty Akikazu Hashimoto Horatiu Nastase |
| author_sort | Soumangsu Chakraborty |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-b94f2ab3cc9d4b3d8f8cb0765ee47435 |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-b94f2ab3cc9d4b3d8f8cb0765ee474352025-01-05T12:06:43ZengSpringerOpenJournal of High Energy Physics1029-84792024-12-0120241212210.1007/JHEP12(2024)143Path integral of free fields and the determinant of Laplacian in warped space-timeSoumangsu Chakraborty0Akikazu Hashimoto1Horatiu Nastase2Université Paris-Saclay, CNRS, CEA, Institut de Physique ThéoriqueDepartment of Physics, University of WisconsinInstituto de Física Teórica, UNESP-Universidade Estadual PaulistaAbstract 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.https://doi.org/10.1007/JHEP12(2024)143AdS-CFT CorrespondenceGauge-Gravity CorrespondenceThermal Field Theory |
| spellingShingle | Soumangsu Chakraborty Akikazu Hashimoto Horatiu Nastase Path integral of free fields and the determinant of Laplacian in warped space-time Journal of High Energy Physics AdS-CFT Correspondence Gauge-Gravity Correspondence Thermal Field Theory |
| title | Path integral of free fields and the determinant of Laplacian in warped space-time |
| title_full | Path integral of free fields and the determinant of Laplacian in warped space-time |
| title_fullStr | Path integral of free fields and the determinant of Laplacian in warped space-time |
| title_full_unstemmed | Path integral of free fields and the determinant of Laplacian in warped space-time |
| title_short | Path integral of free fields and the determinant of Laplacian in warped space-time |
| title_sort | path integral of free fields and the determinant of laplacian in warped space time |
| topic | AdS-CFT Correspondence Gauge-Gravity Correspondence Thermal Field Theory |
| url | https://doi.org/10.1007/JHEP12(2024)143 |
| work_keys_str_mv | AT soumangsuchakraborty pathintegraloffreefieldsandthedeterminantoflaplacianinwarpedspacetime AT akikazuhashimoto pathintegraloffreefieldsandthedeterminantoflaplacianinwarpedspacetime AT horatiunastase pathintegraloffreefieldsandthedeterminantoflaplacianinwarpedspacetime |