Four-Dimensional Spin Foam Perturbation Theory
We define a four-dimensional spin-foam perturbation theory for the BF-theory with a B∧B potential term defined for a compact semi-simple Lie group G on a compact orientable 4-manifold M. This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. W...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147406 |
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| Cite this: | Four-Dimensional Spin Foam Perturbation Theory / J.F. Martins, A. Mikovic // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 27 назв. — англ. |
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Martins, J.F. Mikovic, A. 2019-02-14T17:48:13Z 2019-02-14T17:48:13Z 2011 Four-Dimensional Spin Foam Perturbation Theory / J.F. Martins, A. Mikovic // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81T25; 81T45; 57R56 DOI: http://dx.doi.org/10.3842/SIGMA.2011.094 https://nasplib.isofts.kiev.ua/handle/123456789/147406 We define a four-dimensional spin-foam perturbation theory for the BF-theory with a B∧B potential term defined for a compact semi-simple Lie group G on a compact orientable 4-manifold M. This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group Uq(g) where g is the Lie algebra of G and q is a root of unity. The Chain-Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners Λ⊗Λ→A, where A is the adjoint representation of g, is 1-dimensional for each irrep Λ. We calculate the partition function Z in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold M. We prove that the first-order perturbative contribution vanishes for finite triangulations, so that we define a dilute-gas limit by using the second-order contribution. We show that Z is an analytic continuation of the Crane-Yetter partition function. Furthermore, we relate Z to the partition function for the F∧F theory. This work was partially supported FCT (Portugal) under the projects PTDC/MAT/099880/2008, PTDC/MAT/098770/2008, PTDC/MAT/101503/2008. This work was also partially supported by CMA/FCT/UNL, through the project PEst OE/MAT/UI0297/2011. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Four-Dimensional Spin Foam Perturbation Theory Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Four-Dimensional Spin Foam Perturbation Theory |
| spellingShingle |
Four-Dimensional Spin Foam Perturbation Theory Martins, J.F. Mikovic, A. |
| title_short |
Four-Dimensional Spin Foam Perturbation Theory |
| title_full |
Four-Dimensional Spin Foam Perturbation Theory |
| title_fullStr |
Four-Dimensional Spin Foam Perturbation Theory |
| title_full_unstemmed |
Four-Dimensional Spin Foam Perturbation Theory |
| title_sort |
four-dimensional spin foam perturbation theory |
| author |
Martins, J.F. Mikovic, A. |
| author_facet |
Martins, J.F. Mikovic, A. |
| publishDate |
2011 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We define a four-dimensional spin-foam perturbation theory for the BF-theory with a B∧B potential term defined for a compact semi-simple Lie group G on a compact orientable 4-manifold M. This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group Uq(g) where g is the Lie algebra of G and q is a root of unity. The Chain-Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners Λ⊗Λ→A, where A is the adjoint representation of g, is 1-dimensional for each irrep Λ. We calculate the partition function Z in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold M. We prove that the first-order perturbative contribution vanishes for finite triangulations, so that we define a dilute-gas limit by using the second-order contribution. We show that Z is an analytic continuation of the Crane-Yetter partition function. Furthermore, we relate Z to the partition function for the F∧F theory.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147406 |
| citation_txt |
Four-Dimensional Spin Foam Perturbation Theory / J.F. Martins, A. Mikovic // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 27 назв. — англ. |
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AT martinsjf fourdimensionalspinfoamperturbationtheory AT mikovica fourdimensionalspinfoamperturbationtheory |
| first_indexed |
2025-12-07T19:30:31Z |
| last_indexed |
2025-12-07T19:30:31Z |
| _version_ |
1850879069476880384 |