Symplectic Geometry of character varieties and $SU(2)$ Lattice Gauge Theory I
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866911833326092288 |
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| author | Ramadas, T. R. |
| author_facet | Ramadas, T. R. |
| contents | Given a finite connected graph $Λ$, the space of $SU(2)$ lattice gauge-fields on $Λ$, modulo gauge transformations, is a Lagrangian submanifold -- with mild singularities -- of the $SU(2)$ character variety (= phase-space of Chern-Simons theory) of an associated surface. We present evidence that, in the limit of large $Λ$, integration over the character variety with respect to the Liouville measure approximates lattice-theoretic integrals. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over the character variety. A continuous version of the Verlinde algebra facilitates computations. In two dimensions we recover standard expressions. For the theory on a 3-dimensional periodic lattice $Λ$ with Migdal action we get a very pleasant expression for the symplectic partition function, and with the Wilson action a more elaborate one. Each is a sum of a series with positive terms. One can also write down expressions for plaquette-plaquette correlations and 't Hooft loops. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_07455 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Symplectic Geometry of character varieties and $SU(2)$ Lattice Gauge Theory I Ramadas, T. R. High Energy Physics - Theory Mathematical Physics Symplectic Geometry Given a finite connected graph $Λ$, the space of $SU(2)$ lattice gauge-fields on $Λ$, modulo gauge transformations, is a Lagrangian submanifold -- with mild singularities -- of the $SU(2)$ character variety (= phase-space of Chern-Simons theory) of an associated surface. We present evidence that, in the limit of large $Λ$, integration over the character variety with respect to the Liouville measure approximates lattice-theoretic integrals. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over the character variety. A continuous version of the Verlinde algebra facilitates computations. In two dimensions we recover standard expressions. For the theory on a 3-dimensional periodic lattice $Λ$ with Migdal action we get a very pleasant expression for the symplectic partition function, and with the Wilson action a more elaborate one. Each is a sum of a series with positive terms. One can also write down expressions for plaquette-plaquette correlations and 't Hooft loops. |
| title | Symplectic Geometry of character varieties and $SU(2)$ Lattice Gauge Theory I |
| topic | High Energy Physics - Theory Mathematical Physics Symplectic Geometry |
| url | https://arxiv.org/abs/2206.07455 |