On deformation quantization of the space of connections on a two manifold and Chern Simons Gauge Theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910815909576704 |
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| author | Weitsman, Jonathan |
| author_facet | Weitsman, Jonathan |
| contents | We use recent progress on Chern-Simons gauge theory in three dimensions [18] to give explicit, closed form formulas for the star product on some functions on the affine space ${\mathcal A}(Σ)$ of (smooth) connections on the trivialized principal $G$-bundle on a compact, oriented two manifold $Σ.$ These formulas give a close relation between knot invariants, such as the Kauffman bracket polynomial, and the Jones and HOMFLY polynomials, arising in Chern Simons gauge theory, and deformation quantization of ${\mathcal A}(Σ).$ This relation echoes the relation between the manifold invariants of Witten [20] and Reshetikhin-Turaev [16] and {\em geometric} quantization of this space (or its symplectic quotient by the action of the gauge group). In our case this relation arises from explicit algebraic formulas arising from the (mathematically well-defined) functional integrals of [18]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_16569 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On deformation quantization of the space of connections on a two manifold and Chern Simons Gauge Theory Weitsman, Jonathan Differential Geometry High Energy Physics - Theory Quantum Algebra Symplectic Geometry We use recent progress on Chern-Simons gauge theory in three dimensions [18] to give explicit, closed form formulas for the star product on some functions on the affine space ${\mathcal A}(Σ)$ of (smooth) connections on the trivialized principal $G$-bundle on a compact, oriented two manifold $Σ.$ These formulas give a close relation between knot invariants, such as the Kauffman bracket polynomial, and the Jones and HOMFLY polynomials, arising in Chern Simons gauge theory, and deformation quantization of ${\mathcal A}(Σ).$ This relation echoes the relation between the manifold invariants of Witten [20] and Reshetikhin-Turaev [16] and {\em geometric} quantization of this space (or its symplectic quotient by the action of the gauge group). In our case this relation arises from explicit algebraic formulas arising from the (mathematically well-defined) functional integrals of [18]. |
| title | On deformation quantization of the space of connections on a two manifold and Chern Simons Gauge Theory |
| topic | Differential Geometry High Energy Physics - Theory Quantum Algebra Symplectic Geometry |
| url | https://arxiv.org/abs/2405.16569 |