The Self-Duality Equations on a Riemann Surface and Four-Dimensional Chern-Simons Theory
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914349184974848 |
|---|---|
| author | Bittleston, Roland Mason, Lionel Moosavian, Seyed Faroogh |
| author_facet | Bittleston, Roland Mason, Lionel Moosavian, Seyed Faroogh |
| contents | We construct a Lagrangian formulation of Hitchin's self-duality equations on a Riemann surface $Σ$ using potentials for the connection and Higgs field. This two-dimensional action is then obtained from a four-dimensional Chern-Simons theory on $Σ\times \mathbb{CP}^1$ with an appropriate choice of meromorphic 1-form on $\mathbb{CP}^1$ and boundary conditions at its poles. We show that the symplectic structure induced by the four-dimensional theory coincides with the canonical symplectic form on the Hitchin moduli space in the complex structure corresponding to the moduli space of Higgs bundles. We further provide a direct construction of Hitchin Hamiltonians in terms of the four-dimensional gauge field. Exploiting the freedom in the choice of the meromorphic one-form, we construct a family of four-dimensional Chern-Simons theories depending on a $\mathbb{CP}^1$-valued parameter. Upon reduction to two dimensions, these descend to a corresponding family of two-dimensional actions on $Σ$ whose field equations are again Hitchin's equations. Furthermore, we obtain a family of symplectic structures from our family of four-dimensional theories and show that they agree with the hyperkähler family of symplectic forms on the Hitchin moduli space, thereby identifying the $\mathbb{CP}^1$-valued parameter with the twistor parameter of the Hitchin moduli space. Our results place Hitchin's equations and their integrable structure within the framework of four-dimensional Chern-Simons theory and make the role of the twistor parameter manifest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05309 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Self-Duality Equations on a Riemann Surface and Four-Dimensional Chern-Simons Theory Bittleston, Roland Mason, Lionel Moosavian, Seyed Faroogh High Energy Physics - Theory Mathematical Physics Exactly Solvable and Integrable Systems We construct a Lagrangian formulation of Hitchin's self-duality equations on a Riemann surface $Σ$ using potentials for the connection and Higgs field. This two-dimensional action is then obtained from a four-dimensional Chern-Simons theory on $Σ\times \mathbb{CP}^1$ with an appropriate choice of meromorphic 1-form on $\mathbb{CP}^1$ and boundary conditions at its poles. We show that the symplectic structure induced by the four-dimensional theory coincides with the canonical symplectic form on the Hitchin moduli space in the complex structure corresponding to the moduli space of Higgs bundles. We further provide a direct construction of Hitchin Hamiltonians in terms of the four-dimensional gauge field. Exploiting the freedom in the choice of the meromorphic one-form, we construct a family of four-dimensional Chern-Simons theories depending on a $\mathbb{CP}^1$-valued parameter. Upon reduction to two dimensions, these descend to a corresponding family of two-dimensional actions on $Σ$ whose field equations are again Hitchin's equations. Furthermore, we obtain a family of symplectic structures from our family of four-dimensional theories and show that they agree with the hyperkähler family of symplectic forms on the Hitchin moduli space, thereby identifying the $\mathbb{CP}^1$-valued parameter with the twistor parameter of the Hitchin moduli space. Our results place Hitchin's equations and their integrable structure within the framework of four-dimensional Chern-Simons theory and make the role of the twistor parameter manifest. |
| title | The Self-Duality Equations on a Riemann Surface and Four-Dimensional Chern-Simons Theory |
| topic | High Energy Physics - Theory Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2601.05309 |