Tunnels under geometries (or instantons know their algebras)
Abstract In the tight binding model with multiple degenerate vacua we might treat wave function overlaps as instanton tunnelings between different wells (vacua). An amplitude for such a tunneling process might be constructed as T i → j ∼ e − S inst v j + v i − $$ {\textrm{T}}_{i\to j}\sim {e}^{-{S}_...
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2025-05-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP05(2025)132 |
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| author | Dmitry Galakhov Alexei Morozov |
| author_facet | Dmitry Galakhov Alexei Morozov |
| author_sort | Dmitry Galakhov |
| collection | DOAJ |
| description | Abstract In the tight binding model with multiple degenerate vacua we might treat wave function overlaps as instanton tunnelings between different wells (vacua). An amplitude for such a tunneling process might be constructed as T i → j ∼ e − S inst v j + v i − $$ {\textrm{T}}_{i\to j}\sim {e}^{-{S}_{\textrm{inst}}}{\textbf{v}}_j^{+}{\textbf{v}}_i^{-} $$ , where there is canonical instanton action suppression, and v i − $$ {\textbf{v}}_i^{-} $$ annihilates a particle in the i th vacuum, whereas v j + $$ {\textbf{v}}_j^{+} $$ creates a particle in the j th vacuum. Adiabatic change of the wells leads to a Berry-phase evolution of the couplings, which is described by the zero-curvature Gauss-Manin connection, i.e. by a quantum R-matrix. Zero-curvature is actually a consequence of level repulsion or topological protection, and its implication is the Yang-Baxter relation for the R-matrices. In the simplest case the story is pure Abelian and not very exciting. But when the model becomes more involved, incorporates supersymmetry, gauge and other symmetries, such amplitudes obtain more intricate structures. Operators v i − $$ {\textbf{v}}_i^{-} $$ , v j + $$ {\textbf{v}}_j^{+} $$ might also evolve from ordinary Heisenberg operators into a more sophisticated algebraic object — a “tunneling algebra”. The result for the tunneling algebra would depend strongly on geometry of the QFT we started with, and, unfortunately, at the moment we are unable to solve the reverse engineering problem. In this note we revise few successful cases of the aforementioned correspondence: quantum algebras U q ( g $$ \mathfrak{g} $$ ) and affine Yangians Y( g ̂ $$ \hat{\mathfrak{g}} $$ ). For affine Yangians we demonstrate explicitly how instantons “perform” equivariant integrals over associated quiver moduli spaces appearing in the alternative geometric construction. |
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| language | English |
| publishDate | 2025-05-01 |
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| series | Journal of High Energy Physics |
| spelling | doaj-art-0a10a4b2c5424a7ab6cde2a2e7ff69212025-08-20T02:30:49ZengSpringerOpenJournal of High Energy Physics1029-84792025-05-012025515310.1007/JHEP05(2025)132Tunnels under geometries (or instantons know their algebras)Dmitry Galakhov0Alexei Morozov1Moscow Institute of Physics and TechnologyMoscow Institute of Physics and TechnologyAbstract In the tight binding model with multiple degenerate vacua we might treat wave function overlaps as instanton tunnelings between different wells (vacua). An amplitude for such a tunneling process might be constructed as T i → j ∼ e − S inst v j + v i − $$ {\textrm{T}}_{i\to j}\sim {e}^{-{S}_{\textrm{inst}}}{\textbf{v}}_j^{+}{\textbf{v}}_i^{-} $$ , where there is canonical instanton action suppression, and v i − $$ {\textbf{v}}_i^{-} $$ annihilates a particle in the i th vacuum, whereas v j + $$ {\textbf{v}}_j^{+} $$ creates a particle in the j th vacuum. Adiabatic change of the wells leads to a Berry-phase evolution of the couplings, which is described by the zero-curvature Gauss-Manin connection, i.e. by a quantum R-matrix. Zero-curvature is actually a consequence of level repulsion or topological protection, and its implication is the Yang-Baxter relation for the R-matrices. In the simplest case the story is pure Abelian and not very exciting. But when the model becomes more involved, incorporates supersymmetry, gauge and other symmetries, such amplitudes obtain more intricate structures. Operators v i − $$ {\textbf{v}}_i^{-} $$ , v j + $$ {\textbf{v}}_j^{+} $$ might also evolve from ordinary Heisenberg operators into a more sophisticated algebraic object — a “tunneling algebra”. The result for the tunneling algebra would depend strongly on geometry of the QFT we started with, and, unfortunately, at the moment we are unable to solve the reverse engineering problem. In this note we revise few successful cases of the aforementioned correspondence: quantum algebras U q ( g $$ \mathfrak{g} $$ ) and affine Yangians Y( g ̂ $$ \hat{\mathfrak{g}} $$ ). For affine Yangians we demonstrate explicitly how instantons “perform” equivariant integrals over associated quiver moduli spaces appearing in the alternative geometric construction.https://doi.org/10.1007/JHEP05(2025)132Quantum GroupsSolitons Monopoles and InstantonsTopological Strings |
| spellingShingle | Dmitry Galakhov Alexei Morozov Tunnels under geometries (or instantons know their algebras) Journal of High Energy Physics Quantum Groups Solitons Monopoles and Instantons Topological Strings |
| title | Tunnels under geometries (or instantons know their algebras) |
| title_full | Tunnels under geometries (or instantons know their algebras) |
| title_fullStr | Tunnels under geometries (or instantons know their algebras) |
| title_full_unstemmed | Tunnels under geometries (or instantons know their algebras) |
| title_short | Tunnels under geometries (or instantons know their algebras) |
| title_sort | tunnels under geometries or instantons know their algebras |
| topic | Quantum Groups Solitons Monopoles and Instantons Topological Strings |
| url | https://doi.org/10.1007/JHEP05(2025)132 |
| work_keys_str_mv | AT dmitrygalakhov tunnelsundergeometriesorinstantonsknowtheiralgebras AT alexeimorozov tunnelsundergeometriesorinstantonsknowtheiralgebras |