Difference equations: From Berry connections to the Coulomb branch
In recent work, we demonstrated that a spectral variety for the Berry connection of a 2d $\mathcal{N}=(2,2)$ GLSM with Kähler vacuum moduli space $X$ and Abelian flavour symmetry is the support of a sheaf induced by a certain action on the equivariant quantum cohomology of $X$. This action could be...
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2025-02-01
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| author | Andrea E. V. Ferrari, Daniel Zhang |
| author_facet | Andrea E. V. Ferrari, Daniel Zhang |
| author_sort | Andrea E. V. Ferrari, Daniel Zhang |
| collection | DOAJ |
| description | In recent work, we demonstrated that a spectral variety for the Berry connection of a 2d $\mathcal{N}=(2,2)$ GLSM with Kähler vacuum moduli space $X$ and Abelian flavour symmetry is the support of a sheaf induced by a certain action on the equivariant quantum cohomology of $X$. This action could be quantised to first-order matrix difference equations obeyed by brane amplitudes, and by taking the conformal limit, vortex partition functions. In this article, we elucidate how some of these results may be recovered from a 3d perspective, by placing the 2d theory at a boundary and gauging the flavour symmetry via a bulk A-twisted 3d $\mathcal{N}=4$ gauge theory (a sandwich construction). We interpret the above action as that of the bulk Coulomb branch algebra on boundary twisted chiral operators. This relates our work to recent constructions of actions of Coulomb branch algebras on quantum equivariant cohomology, providing a novel correspondence between these actions and spectral data of generalised periodic monopoles. The effective IR description of the 2d theory in terms of a twisted superpotential allows for explicit computations of these actions, which we demonstrate for Abelian GLSMs. |
| format | Article |
| id | doaj-art-e3b3751e800a4726be3227e060ce04e9 |
| institution | Kabale University |
| issn | 2542-4653 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | SciPost |
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| series | SciPost Physics |
| spelling | doaj-art-e3b3751e800a4726be3227e060ce04e92025-02-05T10:46:15ZengSciPostSciPost Physics2542-46532025-02-0118204510.21468/SciPostPhys.18.2.045Difference equations: From Berry connections to the Coulomb branchAndrea E. V. Ferrari, Daniel ZhangIn recent work, we demonstrated that a spectral variety for the Berry connection of a 2d $\mathcal{N}=(2,2)$ GLSM with Kähler vacuum moduli space $X$ and Abelian flavour symmetry is the support of a sheaf induced by a certain action on the equivariant quantum cohomology of $X$. This action could be quantised to first-order matrix difference equations obeyed by brane amplitudes, and by taking the conformal limit, vortex partition functions. In this article, we elucidate how some of these results may be recovered from a 3d perspective, by placing the 2d theory at a boundary and gauging the flavour symmetry via a bulk A-twisted 3d $\mathcal{N}=4$ gauge theory (a sandwich construction). We interpret the above action as that of the bulk Coulomb branch algebra on boundary twisted chiral operators. This relates our work to recent constructions of actions of Coulomb branch algebras on quantum equivariant cohomology, providing a novel correspondence between these actions and spectral data of generalised periodic monopoles. The effective IR description of the 2d theory in terms of a twisted superpotential allows for explicit computations of these actions, which we demonstrate for Abelian GLSMs.https://scipost.org/SciPostPhys.18.2.045 |
| spellingShingle | Andrea E. V. Ferrari, Daniel Zhang Difference equations: From Berry connections to the Coulomb branch SciPost Physics |
| title | Difference equations: From Berry connections to the Coulomb branch |
| title_full | Difference equations: From Berry connections to the Coulomb branch |
| title_fullStr | Difference equations: From Berry connections to the Coulomb branch |
| title_full_unstemmed | Difference equations: From Berry connections to the Coulomb branch |
| title_short | Difference equations: From Berry connections to the Coulomb branch |
| title_sort | difference equations from berry connections to the coulomb branch |
| url | https://scipost.org/SciPostPhys.18.2.045 |
| work_keys_str_mv | AT andreaevferraridanielzhang differenceequationsfromberryconnectionstothecoulombbranch |