A work of Manton showed how skymions may be viewed as maps between riemannian manifolds minimising an energy functional, with topologically non-trivial global minimisers given precisely by isometries. We consider a generalisation of this energy functional to gauged skyrmions, valid for a broad class of space and target 3-manifolds where the target is equipped with an isometric $G$-action. We show that the energy is bounded below by an equivariant version of the degree of a map, describe the associated BPS equations, and discuss and classify solutions in the cases where $G={rm U}(1)$ and $G={rm SU}(2)$.
{"title":"Geometry of Gauged Skyrmions","authors":"Josh Cork, Derek Harland","doi":"10.3842/sigma.2023.071","DOIUrl":"https://doi.org/10.3842/sigma.2023.071","url":null,"abstract":"A work of Manton showed how skymions may be viewed as maps between riemannian manifolds minimising an energy functional, with topologically non-trivial global minimisers given precisely by isometries. We consider a generalisation of this energy functional to gauged skyrmions, valid for a broad class of space and target 3-manifolds where the target is equipped with an isometric $G$-action. We show that the energy is bounded below by an equivariant version of the degree of a map, describe the associated BPS equations, and discuss and classify solutions in the cases where $G={rm U}(1)$ and $G={rm SU}(2)$.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135406828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Isomonodromy for the fifth Painlevé equation ${rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlevé spaces. This involves explicit formulas for Stokes matrices and parabolic structures. The rank 4 Lax pair for ${rm P}_5$, introduced by Noumi-Yamada et al., is shown to be induced by a natural fine moduli space of connections of rank 4. As a by-product one obtains a polynomial Hamiltonian for ${rm P}_5$, equivalent to the one of Okamoto.
{"title":"Moduli Spaces for the Fifth Painlevé Equation","authors":"Marius van der Put, Jaap Top","doi":"10.3842/sigma.2023.068","DOIUrl":"https://doi.org/10.3842/sigma.2023.068","url":null,"abstract":"Isomonodromy for the fifth Painlevé equation ${rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlevé spaces. This involves explicit formulas for Stokes matrices and parabolic structures. The rank 4 Lax pair for ${rm P}_5$, introduced by Noumi-Yamada et al., is shown to be induced by a natural fine moduli space of connections of rank 4. As a by-product one obtains a polynomial Hamiltonian for ${rm P}_5$, equivalent to the one of Okamoto.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134903223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers.
{"title":"The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces","authors":"Theo Douvropoulos, Matthieu Josuat-Vergès","doi":"10.3842/sigma.2023.069","DOIUrl":"https://doi.org/10.3842/sigma.2023.069","url":null,"abstract":"The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134903235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Indranil Biswas, Sebastian Heller, Laura P. Schaposnik
Through the action of an anti-holomorphic involution $sigma$ (a real structure) on a Riemann surface $X$, we consider the induced actions on ${rm SL}(r,mathbb{C})$-opers and study the real slices fixed by such actions. By constructing this involution for different descriptions of the space of ${rm SL}(r,mathbb{C})$-opers, we are able to give a natural parametrization of the fixed point locus via differentials on the Riemann surface, which in turn allows us to study their geometric properties.
{"title":"Real Slices of ${rm SL}(r,mathbb{C})$-Opers","authors":"Indranil Biswas, Sebastian Heller, Laura P. Schaposnik","doi":"10.3842/sigma.2023.067","DOIUrl":"https://doi.org/10.3842/sigma.2023.067","url":null,"abstract":"Through the action of an anti-holomorphic involution $sigma$ (a real structure) on a Riemann surface $X$, we consider the induced actions on ${rm SL}(r,mathbb{C})$-opers and study the real slices fixed by such actions. By constructing this involution for different descriptions of the space of ${rm SL}(r,mathbb{C})$-opers, we are able to give a natural parametrization of the fixed point locus via differentials on the Riemann surface, which in turn allows us to study their geometric properties.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":"234 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135308491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Inspired by the work of J-L. Loday and M. Ronco, we build free tridendriform algebras over reduced trees and we show that they have a coproduct satisfying some compatibilities with the tridendriform products. Its graded dual is the opposite bialgebra of TSym introduced by N. Bergeron et al., which is described by the lightening splitting of a tree. In particular, we can split the product in three pieces and the coproduct in two pieces with Hopf compatibilities. We generate its codendriform primitives and count its coassociative primitives thanks to L. Foissy's work.
{"title":"Tridendriform Structures","authors":"Pierre Catoire","doi":"10.3842/sigma.2023.066","DOIUrl":"https://doi.org/10.3842/sigma.2023.066","url":null,"abstract":"Inspired by the work of J-L. Loday and M. Ronco, we build free tridendriform algebras over reduced trees and we show that they have a coproduct satisfying some compatibilities with the tridendriform products. Its graded dual is the opposite bialgebra of TSym introduced by N. Bergeron et al., which is described by the lightening splitting of a tree. In particular, we can split the product in three pieces and the coproduct in two pieces with Hopf compatibilities. We generate its codendriform primitives and count its coassociative primitives thanks to L. Foissy's work.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":"129 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135353712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present the exact realization of the extended Snyder model. Using similarity transformations, we construct realizations of the original Snyder and the extended Snyder models. Finally, we present the exact new realization of the $kappa$-deformed extended Snyder model.
{"title":"Realizations of the Extended Snyder Model","authors":"Tea Martinić Bilać, Stjepan Meljanac","doi":"10.3842/sigma.2023.065","DOIUrl":"https://doi.org/10.3842/sigma.2023.065","url":null,"abstract":"We present the exact realization of the extended Snyder model. Using similarity transformations, we construct realizations of the original Snyder and the extended Snyder models. Finally, we present the exact new realization of the $kappa$-deformed extended Snyder model.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134912482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a connection between 3d-5d exponential networks and exact WKB for difference equations associated to five dimensional Seiberg-Witten curves, or equivalently, to quantum mirror curves to toric Calabi-Yau threefolds $X$: the singularities in the Borel planes of local solutions to such difference equations correspond to central charges of 3d-5d BPS KK-modes. It follows that there should be distinguished local solutions of the difference equation in each domain of the complement of the exponential network, and these solutions jump at the walls of the network. We verify and explore this picture in two simple examples of 3d-5d systems, corresponding to taking the toric Calabi-Yau $X$ to be either $mathbb{C}^3$ or the resolved conifold. We provide the full list of local solutions in each sector of the Borel plane and in each domain of the complement of the exponential network, and find that local solutions in disconnected domains correspond to non-perturbative open topological string amplitudes on $X$ with insertions of branes at different positions of the toric diagram. We also study the Borel summation of the closed refined topological string free energy on $X$ and the corresponding non-perturbative effects, finding that central charges of 5d BPS KK-modes are related to the singularities in the Borel plane.
{"title":"Exponential Networks, WKB and Topological String","authors":"Alba Grassi, Qianyu Hao, Andrew Neitzke","doi":"10.3842/sigma.2023.064","DOIUrl":"https://doi.org/10.3842/sigma.2023.064","url":null,"abstract":"We propose a connection between 3d-5d exponential networks and exact WKB for difference equations associated to five dimensional Seiberg-Witten curves, or equivalently, to quantum mirror curves to toric Calabi-Yau threefolds $X$: the singularities in the Borel planes of local solutions to such difference equations correspond to central charges of 3d-5d BPS KK-modes. It follows that there should be distinguished local solutions of the difference equation in each domain of the complement of the exponential network, and these solutions jump at the walls of the network. We verify and explore this picture in two simple examples of 3d-5d systems, corresponding to taking the toric Calabi-Yau $X$ to be either $mathbb{C}^3$ or the resolved conifold. We provide the full list of local solutions in each sector of the Borel plane and in each domain of the complement of the exponential network, and find that local solutions in disconnected domains correspond to non-perturbative open topological string amplitudes on $X$ with insertions of branes at different positions of the toric diagram. We also study the Borel summation of the closed refined topological string free energy on $X$ and the corresponding non-perturbative effects, finding that central charges of 5d BPS KK-modes are related to the singularities in the Borel plane.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134990247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator ${mathcal L} colon F[w] rightarrow F[w]$ where $F$ is the ring of symmetric functions and $w$ is a variable. In this paper, we (1) establish a cyclic decomposition $F[w] cong bigoplus_{lambda} Z(j_{lambda}, {mathcal L})$ into finite-dimensional ${mathcal L}$-cyclic subspaces in which Jack polynomials $j_{lambda}$ may be taken as cyclic vectors and (2) prove that the restriction of ${mathcal L}$ to each $Z(j_{lambda}, {mathcal L})$ has simple spectrum given by the anisotropic contents $[s]$ of the addable corners $s$ of the Young diagram of $lambda$. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated to ${mathcal L}$, both established by Nazarov-Sklyanin. Finally, we conjecture that the ${mathcal L}$-eigenfunctions $psi_{lambda}^s {in F[w]}$ {with eigenvalue $[s]$ and constant term} $psi_{lambda}^s|_{w=0} = j_{lambda}$ are polynomials in the rescaled power sum basis $V_{mu} w^l$ of $F[w]$ with integer coefficients.
{"title":"Spectral Theory of the Nazarov-Sklyanin Lax Operator","authors":"Ryan Mickler, Alexander Moll","doi":"10.3842/sigma.2023.063","DOIUrl":"https://doi.org/10.3842/sigma.2023.063","url":null,"abstract":"In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator ${mathcal L} colon F[w] rightarrow F[w]$ where $F$ is the ring of symmetric functions and $w$ is a variable. In this paper, we (1) establish a cyclic decomposition $F[w] cong bigoplus_{lambda} Z(j_{lambda}, {mathcal L})$ into finite-dimensional ${mathcal L}$-cyclic subspaces in which Jack polynomials $j_{lambda}$ may be taken as cyclic vectors and (2) prove that the restriction of ${mathcal L}$ to each $Z(j_{lambda}, {mathcal L})$ has simple spectrum given by the anisotropic contents $[s]$ of the addable corners $s$ of the Young diagram of $lambda$. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated to ${mathcal L}$, both established by Nazarov-Sklyanin. Finally, we conjecture that the ${mathcal L}$-eigenfunctions $psi_{lambda}^s {in F[w]}$ {with eigenvalue $[s]$ and constant term} $psi_{lambda}^s|_{w=0} = j_{lambda}$ are polynomials in the rescaled power sum basis $V_{mu} w^l$ of $F[w]$ with integer coefficients.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136071810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct polynomial solutions modulo $p^s$ of the differential KZ and dynamical equations where $p$ is an odd prime number.
构造以p^s$为模的微分KZ和p$为奇素数的动力学方程的多项式解。
{"title":"Polynomial Solutions Modulo $p^s$ of Differential KZ and Dynamical Equations","authors":"P. Etingof, A. Varchenko","doi":"10.3842/SIGMA.2023.061","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.061","url":null,"abstract":"We construct polynomial solutions modulo $p^s$ of the differential KZ and dynamical equations where $p$ is an odd prime number.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45417855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Adam, C. Halcrow, K. Oleś, T. Romańczukiewicz, A. Wereszczyński
We apply the collective coordinate model framework to describe collisions of a kink and an antikink with nonzero total momentum, i.e., when the solitons possess different velocities. The minimal moduli space with only two coordinates (the mutual distance and the position of the center of mass) is of a wormhole type, whose throat shrinks to a point for symmetric kinks. In this case, a singularity is formed. For non-zero momentum, it prohibits solutions where the solitons pass through each other. We show that this unphysical feature can be cured by enlarging the dimension of the moduli space, e.g., by the inclusion of internal modes.
{"title":"Moduli Space for Kink Collisions with Moving Center of Mass","authors":"C. Adam, C. Halcrow, K. Oleś, T. Romańczukiewicz, A. Wereszczyński","doi":"10.3842/SIGMA.2023.054","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.054","url":null,"abstract":"We apply the collective coordinate model framework to describe collisions of a kink and an antikink with nonzero total momentum, i.e., when the solitons possess different velocities. The minimal moduli space with only two coordinates (the mutual distance and the position of the center of mass) is of a wormhole type, whose throat shrinks to a point for symmetric kinks. In this case, a singularity is formed. For non-zero momentum, it prohibits solutions where the solitons pass through each other. We show that this unphysical feature can be cured by enlarging the dimension of the moduli space, e.g., by the inclusion of internal modes.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49590663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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