Nan-Kuo Ho;Lisa C Jeffrey;Paul Selick;Eugene Z Xia
We study the topology of the SU(2)-representation variety of the compact oriented surface of genus 2 with one boundary component about which the holonomy is a generator of the center of SU(2).
{"title":"Flat Connections and the Commutator Map for SU(2)","authors":"Nan-Kuo Ho;Lisa C Jeffrey;Paul Selick;Eugene Z Xia","doi":"10.1093/qmath/haaa070","DOIUrl":"https://doi.org/10.1093/qmath/haaa070","url":null,"abstract":"We study the topology of the SU(2)-representation variety of the compact oriented surface of genus 2 with one boundary component about which the holonomy is a generator of the center of SU(2).","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"163-197"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa070","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We review the theory of JNR, mass $frac{1}{2}$ hyperbolic monopoles in particular their spectral curves and rational maps. These are used to establish conditions for a spectral curve to be the spectral curve of a JNR monopole and to show that the rational map of a JNR monopole arises by scattering using results of Atiyah. We show that for JNR monopoles the holomorphic sphere has a remarkably simple form and show that this can be used to give a formula for the energy density at infinity. In conclusion, we illustrate some examples of the energy density at infinity of JNR monopoles.
我们回顾了JNR, mass $frac{1}{2}$双曲单极子的理论，特别是它们的谱曲线和有理图。这些被用来建立谱曲线成为JNR单极子谱曲线的条件，并利用Atiyah的结果证明了JNR单极子的有理图是由散射产生的。我们证明了JNR单极子的全纯球有一个非常简单的形式，并证明了这可以用来给出无穷远处能量密度的公式。最后，我们举例说明了JNR单极子在无穷远处的能量密度。
{"title":"JNR monopoles","authors":"Michael K Murray;Paul Norbury","doi":"10.1093/qmath/haaa033","DOIUrl":"https://doi.org/10.1093/qmath/haaa033","url":null,"abstract":"We review the theory of JNR, mass \u0000<tex>$frac{1}{2}$</tex>\u0000 hyperbolic monopoles in particular their spectral curves and rational maps. These are used to establish conditions for a spectral curve to be the spectral curve of a JNR monopole and to show that the rational map of a JNR monopole arises by scattering using results of Atiyah. We show that for JNR monopoles the holomorphic sphere has a remarkably simple form and show that this can be used to give a formula for the energy density at infinity. In conclusion, we illustrate some examples of the energy density at infinity of JNR monopoles.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"387-405"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa033","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is an expository paper about the index of Toeplitz operators, and in particular Boutet de Monvel’s theorem [5]. We prove Boutet de Monvel’s theorem as a corollary of Bott periodicity, and independently of the Atiyah-Singer index theorem.
这是一篇关于Toeplitz算子的指数，特别是关于Boutet de Monvel定理[5]的说明性论文。我们独立于Atiyah-Singer指数定理，证明了Boutet de Monvel定理是Bott周期性的一个推论。
{"title":"The Index Theorem for Toeplitz Operators as a Corollary of Bott Periodicity","authors":"Paul F Baum;Erik Van Erp","doi":"10.1093/qmath/haab008","DOIUrl":"https://doi.org/10.1093/qmath/haab008","url":null,"abstract":"This is an expository paper about the index of Toeplitz operators, and in particular Boutet de Monvel’s theorem [5]. We prove Boutet de Monvel’s theorem as a corollary of Bott periodicity, and independently of the Atiyah-Singer index theorem.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"547-569"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the induced metric on the spherical fixed point set of a circle action on an ALE space and describe it by using the algebraic geometry of rational curves on algebraic surfaces, in particular the lines on a cubic.
{"title":"The central sphere of an ALE space","authors":"Nigel Hitchin","doi":"10.1093/qmath/haaa051","DOIUrl":"https://doi.org/10.1093/qmath/haaa051","url":null,"abstract":"We consider the induced metric on the spherical fixed point set of a circle action on an ALE space and describe it by using the algebraic geometry of rational curves on algebraic surfaces, in particular the lines on a cubic.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"253-276"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa051","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Segal’s Γ-rings provide a natural framework for absolute algebraic geometry. We use G. Almkvist’s global Witt construction to explore the relation with J. Borger ${mathbb F}_1$ -geometry and compute the Witt functor-ring ${mathbb W}_0({mathbb S})$ of the simplest Γ-ring ${mathbb S}$ . We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between λ-rings and the Arithmetic Site. Then, we concentrate on the Arakelov compactification ${overline{{rm Spec,}{mathbb Z}}}$ which acquires a structure sheaf of ${mathbb S}$ -algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor D on ${overline{{rm Spec,}{mathbb Z}}}$ , we show how to associate to D a Γ-space that encodes, in homotopical terms, the Riemann–Roch problem for D.
{"title":"Segal’s Gamma rings and universal arithmetic","authors":"Alain CONNES;Caterina CONSANI","doi":"10.1093/qmath/haaa042","DOIUrl":"https://doi.org/10.1093/qmath/haaa042","url":null,"abstract":"Segal’s Γ-rings provide a natural framework for absolute algebraic geometry. We use G. Almkvist’s global Witt construction to explore the relation with J. Borger \u0000<tex>${mathbb F}_1$</tex>\u0000-geometry and compute the Witt functor-ring \u0000<tex>${mathbb W}_0({mathbb S})$</tex>\u0000 of the simplest Γ-ring \u0000<tex>${mathbb S}$</tex>\u0000. We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between λ-rings and the Arithmetic Site. Then, we concentrate on the Arakelov compactification \u0000<tex>${overline{{rm Spec,}{mathbb Z}}}$</tex>\u0000 which acquires a structure sheaf of \u0000<tex>${mathbb S}$</tex>\u0000-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor D on \u0000<tex>${overline{{rm Spec,}{mathbb Z}}}$</tex>\u0000, we show how to associate to D a Γ-space that encodes, in homotopical terms, the Riemann–Roch problem for D.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"1-29"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Nash-Moser implicit function theorem.
{"title":"Deformations Of Multivalued Harmonic Functions","authors":"Simon Donaldson","doi":"10.1093/qmath/haab018","DOIUrl":"https://doi.org/10.1093/qmath/haab018","url":null,"abstract":"We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Nash-Moser implicit function theorem.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"199-235"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/iel7/8016816/9514696/09519170.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michael Atiyah’s interest in Skyrmions and their relationship to monopoles and instantons is recalled. Some approximate models of Skyrmions with large baryon numbers are then considered. Skyrmions having particularly strong binding are clusters of unit baryon number Skyrmions arranged as truncated tetrahedra. Their baryon numbers, $B = 4 ,, 16 ,, 40 ,, 80 ,, 140 ,, 224$ , are the tetrahedral numbers multiplied by four, agreeing with the magic proton and neutron numbers $2 ,, 8 ,, 20 ,, 40 ,, 70 ,, 112$ occurring in the nuclear shell model in the absence of strong spin-orbit coupling.
Michael Atiyah对Skyrmions的兴趣以及它们与单极子和瞬子的关系被回忆起来。然后考虑了具有大重子数的Skyrmions的一些近似模型。具有特别强结合的Skyrmions是单位重子数Skyrmions排列为截角四面体的簇。它们的重子数$B=4，，16，，40，，80，140，224$是四面体数乘以4，与核壳模型中在没有强自旋-轨道耦合的情况下出现的神奇质子和中子数$2，，8，20，40，70，112$一致。
{"title":"Skyrmions, Tetrahedra and Magic Numbers","authors":"Nicholas S Manton","doi":"10.1093/qmathj/haaa025","DOIUrl":"10.1093/qmathj/haaa025","url":null,"abstract":"Michael Atiyah’s interest in Skyrmions and their relationship to monopoles and instantons is recalled. Some approximate models of Skyrmions with large baryon numbers are then considered. Skyrmions having particularly strong binding are clusters of unit baryon number Skyrmions arranged as truncated tetrahedra. Their baryon numbers, \u0000<tex>$B = 4 ,, 16 ,, 40 ,, 80 ,, 140 ,, 224$</tex>\u0000, are the tetrahedral numbers multiplied by four, agreeing with the magic proton and neutron numbers \u0000<tex>$2 ,, 8 ,, 20 ,, 40 ,, 70 ,, 112$</tex>\u0000 occurring in the nuclear shell model in the absence of strong spin-orbit coupling.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"735-753"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa025","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47013671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain D k ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as nonlinear superpositions of the moduli space of centred SU(2) monopoles, equipped with the Atiyah–Hitchin metric, and k copies of the Taub–NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we show that the constituent Atiyah–Hitchin and Taub–NUT geometries arise as boundary components of the five-dimensional geometry as the adiabatic parameter is taken to zero.
{"title":"Dk Gravitational Instantons as Superpositions of Atiyah–Hitchin and Taub–NUT Geometries","authors":"B J Schroers;M A Singer","doi":"10.1093/qmath/haab002","DOIUrl":"https://doi.org/10.1093/qmath/haab002","url":null,"abstract":"We obtain D\u0000<inf>k</inf>\u0000 ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as nonlinear superpositions of the moduli space of centred SU(2) monopoles, equipped with the Atiyah–Hitchin metric, and k copies of the Taub–NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we show that the constituent Atiyah–Hitchin and Taub–NUT geometries arise as boundary components of the five-dimensional geometry as the adiabatic parameter is taken to zero.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"277-337"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Infinite-dimensional degree theory, especially for Fredholm maps with positive index as developed with Tromba, is combined with Ramer’s unpublished thesis work on finite co-dimensional differential forms. As an illustrative example, the approach of Nicolaescu and Savale to the Gauss–Bonnet–Chern theorem for vector bundles is reworked in this framework. Other examples mentioned are Kokarev and Kuksin’s approach to periodic differential equations and to forced harmonic maps. A discussion about how such forms and their constructions and cohomology relate to constructions for diffusion measures on path and loop spaces is also included.
{"title":"Infinite-Dimensional Degree Theory and Ramer’S Finite Co-Dimensional Differential Forms","authors":"K. D. Elworthy","doi":"10.1093/qmath/haab022","DOIUrl":"https://doi.org/10.1093/qmath/haab022","url":null,"abstract":"Infinite-dimensional degree theory, especially for Fredholm maps with positive index as developed with Tromba, is combined with Ramer’s unpublished thesis work on finite co-dimensional differential forms. As an illustrative example, the approach of Nicolaescu and Savale to the Gauss–Bonnet–Chern theorem for vector bundles is reworked in this framework. Other examples mentioned are Kokarev and Kuksin’s approach to periodic differential equations and to forced harmonic maps. A discussion about how such forms and their constructions and cohomology relate to constructions for diffusion measures on path and loop spaces is also included.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"571-602"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that there is no parity anomaly in M-theory in the low-energy field theory approximation. Our approach is computational. We determine the generators for the 12-dimensional bordism group of pin manifolds with a w 1 -twisted integer lift of w 4 ; these are the manifolds on which Wick-rotated M-theory exists. The anomaly cancellation comes down to computing a specific η-invariant and cubic form on these manifolds. Of interest beyond this specific problem are our expositions of computational techniques for η-invariants, the algebraic theory of cubic forms, Adams spectral sequence techniques and anomalies for spinor fields and Rarita–Schwinger fields.
{"title":"Consistency of M-Theory on Non-Orientable Manifolds","authors":"Daniel S Freed;Michael J Hopkins","doi":"10.1093/qmath/haab007","DOIUrl":"https://doi.org/10.1093/qmath/haab007","url":null,"abstract":"We prove that there is no parity anomaly in M-theory in the low-energy field theory approximation. Our approach is computational. We determine the generators for the 12-dimensional bordism group of pin manifolds with a w\u0000<inf>1</inf>\u0000-twisted integer lift of w\u0000<inf>4</inf>\u0000; these are the manifolds on which Wick-rotated M-theory exists. The anomaly cancellation comes down to computing a specific η-invariant and cubic form on these manifolds. Of interest beyond this specific problem are our expositions of computational techniques for η-invariants, the algebraic theory of cubic forms, Adams spectral sequence techniques and anomalies for spinor fields and Rarita–Schwinger fields.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"603-671"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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